The Project Gutenberg EBook of Encyclopaedia Britannica, 11th Edition,
Volume 11, Slice 3, by Various
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Title: Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3
"Frost" to "Fyzabad"
Author: Various
Release Date: August 13, 2011 [EBook #37064]
Language: English
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*** START OF THIS PROJECT GUTENBERG EBOOK ENCYC. BRITANNICA, VOL 11, SL 3 ***
Produced by Marius Masi, Don Kretz and the Online
Distributed Proofreading Team at https://www.pgdp.net
Transcriber's notes:
(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.
(2) Characters following a carat (^) were printed in superscript.
(3) Side-notes were relocated to function as titles of their respective
paragraphs.
(4) Macrons and breves above letters and dots below letters were not
inserted.
(5) [root] stands for the root symbol; [alpha], [beta], etc. for greek
letters.
(6) RN stands for Real Number symbol and (Pd) for Partial derivative
symbol
(7) The following typographical errors have been corrected:
ARTICLE FRUIT: "The covering of the seed is marked i. n is the
nucellus or perisperm, enclosing the embryo-sac es, in which the
endosperm is formed." 'in' amended from 'is'.
ARTICLE FRUIT: "It occupies the whole cavity of the embryo-sac, or
is formed only at certain portions of it, at the apex, as in
Rhinanthus, ..." 'occupies' amended from 'occupied'.
ARTICLE FUEL: "Owing to the siliceous nature of the ash of straw,
it is desirable to have a means of clearing the grate bars from
slags and clinkers at short intervals, and to use a steam jet to
clear the tubes from similar deposits." 'straw' amended from
'sraw'.
ARTICLE FUEL: "In a research upon the heating power and other
properties of coal for naval use, carried out by the German
admiralty, the results tabulated below were obtained with coals
from different localities." 'from' amended from 'form'.
ARTICLE FUEL: "C + 2H2O = CO2 + 2H2" '+' amended from '='.
ARTICLE FULHAM: "The principal thoroughfares are Fulham Palace Road
running S. from Hammersmith, Fulham Road and King's Road, W. from
Chelsea, converging and leading to Putney Bridge over the Thames;
..." 'converging' amended from 'coverging'.
ARTICLE FULLER, THOMAS: "Their eldest child, John, baptized at
Broadwindsor by his father, 6th June 1641, was afterwards rector of
Sidney Sussex College, edited the Worthies of England, 1662, and
became rector of Great Wakering, Essex, where he died in 1687."
added 'rector'.
ARTICLE FUNCTION: "When Taylor's theorem leads to a representation
of the function by means of an infinite series, the function is
said to be "analytic" (cf. S 21)." 'Taylor's' amended from
'Talyor's'.
ARTICLE FUNCTION: "or a series which converges uniformly may be
integrated term by term." 'uniformly' amended from 'unformly'.
ARTICLE FUNGI: "Parasitism.--Some fungi, though able to live as
saprophytes, occasionally enter the body of living plants, and are
thus termed facultative parasites." 'Parasitism' amended from
'Parasatism'.
ARTICLE FUNGI: "Recent experiments have shown that the difficulties
of getting orchid seeds to germinate are due to the absence of the
necessary fungus, which must be in readiness to infect the young
seedling immediately after it emerges from the seed." added
'after'.
ENCYCLOPAEDIA BRITANNICA
A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION
ELEVENTH EDITION
VOLUME XI, SLICE III
FROST to FYZABAD
ARTICLES IN THIS SLICE:
FROST FULMAR
FROSTBITE FULMINIC ACID
FROSTBURG FULTON, ROBERT
FROTHINGHAM, OCTAVIUS BROOKS FULTON (Missouri, U.S.A.)
FROUDE, JAMES ANTHONY FULTON (New York, U.S.A.)
FRUCTOSE FUM
FRUGONI, CARLO INNOCENZIO MARIA FUMARIC AND MALEIC ACIDS
FRUIT FUMAROLE
FRUIT AND FLOWER FARMING FUMIGATION
FRUMENTIUS FUMITORY
FRUNDSBERG, GEORG VON FUNCHAL
FRUSTUM FUNCTION
FRUYTIERS, PHILIP FUNDY, BAY OF
FRY FUNERAL RITES
FRY, SIR EDWARD FUNGI
FRY, ELIZABETH FUNJ
FRYXELL, ANDERS FUNKIA
FUAD PASHA FUNNEL
FUCHOW FUR
FUCHS, JOHANN NEPOMUK VON FURAZANES
FUCHS, LEONHARD FURETIERE, ANTOINE
FUCHSIA FURFOOZ
FUCHSINE FURFURANE
FUCINO, LAGO DI FURIES
FUEL FURLONG
FUENTE OVEJUNA FURNACE
FUENTERRABIA FURNEAUX, TOBIAS
FUERO FURNES
FUERTEVENTURA FURNESS, HORACE HOWARD
FUGGER FURNESS
FUGITIVE SLAVE LAWS FURNISS, HARRY
FUGLEMAN FURNITURE
FUGUE FURNIVALL, FREDERICK JAMES
FUHRICH, JOSEPH VON FURSE, CHARLES WELLINGTON
FUJI FURST, JULIUS
FU-KIEN FURSTENBERG
FUKUI FURSTENWALDE
FUKUOKA FURTH
FULA FURTWANGLER, ADOLF
FULCHER OF CHARTRES FURZE
FULDA FUSARO, LAGO
FULGENTIUS, FABIUS PLANCIADES FUSELI, HENRY
FULGINIAE FUSEL OIL
FULGURITE FUSIBLE METAL
FULHAM FUSILIER
FULK (king of Jerusalem) FUSION
FULK (archbishop of Reims) FUSSEN
FULKE, WILLIAM FUST, JOHANN
FULK NERRA FUSTEL DE COULANGES, NUMA DENIS
FULLEBORN, GEORG GUSTAV FUSTIAN
FULLER, ANDREW FUSTIC
FULLER, GEORGE FUTURES
FULLER, MARGARET FUX, JOHANN JOSEPH
FULLER, MELVILLE WESTON FUZE
FULLER, THOMAS FYNE, LOCH
FULLER, WILLIAM FYRD
FULLER'S EARTH FYT, JOHANNES
FULLERTON, LADY GEORGIANA FYZABAD
FROST (a common Teutonic word, cf. Dutch, _vorst_, Ger. _Frost_, from
the common Teutonic verb meaning "to freeze," Dutch, _vriezen_, Ger.
_frieren_; the Indo-European root is seen in Lat. _pruina_, hoar-frost,
cf. _prurire_, to itch, burn, _pruna_, burning coal, Sansk. _plush_, to
burn), in meteorology, the act, or agent of the process, of freezing;
hence the terms "hoar-frost" and "white-frost" applied to visible frozen
vapour formed on exposed surfaces. A frost can only occur when the
surface temperature falls below 32 deg. F., the freezing-point of water;
if the temperature be between 28 deg. and 32 deg. it is a "light frost,"
if below 28 deg. it is a "heavy," "killing" or "black frost"; the term
"black frost" is also used when no hoar-frost is present. The number of
degrees below freezing-point is termed "degrees of frost." As soon as a
mass of air is cooled to its dew-point, water begins to be precipitated
in the form of rain, dew, snow or hail. Hoarfrost is only formed at the
immediate surface of the land if the latter be at a temperature below 32
deg., and this may occur even when the temperature of the air a few feet
above the ground is 12 deg.-16 deg. above the freezing-point. The
heaviest hoar-frosts are formed under weather conditions similar to
those under which the heaviest summer dews occur, namely, clear and calm
nights, when there is no cloud to impede the radiation of heat from the
surface of the land, which thereby becomes rapidly and completely
cooled. The danger of frost is minimized when the soil is very moist, as
for example after 10-12 mm. of rain; and it is a practice in America to
flood fields on the receipt of a frost warning, radiation being checked
by the light fog sheets which develop over moist soils, just as a
cloud-layer in the upper atmosphere impedes radiation on a grand scale.
A layer of smoke will also impede radiation locally, and to this end
smoky fires are sometimes lit in such positions that the smoke may drift
over planted ground which it is desirable to preserve from frost.
Similarly, frost may occur in open country when a town, protected by its
smoke-cloud above, is free of it. In a valley with fairly high and steep
flanks frost sometimes occurs locally at the bottom, because the layer
of air cooled by contact with the cold surface of the higher ground is
heavier than that not so cooled, and therefore tends to flow or settle
downwards along the slope of the land. When meteorological
considerations point to a frost, an estimate of the night temperature
may be obtained by multiplying the difference between the readings of
the wet and dry bulb thermometer by 2.5 and subtracting the result from
the dry bulb temperature. This rule applies when the evening air is at
about 50 deg. and 30.1 in. pressure, the sky being clear. An instrument
has been devised in France for the prediction of frost. It consists of a
wet bulb and a dry bulb thermometer, mounted on a board on which is also
a scale of lines corresponding to degrees of the dry bulb, and a pointer
traversing a scale graduated according to degrees of the wet bulb.
Observations for the night are taken about half an hour before sunset.
By means of the pointer and scale, the point may be found at which the
line of the dry-bulb reading meets the pointer set to the reading of the
wet bulb. The scale is further divided by colours so that the observed
point may fall within one of three zones, indicating certain frost,
probable frost or no probability of frost.
FROSTBITE, a form of mortification (q.v.), due to the action of extreme
cold in cutting off the blood-supply from the fingers, toes, nose, ears,
&c. In comparatively trifling forms it occurs as "chaps" and
"chilblains," but the term frostbite is usually applied only to more
severe cases, where the part affected becomes in danger of gangrene. An
immediate application of snow, or ice-water, will restore the
circulation; the application of heat would cause inflammation. But if
the mortification has gone too far for the circulation to be restored,
the part will be lost, and surgical treatment may be necessary.
FROSTBURG, a town of Allegany county, Maryland, U.S.A., 11 m. W. of
Cumberland. Pop. (1890) 3804; (1900) 5274 (578 foreign-born and 236
negroes); (1910) 6028. It is served by the Cumberland & Pennsylvania
railway and the Cumberland & Westernport electric railway. The town is
about 2000 ft. above sea-level on a plateau between the Great Savage and
Dans mountains, and its delightful scenery and air have made it
attractive as a summer resort. It is the seat of the second state normal
school, opened in 1904. Frostburg is in the midst of the coal region of
the state, and is itself almost completely undermined; it has planing
mills and manufactures large quantities of fire-brick. The municipality
owns and operates its waterworks. Natural gas is piped to Frostburg from
the West Virginia fields, 120 m. away. Frostburg was first settled in
1812; was called Mount Pleasant until about 1830, when the present name
was substituted in honour of Meshech Frost, one of the town's founders;
and was incorporated in 1870.
FROTHINGHAM, OCTAVIUS BROOKS (1822-1895), American clergyman and author,
was born in Boston on the 26th of November 1822, son of Nathaniel
Langdon Frothingham (1793-1870), a prominent Unitarian preacher of
Boston, and through his mother's family related to Phillips Brooks. He
graduated from Harvard College in 1843 and from the Divinity School in
1846. He was pastor of the North Unitarian church of Salem,
Massachusetts, in 1847-1855. From 1855 to 1860 he was pastor of a new
Unitarian society in Jersey City, where he gave up the Lord's Supper,
thinking that it ministered to self-satisfaction; and it was as a
radical Unitarian that he became pastor of another young church in New
York City in 1860. Indeed in 1864 he was recognized as leader of the
radicals after his reply to Dr Hedge's address to the graduating
students of the Divinity School on _Anti-Supernaturalism in the Pulpit_.
In 1865, when he had practically given up "transcendentalism," his
church building was sold and his congregation began to worship in Lyric
Hall under the name of the Independent Liberal Church; in 1875 they
removed to the Masonic Temple, but four years later ill-health compelled
Frothingham's resignation, and the church dissolved. Paralysis
threatened him and he never fully recovered his health; in 1881 he
returned to Boston, where he died on the 27th of November 1895. To this
later period of his life belongs his best literary work. While he was in
New York he was for a time art critic of the _Tribune_. Always himself
on the unpopular side and an able but thoroughly fair critic of the
majority, he habitually under-estimated his own worth; he was not only
an anti-slavery leader when abolition was not popular even in New
England, and a radical and rationalist when it was impossible for him to
stay conveniently in the Unitarian Church, but he was the first
president of the National Free Religious Association (1867) and an early
and ardent disciple of Darwin and Spencer. To his radical views he was
always faithful. It is a mistake to say that he grew more conservative
in later years; but his judgment grew more generous and catholic. He was
a greater orator than man of letters, and his sermons in New York were
delivered to large audiences, averaging one thousand at the Masonic
Temple, and were printed each week; in eloquence and in the charm of his
spoken word he was probably surpassed in his day by none save George
William Curtis. Personally he seemed cold and distant, partly because of
his impressive appearance, and partly because of his own modesty, which
made him backward in seeking friendships.
His principal published works are: _Stories from the Life of the
Teacher_ (1863), _A Child's Book of Religion_ (1866), and other works
of religious teaching for children; several volumes of sermons;
_Beliefs of Unbelievers_ (1876), _The Cradle of the Christ: a Study in
Primitive Christianity_ (1877), _The Spirit of New Faith_ (1877), _The
Rising and the Setting Faith_ (1878), and other expositions of the
"new faith" he preached; _Life of Theodore Parker_ (1874),
_Transcendentalism in New England_ (1876), which is largely
biographical, _Gerrit Smith, a Biography_ (1878), _George Ripley_
(1882), in the "American Men of Letters" series, _Memoir of William
Henry Channing_ (1886), _Boston Unitarianism, 1820-1850_ (1890),
really a biography of his father; and _Recollections and Impressions,
1822-1890_ (1891).
FROUDE, JAMES ANTHONY (1818-1894), English historian, son of R.H.
Froude, archdeacon of Totnes, was born at Dartington, Devon, on the 23rd
of April 1818. He was educated at Westminster and Oriel College, Oxford,
then the centre of the ecclesiastical revival. He obtained a second
class and the chancellor's English essay prize, and was elected a fellow
of Exeter College (1842). His elder brother, Richard Hurrell Froude
(1803-1836), had been one of the leaders of the High Church movement at
Oxford. Froude joined that party and helped J.H. Newman, afterwards
cardinal, in his _Lives of the English Saints_. He was ordained deacon
in 1845. By that time his religious opinions had begun to change, he
grew dissatisfied with the views of the High Church party, and came
under the influence of Carlyle's teaching. Signs of this change first
appeared publicly in his _Shadows of the Clouds_, a volume containing
two stories of a religious sort, which he published in 1847 under the
pseudonym of "Zeta," and his complete desertion of his party was
declared a year later in his _Nemesis of Faith_, an heretical and
unpleasant book, of which the earlier part seems to be autobiographical.
On the demand of the college he resigned his fellowship at Oxford, and
mainly at least supported himself by writing, contributing largely to
_Fraser's Magazine_ and the _Westminster Review_. The excellence of his
style was soon generally recognized. The first two volumes of his
_History of England from the Fall of Wolsey to the Defeat of the Spanish
Armada_ appeared in 1856, and the work was completed in 1870. As an
historian he is chiefly remarkable for literary excellence, for the art
with which he represents his conception of the past. He condemns a
scientific treatment of history and disregards its philosophy. He held
that its office was simply to record human actions and that it should be
written as a drama. Accordingly he gives prominence to the personal
element in history. His presentations of character and motives, whether
truthful or not, are undeniably fine; but his doctrine that there should
be "no theorizing" about history tended to narrow his survey, and
consequently he sometimes, as in his remarks on the foreign policy of
Elizabeth, seems to misapprehend the tendencies of a period on which he
is writing.
Froude's work is often marred by prejudice and incorrect statements. He
wrote with a purpose. The keynote of his _History_ is contained in his
assertion that the Reformation was "the root and source of the expansive
force which has spread the Anglo-Saxon race over the globe." Hence he
overpraises Henry VIII. and others who forwarded the movement, and
speaks too harshly of some of its opponents. So too, in his _English in
Ireland_ (1872-1874), which was written to show the futility of attempts
to conciliate the Irish, he aggravates all that can be said against the
Irish, touches too lightly on English atrocities, and writes unjustly of
the influence of Roman Catholicism. A strong anti-clerical prejudice is
manifest in his historical work generally, and is doubtless the result
of the change in his views on Church matters and his abandonment of the
clerical profession. Carlyle's influence on him may be traced both in
his admiration for strong rulers and strong government, which led him to
write as though tyranny and brutality were excusable, and in his
independent treatment of character. His rehabilitation of Henry VIII.
was a useful protest against the idea that the king was a mere
sanguinary profligate, but his representation of him as the self-denying
minister of his people's will is erroneous, and is founded on the false
theory that the preambles of the acts of Henry's parliaments represented
the opinions of the educated laymen of England. As an advocate he
occasionally forgets that sobriety of judgment and expression become an
historian. He was not a judge of evidence, and seems to have been
unwilling to admit the force of any argument or the authority of any
statement which militated against his case. In his _Divorce of Catherine
of Aragon_ (1891) he made an unfortunate attempt to show that certain
fresh evidence on the subject, brought forward by Dr Gairdner, Dr
Friedmann and others, was not inconsistent with the views which he has
expressed in his _History_ nearly forty years before. He worked
diligently at original manuscript authorities at Simancas, the Record
Office and Hatfield House; but he used his materials carelessly, and
evidently brought to his investigation of them a mind already made up as
to their significance. His _Life of Caesar_ (1879), a glorification of
imperialism, betrays an imperfect acquaintance with Roman politics and
the life of Cicero; and of his two pleasant books of travel, _The
English in the West Indies_ (1888) shows that he made little effort to
master his subject, and _Oceana_ (1886), the record of a tour in
Australia and New Zealand, among a multitude of other blunders, notes
the prosperity of the working-classes in Adelaide at the date of his
visit, when, in fact, owing to a failure in the wheat-crop, hundreds
were then living on charity. He was constitutionally inaccurate, and
seems to have been unable to represent the exact sense of a document
which lay before him, or even to copy from it correctly. Historical
scholars ridiculed his mistakes, and Freeman, the most violent of his
critics, never let slip a chance of hitting at him in the _Saturday
Review_. Froude's temperament was sensitive, and he suffered from these
attacks, which were often unjust and always too savage in tone. The
literary quarrel between him and Freeman excited general interest when
it blazed out in a series of articles which Freeman wrote in the
_Contemporary Review_ (1878-1879) on Froude's _Short Study_ of Thomas
Becket.
Notwithstanding its defects, Froude's _History_ is a great achievement;
it presents an important and powerful account of the Reformation period
in England, and lays before us a picture of the past magnificently
conceived, and painted in colours which will never lose their freshness
and beauty. As with Froude's work generally, its literary merit is
remarkable; it is a well-balanced and orderly narrative, coherent in
design and symmetrical in execution. Though it is perhaps needlessly
long, the thread of the story is never lost amid a crowd of details;
every incident is made subordinate to the general idea, appears in its
appropriate place, and contributes its share to the perfection of the
whole. The excellence of its form is matched by the beauty of its style,
for Froude was a master of English prose. The most notable
characteristic of his style is its graceful simplicity; it is never
affected or laboured; his sentences are short and easy, and follow one
another naturally. He is always lucid. He was never in doubt as to his
own meaning, and never at a loss for the most appropriate words in which
to express it. Simple as his language is, it is dignified and worthy of
its subject. Nowhere perhaps does his style appear to more advantage
than in his four series of essays entitled _Short Studies on Great
Subjects_ (1867-1882), for it is seen there unfettered by the
obligations of narrative. Yet his narrative is admirably told. For the
most part flowing easily along, it rises on fit occasions to splendour,
picturesque beauty or pathos. Few more brilliant pieces of historical
writing exist than his description of the coronation procession of Anne
Boleyn through the streets of London, few more full of picturesque power
than that in which he relates how the spire of St Paul's was struck by
lightning; and to have once read is to remember for ever the touching
and stately words in which he compares the monks of the London
Charterhouse preparing for death with the Spartans at Thermopylae.
Proofs of his power in the sustained narration of stirring events are
abundant; his treatment of the Pilgrimage of Grace, of the sea fight at
St Helens and the repulse of the French invasion, and of the murder of
Rizzio, are among the most conspicuous examples of it. Nor is he less
successful when recording pathetic events, for his stories of certain
martyrdoms, and of the execution of Mary queen of Scots, are told with
exquisite feeling and in language of well-restrained emotion. And his
characters are alive. We may not always agree with his portraiture, but
the men and women whom he saw exist for us instinct with the life with
which he endows them and animated by the motives which he attributes to
them. His successes must be set against his failures. At the least he
wrote a great history, one which can never be disregarded by future
writers on his period, be their opinions what they may; which attracts
and delights a multitude of readers, and is a splendid example of
literary form and grace in historical composition.
The merits of his work met with full recognition. Each instalment of his
_History_, in common with almost everything which he wrote, was widely
read, and in spite of some adverse criticisms was received with eager
applause. In 1868 he was elected rector of St Andrews University,
defeating Disraeli by a majority of fourteen. He was warmly welcomed in
the United States, which he visited in 1872, but the lectures on Ireland
which he delivered there caused much dissatisfaction. On the death of
his adversary Freeman in 1892, he was appointed, on the recommendation
of Lord Salisbury, to succeed him as regius professor of modern history
at Oxford. Except to a few Oxford men, who considered that historical
scholarship should have been held to be a necessary qualification for
the office, his appointment gave general satisfaction. His lectures on
Erasmus and other 16th-century subjects were largely attended. With some
allowance for the purpose for which they were originally written, they
present much the same characteristics as his earlier historical books.
His health gave way in the summer of 1894, and he died on the 20th of
October.
His long life was full of literary work. Besides his labours as an
author, he was for fourteen years editor of _Fraser's Magazine_. He was
one of Carlyle's literary executors, and brought some sharp criticism
upon himself by publishing Carlyle's _Reminiscences_ and the _Memorials
of Jane Welsh Carlyle_, for they exhibited the domestic life and
character of his old friend in an unpleasant light. Carlyle had given
the manuscripts to him, telling him that he might publish them if he
thought it well to do so, and at the close of his life agreed to their
publication. Froude therefore declared that in giving them to the world
he was carrying out his friend's wish by enabling him to make a
posthumous confession of his faults. Besides publishing these
manuscripts he wrote a _Life of Carlyle_. His earlier study of Irish
history afforded him suggestions for a historical novel entitled _The
Two Chiefs of Dunboy_ (1889). In spite of one or two stirring scenes it
is a tedious book, and its personages are little more than machines for
the enunciation of the author's opinions and sentiments. Though Froude
had some intimate friends he was generally reserved. When he cared to
please, his manners and conversation were charming. Those who knew him
well formed a high estimate of his ability in practical affairs. In 1874
Lord Carnarvon, then colonial secretary, sent Froude to South Africa to
report on the best means of promoting a confederation of its colonies
and states, and in 1875 he was again sent to the Cape as a member of a
proposed conference to further confederation. Froude's speeches in South
Africa were rather injudicious, and his mission was a failure (see SOUTH
AFRICA: _History_). He was twice married. His first wife, a daughter of
Pascoe Grenfell and sister of Mrs Charles Kingsley, died in 1860; his
second, a daughter of John Warre, M.P. for Taunton, died in 1874.
Froude's _Life_, by Herbert Paul, was published in 1905. (W. Hu.)
FRUCTOSE, LAEVULOSE, or FRUIT-SUGAR, a carbohydrate of the formula
C6H12O6. It is closely related to ordinary d-glucose, with which it
occurs in many fruits, starches and also in honey. It is a hydrolytic
product of inulin, from which it may be prepared; but it is more usual
to obtain it from "invert sugar," the mixture obtained by hydrolysing
cane sugar with sulphuric acid. Cane sugar then yields a syrupy mixture
of glucose and fructose, which, having been freed from the acid and
concentrated, is mixed with water, cooled in ice and calcium hydroxide
added. The fructose is precipitated as a saccharate, which is filtered,
suspended in water and decomposed by carbon dioxide. The liquid is
filtered, the filtrate concentrated, and the syrup so obtained washed
with cold alcohol. On cooling the fructose separates. It may be obtained
as a syrup, as fine, silky needles, a white crystalline powder, or as a
granular crystalline, somewhat hygroscopic mass. When anhydrous it melts
at about 95 deg. C. It is readily soluble in water and in dilute alcohol,
but insoluble in absolute alcohol. It is sweeter than cane sugar and is
more easily assimilated. It has been employed under the name diabetin as
a sweetening agent for diabetics, since it does not increase the
sugar-content of the urine; other medicinal applications are in phthisis
(mixed with quassia or other bitter), and for children suffering from
tuberculosis or scrofula in place of cane sugar or milk-sugar.
Chemically, fructose is an oxyketone or ketose, its structural formula
being CH2OH.(CH.OH)3.CO.CH2OH; this result followed from its conversion
by H. Kiliani into methylbutylacetic acid. The form described above is
_laevo_-rotatory, but it is termed d-fructose, since it is related to
d-glucose. Solutions exhibit mutarotation, fresh solutions having a
specific rotation of -104.0 deg., which gradually diminishes to -92 deg.
It was synthesized by Emil Fischer, who found the synthetic sugar which
he named [alpha]-acrose to be (d + l)-fructose, and by splitting this
mixture he obtained both the d and l forms. Fructose resembles d-glucose
in being fermentable by yeast (it is the one ketose which exhibits this
property), and also in its power of reducing alkaline copper and silver
solutions; this latter property is assigned to the readiness with which
hydroxyl and ketone groups in close proximity suffer oxidation. For the
structural (stereochemical) relations of fructose see SUGAR.
FRUGONI, CARLO INNOCENZIO MARIA (1692-1768), Italian poet, was born at
Genoa on the 21st of November 1692. He was originally destined for the
church and at the age of fifteen, in opposition to his strong wishes,
was shut up in a convent; but although in the following year he was
induced to pronounce monastic vows, he had no liking for this life. He
acquired considerable reputation as an elegant writer both of Latin and
Italian prose and verse; and from 1716 to 1724 he filled the chairs of
rhetoric at Brescia, Rome, Genoa, Bologna and Modena successively,
attracting by his brilliant fluency a large number of students at each
university. Through Cardinal Bentivoglio he was recommended to Antonio
Farnese, duke of Parma, who appointed him his poet laureate; and he
remained at the court of Parma until the death of Antonio, after which
he returned to Genoa. Shortly afterwards, through the intercession of
Bentivoglio, he obtained from the pope the remission of his monastic
vows, and ultimately succeeded in recovering a portion of his paternal
inheritance. After the peace of Aix-la-Chapelle he returned to the court
of Parma, and there devoted the later years of his life chiefly to
poetical composition. He died on the 20th of December 1768. As a poet
Frugoni was one of the best of the school of the Arcadian Academy, and
his lyrics and pastorals had great facility and elegance.
His collected works were published at Parma in 10 vols. in 1799, and a
more complete edition appeared at Lucca in the same year in 15 vols. A
selection from his works was published at Brescia in 1782, in 4 vols.
FRUIT (through the French from the Lat. _fructus_; _frui_, to enjoy), in
its widest sense, any product of the soil that can be enjoyed by man or
animals; the word is so used constantly in the Bible, and extended, as a
Hebraism, to offspring or progeny of man and of animals, in such
expressions as "the fruit of the body," "of the womb," "fruit of thy
cattle" (Deut. xxviii. 4), &c., and generally to the product of any
action or effort. Between this wide and frequently figurative use of the
word and its application in the strict botanical sense treated below,
there is a popular meaning, regarding the objects denoted by the word
entirely from the standpoint of edibility, and differentiating them
roughly from those other products of the soil, which, regarded
similarly, are known as vegetables. In this sense "fruit" is applied to
such seed-envelopes of plants as are edible, either raw or cooked, and
are usually sweet, juicy or of a refreshing flavour. But applications of
the word in this sense are apt to be loose and shifting according to the
fashion of the time.
Fruit, in the botanical sense, is developed from the flower as the
result of fertilization of the ovule. After fertilization various
changes take place in the parts of the flower. Those more immediately
concerned in the process, the anther and stigma, rapidly wither and
decay, while the filaments and style often remain for some time; the
floral envelopes become dry, the petals fall, and the sepals are either
deciduous, or remain persistent in an altered form; the ovary becomes
enlarged, forming the _pericarp_; and the ovules are developed as the
seeds, containing the embryo-plant. The term fruit is strictly applied
to the mature pistil or ovary, with the seeds in its interior; but it
often includes other parts of the flower, such as the bracts and floral
envelopes. Thus the fruit of the hazel and oak consists of the ovary
enveloped by the bracts; that of the apple and pear, of the ovary and
floral receptacle; and that of the pine-apple, of the whole
inflorescence. Such fruits are sometimes distinguished as _pseudocarps_.
In popular language, the fruit includes all those parts which exhibit a
striking change as the result of fertilization. In general, the fruit is
not ripened unless fertilization has been effected; but cases occur as
the result of cultivation in which the fruit swells and becomes to all
appearance perfect, while no seeds are produced. Thus, there are
seedless oranges, grapes and pineapples. When the ovules are
unfertilized, it is common to find that the ovary withers and does not
come to maturity; but in the case of bananas, plantains and bread-fruit,
the non-development of seeds seems to lead to a larger growth and a
greater succulence of fruit.
The fruit, like the ovary, may be formed of a single carpel or of
several. It may have one cell or cavity, being _unilocular_; or many,
_multilocular_, &c. The number and nature of the divisions depend on
the number of carpels and the extent to which their edges are folded
inwards. The appearances presented by the ovary do not always remain
permanent in the fruit. Great changes are observed to take place, not
merely as regards the increased size of the ovary, its softening or
hardening, but also in its internal structure, owing to the
suppression, additional formation or enlargement of parts. Thus, in
the ash (fig. 1) an ovary with two cells, each containing an ovule
attached to a central placenta, is changed into a unilocular fruit
with one seed; one ovule becomes abortive, while the other, g,
gradually enlarging until the septum is pushed to one side, unites
with the walls of the cell, and the placenta appears to be parietal.
In the oak and hazel, an ovary with three and two cells respectively,
and two ovules in each, produces a one-celled fruit with one seed. In
the coco-nut, a trilocular and triovular ovary produces a one-celled,
one-seeded fruit. This abortion may depend on the pressure caused by
the development of certain ovules, or it may proceed from
non-fertilization of all the ovules and consequent non-enlargement of
the carpels. Again, by the growth of the placenta, or the folding
inwards of parts of the carpels, divisions occur in the fruit which
did not exist in the ovary. In _Cathartocarpus Fistula_ a one-celled
ovary is changed into a fruit having each of its seeds in a separate
cell, in consequence of spurious dissepiments being produced
horizontal from the inner wall of the ovary. In flax (_Linum_) by the
folding inwards of the back of the carpels a five-celled ovary becomes
a ten-celled fruit. In _Astragalus_ the folding inwards of the dorsal
suture converts a one-celled ovary into a two-celled fruit; and in
_Oxytropis_ the folding of the ventral suture gives rise to a similar
change. The development of cellular or pulpy matter, and the
enlargement of parts not forming whorls of the flower, frequently
alter the appearance of the fruit, and render it difficult to discover
its formation. In the gooseberry (fig. 29), grape, guava, tomato and
pomegranate, the seeds nestle in pulp formed by the placentas. In the
orange the pulpy matter surrounding the seeds is formed by succulent
cells, which are produced from the inner partitioned lining of the
pericarp. In the strawberry the receptacle becomes succulent, and
bears the mature carpels on its convex surface (fig. 2); in the rose
there is a fleshy hollow receptacle which bears the carpels on its
concave surface (fig. 3). In the juniper the scaly bracts grow up
round the seeds and become succulent, and in the fig (fig. 4) the
receptacle becomes succulent and encloses an inflorescence.
[Illustration:
Fig. 1.--Samara or winged fruit of Ash (_Fraxinus_). 1, Entire, with
its wing a; 2, lower portion cut transversely, to show that it
consists of two cells; one of which, l, is abortive, and is reduced
to a very small cavity, while the other is much enlarged and filled
with a seed g.
Fig. 2.--Fruit of the Strawberry (_Fragaria vesca_), consisting of
an enlarged succulent receptacle, bearing on its surface the small
dry seed-like fruits (achenes). (After Duchartre.)
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.
Fig. 3.--Fruit of the Rose cut vertically. s', Fleshy hollowed
receptacle; s, persistent sepals; _fr_, ripe carpels; e, stamens,
withered.
Fig. 4.--Peduncle of Fig (_Ficus Carica_), ending in a hollow
receptacle enclosing numerous male and female flowers.
Fig. 5.--Fruit of Cherry (_Prunus Cerasus_) in longitudinal section.
ep, Epicarp; m, mesocarp; en, endocarp.
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.]
The pericarp consists usually of three layers, the external, or
_epicarp_ (fig. 5, ep); the middle, or _mesocarp_, m; and the
internal, or _endocarp_, _en_. These layers are well seen in such a
fruit as the peach, plum or cherry, where they are separable one from
the other; in them the epicarp forms what is commonly called the skin;
the mesocarp, much developed, forms the flesh or pulp, and hence has
sometimes been called _sarcocarp_; while the endocarp, hardened by the
production of woody cells, forms the _stone_ or _putamen_ immediately
covering the kernel or seed. The pulpy matter found in the interior of
fruits, such as the gooseberry, grape and others, is formed from the
placentas, and must not be confounded with the sarcocarp. In some
fruits, as in the nut, the three layers become blended together and
are indistinguishable. In bladder senna (_Colutea arborescens_) the
pericarp retains its leaf-like appearance, but in most cases it
becomes altered both in consistence and in colour. Thus in the date
the epicarp is the outer brownish skin, the pulpy matter is the
mesocarp or sarcocarp, and the thin papery-like lining is the endocarp
covering the hard seed. In the medlar the endocarp becomes of a stony
hardness. In the melon the epicarp and endocarp are very thin, while
the mesocarp forms the bulk of the fruit, differing in texture and
taste in its external and internal parts. The rind of the orange
consists of epicarp and mesocarp, while the endocarp forms partitions
in the interior, filled with pulpy cells. The part of the pericarp
attached to the peduncle is the base, and the point where the style or
stigma existed is the apex. This latter is not always the apparent
apex, as in the case of the ovary; it may be lateral or even basilar.
The style sometimes remains in a hardened form, rendering the fruit
_apiculate_; at other times it falls off, leaving only traces of its
existence. The presence of the style or stigma serves to distinguish
certain single-seeded pericarps from seeds.
[Illustration:
FIG. 6.--Seed-vessel or capsule of Campion, opening by ten teeth at
the apex. The calyx c is seen surrounding the seed-vessel.
FIG. 7.--Capsule of Poppy, opening by pores p, under the radiating
peltate stigma s.]
Dehiscence of fruits.
When the fruit is mature and the seeds are ripe, the carpels usually
give way either at the ventral or dorsal suture or at both, and so
allow the seeds to escape. The fruit in this case is _dehiscent_. But
some fruits are _indehiscent_, falling to the ground entire, and the
seeds eventually reaching the soil by their decay. By dehiscence the
pericarp becomes divided into different pieces, or _valves_, the fruit
being univalvular, bivalvular or multivalvular, &c., according as
there are one, two or many valves. The splitting extends the whole
length of the fruit, or is partial, the valves forming teeth at the
apex, as in the order Caryophyllaceae (fig. 6). Sometimes the valves
are detached only at certain points, and thus dehiscence takes place
by pores at the apex, as in poppy (fig. 7), or at the base, as in
_Campanula_. Indehiscent fruits are either dry, as the nut, or fleshy,
as the cherry and apple. They are formed of one or several carpels. In
the former case they usually contain only a single seed, which may
become so incorporated with the pericarp as to appear to be naked, as
in the grain of wheat and generally in grasses. In such cases the
presence of the remains of style or stigma determines their true
nature.
[Illustration:
FIG. 8.--Dry dehiscent fruit. The pod (legume) of the Pea; r, the
dorsal suture; b, the ventral; c, calyx; s, seeds.
From Vines' _Students' Text-Book of Botany_, by permission of Swan
Sonnenschein & Co.
FIG. 9.--(1) Fruit or capsule of Meadow Saffron (_Colchicum
autumnale_), dehiscing along the septa (septicidally); (2) same cut
across, showing the three chambers with the seeds attached along the
middle line (axile placentation).
FIG. 10.--Diagram to illustrate the septicidal dehiscence in a
pentalocular capsule. The loculaments l correspond to the number of
the carpels, which separate by splitting through the septa, s.
FIG. 11.--The seed vessel (capsule) of the Flower-de-Luce (_Iris_),
opening in a loculicidal manner. The three valves bear the septa in
the centre, and the opening takes place through the back of the
loculaments. Each valve is formed by the halves of contiguous
carpels.
FIG. 12.--Diagram to illustrate loculicidal dehiscence. The
loculaments l, split at the back, and the valves separate, bearing
the septa s on their centres.
FIG. 13.--Diagram to illustrate septifragal dehiscence, in which the
dehiscence takes place through the back of the loculaments l, and
the valves separate from the septa s, which are left attached to the
placentas in the centre.]
Dehiscent fruits, when composed of single carpels, may open by the
ventral suture only, as in the paeony, hellebore, _Aquilegia_ (fig.
28) and _Caltha_; by the dorsal suture only, as in magnolias and some
_Proteaceae_, or by both together, as in the pea (fig. 8) and bean; in
these cases the dehiscence is _sutural_. When composed of several
united carpels, two types of dehiscence occur--a longitudinal and a
transverse. In the longitudinal the separation may take place by the
dissepiments throughout their length, so that the fruit is resolved
into its original carpels, and each valve represents a carpel, as in
rhododendron, _Colchicum_, &c.; this dehiscence, in consequence of
taking place through the septum, is called _septicidal_ (figs. 9, 10).
The valves separate from their commissure, or central line of union,
carrying the placentas with them, or they leave the latter in the
centre, so as to form with the axis a column of a cylindrical, conical
or prismatic shape. Dehiscence is _loculicidal_ when the union between
the edges of the carpels is persistent, and they dehisce by the dorsal
suture, or through the back of the loculaments, as in the lily and
iris (figs. 11, 12). In these cases each valve consists of a half of
each of two contiguous carpels. The placentas either remain united to
the axis, or they separate from it, being attached to the septa on the
valves. When the outer walls of the carpels break off from the septa,
leaving them attached to the central column, the dehiscence is said to
be _septifragal_ (fig. 13), and where, as in _Linum catharticum_ and
_Calluna_, the splitting takes place first of all in a septicidal
manner, the fruit is described as _septicidally septifragal_; while in
other cases, as in thorn apple (_Datura Stramonium_), where the
splitting is at first loculicidal, the dehiscence is _loculicidally
septifragal_. In all those forms the separation of the valves takes
place either from above downwards or from below upwards. In
_Saxifraga_ a splitting for a short distance of the ventral sutures of
the carpels takes place, so that a large apical pore is formed. In the
fruit of Cruciferae, as wallflower (fig. 14), the valves separate from
the base of the fruit, leaving a central _replum_, or frame, which
supports the false septum formed by a prolongation from the parietal
placentas on opposite sides of the fruit, extending between the
ventral sutures of the carpels. In Orchidaceae (fig. 15) the pericarp,
when ripe, separates into three valves in a loculicidal manner, but
the midribs of the carpels, to which the placentas are attached, often
remain adherent to the axis both at the apex and base after the valves
bearing the seeds have fallen. The other type of dehiscence is
transverse, or _circumscissile_, when the upper part of the united
carpels falls off in the form of a lid or operculum, as in _Anagallis_
and in henbane (_Hyoscyamus_) (fig. 16).
[Illustration:
FIG. 14.--Siliqua or seed-vessel of Wallflower (_Cheiranthus
Cheiri_), opening by two valves, which separate from the base
upwards, leaving the seeds attached to the dissepiment which is
supported by the replum.
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.
FIG. 15.--Capsule of an Orchid (_Xylobium_). v, valve.
FIG. 16.--Seed-vessel of _Anagallisarvensis_, opening by
circumscissile dehiscence.
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.
FIG. 17.--Lomentum of _Hedysarum_ which, when ripe, separates
transversely into single-seeded portions or mericarps.
FIG. 18.--Fruit of _Geranium pratense_, after splitting.]
Sometimes the axis is prolonged beyond the base of the carpels, as in
the mallow and castor-oil plant, the carpels being united to it
throughout their length by their faces, and separating from it without
opening. In the Umbelliferae the two carpels separate from the lower
part of the axis, and remain attached by their apices to a
prolongation of it, called a _carpophore_ or _podocarp_, which splits
into two (fig. 25) and suspends them; hence the fruit is termed a
_cremocarp_, which divides into two _mericarps_. The general term
_schizocarp_ is applied to all dry fruits, which break up into two or
more one-seeded indehiscent mericarps, as in _Hedysarum_ (fig. 17). In
the order Geraniaceae the styles remain attached to a central column,
and the mericarps separate from below upwards, before dehiscing by
their ventral suture (fig. 18). Carpels which separate one from
another in this manner are called _cocci_. They are well seen in the
order Euphorbiaceae, where there are usually three such carpels, and
the fruit is termed tricoccus. In many of them, as _Hura crepitans_,
the cocci separate with great force and elasticity. In many leguminous
plants, such as _Ornithopus_, _Hedysarum_ (fig. 17), _Entada_,
_Coronilla_ and the gum-arabic plant (_Acacia arabica_), the fruit
becomes a schizocarp by the formation of transverse partitions from
the folding in of the sides of the pericarp, and distinct separations
taking place at these partitions.
Fruits are formed by one flower, or are the product of several flowers
combined. In the former case they are either _apocarpous_, of one
mature carpel or of several separate free carpels; or _syncarpous_, of
several carpels, more or less completely united. When the fruit is
composed of the ovaries of several flowers united, it is usual to find
the bracts and floral envelopes also joined with them, so as to form
one mass; hence such fruits are known as multiple, confluent or
_anthocarpous_. The term simple is applied to fruits which are formed
by the ovary of a single flower, whether they are composed of one or
several carpels, and whether these carpels are separate or combined.
[Illustration: From Vines' _Students' Text-Book of Botany_, by
permission of Swan Sonnenschein & Co.
FIG. 19.--Dry one-seeded fruit of dock (_Rumex_) cut vertically. ov,
Pericarp formed from ovary wall; s, seed; e, endosperm; pl, embryo
with radicle pointing upwards and cotyledons downwards--enlarged.
FIG. 20.--Achene of _Ranunculus arvensis_ in longitudinal section;
e, endosperm; pl, embryo. (After Baillon, enlarged.)
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.
FIG. 21.--Fruit of Common Sycamore (_Acer Pseudoplatanus_), dividing
into two mericarps m; s, pedicel; fl, wings (nat. size).]
Dispersal of fruit or seed.
The object of the fruit in the economy of the plant is the protection
and nursing of the developing seed and the dispersion of the ripe
seeds. Hence, generally, one-seeded fruits are indehiscent, while
fruits containing more than one seed open to allow of the dispersal of
the seeds over as wide an area as possible. The form, colour,
structure and method of dehiscence of fruits and the form of the
contained seeds are intimately associated with the means of dispersal,
which fall into several categories. (1) By a mechanism residing in the
fruit. Thus many fruits open suddenly when they are dry, and the seeds
are ejected by the twisting or curving of the valves, or in some other
way; e.g. in gorse, by the spiral curving of the valves; in
_Impatiens_, by the twisting of the cocci; in squirting cucumber, by
the pressure exerted on the pulpy contents by the walls of the
pericarp. (2) By aid of various external agencies such as water.
Fruits or seeds are sometimes sufficiently buoyant to float for a long
time on sea- or fresh-water; e.g. coco-nut, by means of its thick,
fibrous coat (mesocarp), is carried hundreds of miles in the sea, the
tough, leathery outer coat (epicarp) preventing it from becoming
water-soaked. Fruits and seeds of West Indian plants are thrown up on
the coasts of north-west Europe, having been carried by the Gulf
Stream, and will often germinate; many are rendered buoyant by
air-containing cavities, and the embryo is protected from the seawater
by the tough coat of fruit or seed. Water-lily seeds are surrounded
with a spongy tissue when set free from the fruit, and float for some
distance before dropping to the bottom. (3) The most general agent in
the dispersal of seeds is the wind or currents of air--the fruit or
seed being rendered buoyant by wing-developments as in fruits of ash
(fig. 1) or maple (fig. 21), seeds of pines and firs, or many members
of the order Bignoniaceae; or hair-developments as in fruits of
clematis, where the style forms a feathery appendage, fruits of many
Compositae (dandelion, thistle, &c.), which are crowned by a plumose
pappus, or seeds of willow and poplar, or _Asclepias_ (fig. 36), which
bear tufts of silky hairs; to this category belong bladder-like
fruits, such as bladder-senna, which are easily rolled by the wind, or
cases like the so-called rose of Jericho, a small cruciferous plant
(_Anastatica hierocuntica_), where the plant dries up after developing
its fruits and becomes detached from the ground; the branches curl
inwards, and the whole plant is rolled over the dry ground by the
wind. The wind also aids the dispersal of the seeds in the case of
fruits which open by small teeth (many Caryophyllaceae [fig. 6]) or
pores (poppy [fig. 7], _Campanula_, &c.); the seeds are in these cases
small and numerous, and are jerked through the pores when the
capsules, which are generally borne on long, dry stems or stalks, are
shaken by the wind. (4) In other cases members of the animal world aid
in seed-dispersal. Fruits often bear stiff hairs or small hooks, which
cling to the coat of an animal or the feathers of a bird; such are
fruits of cleavers (_Galium Aparine_), a common hedge-row plant,
_Ranunculus arvensis_ (fig. 20), carrot, _Geum_, &c.; or the fruit or
seed has an often bright-coloured, fleshy covering, which is sought by
birds as food, as in stone-fruits such as plum, cherry (fig. 5), &c.,
where the seed is protected from injury in the mouth or stomach of the
animal by the hard endocarp; or the hips of the rose (fig. 3), where
the succulent scarlet "fruit" (the swollen receptacle) envelops a
number of small dry true fruits (achenes), which cling by means of
stiff hairs to the beak of the bird.
[Illustration:
FIG. 22.--Vertical section of a grain of wheat, showing embryo below
at the base of the endosperm e; s, scutellum separating embryo from
endosperm; f.l, foliage leaf; p.s, sheath of plumule; p.r, primary
root; s.p.r, sheath of primary root.
FIG. 23.--Fruit of Comfrey (_Symphytum_) surrounded by persistent
calyx, c. The style s appears to arise from the base of the carpels,
enlarged.
FIG. 24.--Ovary of _Foeniculum officinale_ with pendulous ovules, in
longitudinal section. (After Berg and Schmidt, magnified.)
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.
FIG. 25.--Fruit of _Carum Carui_. A, Ovary of the flower; B, ripe
fruit. The two carpels have separated so as to form two mericarps
(m). Part of the septum constitutes the carpophore (a). p, Top of
flower-stalk; d, disk on top of ovary; n, stigma.
From Vines' _Students' Text-Book of Botany_, by permission of Swan
Sonnenschein & Co.]
Forms of fruit.
Simple fruits have either a _dry_ or _succulent_ pericarp. The
_achene_ is a dry, one-seeded, indehiscent fruit, the pericarp of
which is closely applied to the seed, but separable from it. It is
solitary, forming a single fruit, as in the dock (fig. 19) and in the
cashew, where it is supported on a fleshy peduncle; or _aggregate_, as
in _Ranunculus_ (fig. 20), where several achenes are placed on a
common elevated receptacle. In the strawberry the achenes (fig. 2) are
aggregated on a convex succulent receptacle. In the rose they are
supported on a concave receptacle (fig. 3), and in the fig the
succulent receptacle completely encloses the achenes (fig. 4). In
_Dorstenia_ the achenes are situated on a flat or slightly concave
receptacle. Hence what in common language are called the seeds of the
strawberry, rose and fig, are in reality ripe carpels. The styles
occasionally remain attached to the achenes in the form of feathery
appendages, as in _Clematis_. In Compositae, the fruit is an inferior
achene (_cypsela_), to which the pappus (modified calyx) remains
adherent. Such is also the nature of the fruit in Dipsacaceae (e.g.
scabious). When the pericarp is thin, and appears like a bladder
surrounding the seed, the achene is termed a _utricle_, as in
Amarantaceae. When the pericarp is extended in the form of a winged
appendage, a _samara_ or _samaroid achene_ is produced, as in the ash
(fig. 1) and common sycamore (fig. 21). In these cases there are
usually two achenes united, one of which, however, as in _Fraxinus_
(fig. 1), may be abortive. The wing surrounds the fruit longitudinally
in the elm. When the pericarp becomes so incorporated with the seed as
to be inseparable from it, as in grains of wheat (fig. 22), maize,
oats and other grasses, then the name _caryopsis_ is given. The
one-seeded portions (mericarps) of schizocarps often take the form of
achenes, e.g. the mericarps of the mallows or of umbellifers (figs.
24, 25). In Labiatae and Boraginaceae (e.g. comfrey, fig. 23), where
the bicarpellary ovary becomes our one-seeded portions in the fruit,
the partial fruits are of the nature of achenes or nutlets according
to the texture (leathery or hard) of the pericarp.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 26.--Cupule of _Quercus Aegilops_. cp, cupule; gl, fruit.
(After Duchartre.)]
The _nut_ or _glans_ is a dry one-celled indehiscent fruit with a
hardened pericarp, often surrounded by bracts at the base, and, when
mature, containing only one seed. In the young state the ovary often
contains two or more ovules, but only one comes to maturity. It is
illustrated by the fruits of the hazel and chestnut, which are covered
by leafy bracts, in the form of a _husk_, and by the acorn, in which
the bracts and receptacle form a _cupula_ or _cup_ (fig. 26). The
parts of the pericarp of the nut are united so as to appear one. In
common language the term nut is very vaguely applied both to fruit and
seeds.
The _drupe_ is a succulent usually one-seeded indehiscent fruit, with
a pericarp easily distinguishable into epicarp, mesocarp and endocarp.
This term is applied to such fruits as the cherry (fig. 5), peach,
plum, apricot or mango. The endocarp is usually hard, forming the
stone (putamen) of the fruit, which encloses the kernel or seed. The
mesocarp is generally pulpy and succulent, so as to be truly a
sarcocarp, as in the peach, but it is sometimes of a tough texture, as
in the almond, and at other times is more or less fibrous, as in the
coco-nut. In the almond there are often two ovules formed, only one of
which comes to perfection. In the raspberry and bramble several small
drupes or _drupels_ are aggregated so as to constitute an _etaerio_.
The _follicle_ is a dry unilocular many-seeded fruit, formed from one
carpel and dehiscing by the ventral suture. It is rare to meet with a
solitary follicle forming the fruit. There are usually several
aggregated together, either in a whorl on a shortened receptacle, as
in hellebore, aconite, larkspur, columbine (figs. 27, 28) or the order
Crassulaceae, or in a spiral manner on an elongated receptacle, as in
_Magnolia_ and _Banksia_. Occasionally, follicles dehisce by the
dorsal suture, as in _Magnolia grandiflora_ and _Banksia_.
[Illustration:
FIG. 27.--Fruit of Columbine (_Aquilegia_), formed of five
follicles.
FIG. 28.--Single follicle, showing dehiscence by the ventral suture.
FIG. 29.--Transverse section of berry of Gooseberry, showing the
seeds attached to the parietal placentas and immersed in pulp, which
is formed partly from the endocarp, partly from the seed-coat.
FIG. 30.--Section of the fruit of the Apple (_Pyrus Malus_), or
pome, consisting of a fleshy covering formed by the floral
receptacle and the true fruit or core with five cavities with
seeds.]
The _legume_ or _pod_ is a dry monocarpellary unilocular many-seeded
fruit, formed from one carpel, dehiscing both by the ventral and the
dorsal suture. It characterizes leguminous plants, as the bean and pea
(fig. 8). In the bladder-senna it forms an inflated legume. In some
Leguminosae, as _Arachis_, _Cathartocarpus Fistula_ and the tamarind,
the fruit must be considered a legume, although it does not dehisce.
The first of these plants produces its fruit underground, and is
called earth-nut; the second has a partitioned legume and is
schizocarpic; and both the second and third have pulpy matter
surrounding the seeds. Some legumes are schizocarpic by the formation
of constrictions externally. Such a form is the _lomentum_ or
_lomentaceous legume_ of _Hedysarum_ (fig. 17), _Coronilla_,
_Ornithopus_, _Entada_ and of some Acacias. In _Medicago_ the legume
~~ is twisted like a snail, and in _Caesalpinia coriaria_, or
Divi-divi, it is vermiform or curved like a worm. Sometimes the number
of seeds is reduced, as in _Erythrina monosperma_ and _Geoffroya
superba_, which are one-seeded, and in _Pterocarpus_ and _Dalbergia_,
which are two-seeded.
[Illustration: FIG. 31.--Transverse section of the fruit of the Melon
(_Cucumis Melo_), showing the placentas with the seeds attached to
them. The three carpels forming the pepo are separated by partitions.
From the centre processes pass outwards, ending in the curved
placenta.
The _berry_ (_bacca_) is a term applied generally to all fruits with
seeds immersed in pulp, and includes fruits of very various origin. In
_Actaea_ (baneberry) or _Berberis_ (barberry) it is derived from a
single free carpel; generally, however, it is the product of a
syncarpous ovary, which is superior, as in grape or potato, or
inferior, as in gooseberry (fig. 29) or currant. In the pomegranate
there is a peculiar baccate many-celled inferior fruit, having a tough
rind, enclosing two rows of carpels placed one above the other. The
seeds are immersed in pulp, and are attached irregularly to the wall,
base and centre of the loculi. In the baobab there is a multilocular
syncarpous fruit, in which the seeds are immersed in pulp.
The _pepo_, another indehiscent syncarpous fruit, is illustrated by
the fruit of the gourd, melon (fig. 31) and other Cucurbitaceae. It is
formed of three carpels, surmounted by the calyx; the rind is thick
and fleshy, and there are three or more seed-bearing parietal
placentas, either surrounding a central cavity or prolonged inwards
into it. The fruit of the papaw resembles the pepo, but the calyx is
not superior.
The _hesperidium_ is the name given to such indehiscent fleshy
syncarpous fruits as the orange, lemon and shaddock, in which the
epicarp and mesocarp form a separable rind, and the endocarp sends
prolongations inwards, forming triangular divisions, to the inner
angle of which the seeds are attached, pulpy cells being developed
around them from the wall. Both pepo and hesperidium may be considered
as modifications of the berry.
The _pome_ (fig. 30), seen in the apple, pear, quince, medlar and
hawthorn, is a fleshy indehiscent syncarpous fruit, in the formation
of which the receptacle takes part. The outer succulent part is the
swollen receptacle, the horny core being the true fruit developed from
the usually five carpels and enclosing the seeds. In the medlar the
core (or true pericarp) is of a stony hardness, while the outer
succulent covering is open at the summit. The pome somewhat resembles
the fruit of the rose (fig. 3), where the succulent receptacle
surrounds a number of separate achenes.
The name _capsule_ is applied generally to all dry syncarpous fruits,
which dehisce by valves. It may thus be unilocular or multilocular,
one- or many-seeded. The true valvular capsule is observed in
_Colchicum_ (fig. 9), lily and iris (fig. 11). The _porose capsule_ is
seen in the poppy (fig. 7), _Antirrhinum_ and _Campanula_. In
_Campanula_ the pores occur at the base of the capsule, which becomes
inverted when ripe. When the capsule opens by a lid, or by
circumscissile dehiscence, it is called a _pyxidium_, as in pimpernel
(_Anagallis arvensis_) (fig. 16), henbane and monkey-pot (_Lecythis_).
The capsule assumes a screw-like form in _Helicteres_, and a star-like
form in star-anise (_Illicium anisatum_). In certain instances the
cells of the capsule separate from each other, and open with
elasticity to scatter the seeds. This kind of capsule is met with in
the sandbox tree (_Hura crepitans_) and other Euphorbiaceae, where the
cocci, containing each a single seed, burst asunder with force; and in
Geraniaceae, where the cocci, each containing, when mature, usually
one seed, separate from the carpophore, become curved upwards by their
adherent styles, and open by the ventral suture (fig. 18).
The _siliqua_ is a dry syncarpous bilocular many-seeded fruit, formed
from two carpels, with a false septum, dehiscing by two valves from
below upwards, the valves separating from the placentas and leaving
them united by the septum (fig. 32). The seeds are attached on both
sides of the septum, either in one row or in two. When the fruit is
long and narrow it is a _siliqua_ (fig. 14); when broad and short,
_silicula_ (fig. 33). It occurs in cruciferous plants, as wallflower,
cabbage and cress. In _Glaucium_ and _Eschscholtzia_ (Papaveraceae)
the dissepiment is of a spongy nature. It may become transversely
constricted (_lomentaceous_), as in radish (_Raphanus_) and sea-kale,
and it may be reduced, as in woad (_Isatis_), to a one-seeded
condition.
It sometimes happens that the ovaries of two flowers unite so as to
form a double fruit (_syncarp_). This may be seen in many species of
honeysuckle. But the fruits which are now to be considered consist
usually of the floral envelopes, as well as the ovaries of several
flowers united into one, and are called _multiple_ or _confluent_. The
term _anthocarpous_ has also been applied as indicating that the
floral envelopes as well as the carpels are concerned in the formation
of the fruit.
The _sorosis_ is a succulent multiple fruit formed by the confluence
of a spike of flowers, as in the fruit of the pine-apple (fig. 34),
the bread-fruit and jack-fruit. Similarly the fruit of the mulberry
represents a catkin-like inflorescence.
The _syconus_ is an anthocarpous fruit, in which the receptacle
completely encloses numerous flowers and becomes succulent. The fig
(fig. 4) is of this nature, and what are called its seeds are the
achenes of the numerous flowers scattered over the succulent hollowed
receptacle. In _Dorstenia_ the axis is less deeply hollowed, and of a
harder texture, the fruit exhibiting often very anomalous forms.
The _strobilus_, or _cone_, is a seed-bearing spike, more or less
elongated, covered with scales, each of which may be regarded as
representing a separate flower, and has often two seeds at its base;
the seeds are naked, no ovary being present. This fruit is seen in the
cones of firs, spruces, larches and cedars, which have received the
name of Coniferae, or cone-bearers, on this account. Cone-like fruit
is also seen in most Cycadaceae. The scales of the strobilus are
sometimes thick and closely united, so as to form a more or less
angular and rounded mass, as in the cypress; while in the juniper they
become fleshy, and are so incorporated as to form a globular fruit
like a berry. The dry fruit of the cypress and the succulent fruit of
the juniper have received the name of _galbulus_. In the hop the fruit
is called also a strobilus, but in it the scales are thin and
membranous, and the seeds are not naked but are contained in
pericarps.
[Illustration:
FIG. 32.--Honesty (_Lunaria biennis_), showing the septum after the
carpels have fallen away.
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.
FIG. 33.--Silicula or pouch of shepherd's purse (_Capsella_),
opening by two folded valves, which separate from above downwards.
The partition is narrow, hence the silicula is angustiseptal.
From Strasburger's _Lehrbuch der Botanik_, by permission of Gustav
Fischer.
FIG. 34.--Fruit of the pine-apple (_Ananassa sativa_), developed
from a spike of numerous flowers with bracts, united so as to form a
collective or anthocarpous fruit. The crown of the pine-apple, c,
consists of a series of empty bracts prolonged beyond the fruit.]
The same causes which produce alterations in the other parts of the
flower give rise to anomalous appearances in the fruit. The carpels,
in place of bearing seeds, are sometimes changed into leaves, with
lobes at their margins. Leaves are sometimes produced from the upper
part of the fruit. In the genus _Citrus_, to which the orange and
lemon belong, it is very common to meet with a separation of the
carpels, so as to produce what are called horned oranges and fingered
citrons. In this case a syncarpous fruit has a tendency to become
apocarpous. In the orange we occasionally find a supernumerary row of
carpels produced, giving rise to the appearance of small and imperfect
oranges enclosed within the original one; the navel orange is of this
nature. It sometimes happens that, by the union of flowers, double
fruits are produced. Occasionally a double fruit is produced, not by
the incorporation of two flowers, but by the abnormal development of a
second carpel in the flower.
_Arrangement of Fruits._
A. True fruits--developed from the ovary alone.
1. Pericarp not fleshy or fibrous.
i. Indehiscent--not opening to allow the escape of the
seeds--generally one-seeded. Achene; caryopsis; cypsela;
nut; schizocarp.
ii. Dehiscent--the pericarp splits to allow the escape of the
seeds--generally many-seeded. Follicle; legume; siliqua;
capsule.
2. Pericarp generally differentiated into distinct layers, one
of which is succulent or fibrous. Drupe; berry.
B. Pseudocarps--the development extends beyond the ovary. Pome;
syconus; sorosis.
_The Seed._--The _seed_ is formed from the ovule as the result of
fertilization. It is contained in a seed-vessel formed from the ovary
in the plants called _angiospermous_; while in _gymnospermous_
plants, such as Coniferae and Cycadaceae, it is naked, or, in other
words, has no true pericarp. It sometimes happens in Angiosperms, that
the seed-vessel is ruptured at an early period of growth, so that the
seeds become more or less exposed during their development; this
occurs in mignonette, where the capsule opens at the apex, and in
_Cuphea_, where the placenta bursts through the ovary and floral
envelopes, and appears as an erect process bearing the young seeds.
After fertilization the ovule is greatly changed, in connexion with
the formation of the embryo. In the embryo-sac of most Angiosperms
(q.v.) there is a development of cellular tissue, the endosperm, more
or less filling the embryo-sac. In Gymnosperms (q.v.) the endosperm is
formed preparatory to fertilization. The fertilized egg enlarges and
becomes multicellular, forming the embryo. The embryo-sac enlarges
greatly, displacing gradually the surrounding nucellus, which
eventually forms merely a thin layer around the sac, or completely
disappears. The remainder of the nucellus and the integuments of the
ovules form the seed-coats. In some cases (fig. 35) a delicate inner
coat or _tegmen_ can be distinguished from a tougher outer coat or
_testa_; often, however, the layers are not thus separable. The
consistency of the seed-coat, its thickness, the character of its
surface, &c., vary widely, the variations being often closely
associated with the environment or with the means of seed-dispersal.
An account of the development of the seed from the ovule will be found
in the article ANGIOSPERMS. When the pericarp is dehiscent the
seed-covering is of a strong and often rough character; but when the
pericarp is indehiscent and encloses the seed for a long period, the
outer seed-coat is thin and soft. The cells of the testa are often
coloured, and have projections and appendages of various kinds. Thus
in _Abrus precatorius_ and _Adenanthera pavonina_ it is of a bright
red colour; in French beans it is beautifully mottled; in the almond
it is veined; in the tulip and primrose it is rough; in the snapdragon
it is marked with depressions; in cotton and _Asclepias_ (fig. 36) it
has hairs attached to it; and in mahogany, _Bignonia_, and the pines
and firs it is expanded in the form of wing-like appendages (fig. 37).
In _Collomia_, _Acanthodium_, _Cobaea scandens_ and other seeds, it
contains spiral cells, from which, when moistened with water, the
fibres uncoil in a beautiful manner; and in flax (_Linum_) and others
the cells are converted into mucilage. These structural peculiarities
of the testa in different plants have relation to the scattering of
the seed and its germination upon a suitable nidus. But in some plants
the pericarps assume structures which subserve the same purpose; this
especially occurs in small pericarps enclosing single seeds, as
achenes, caryopsides, &c. Thus in Compositae and valerian, the pappose
limb of the calyx forms a parachute to the pericarp; in Labiatae and
some Compositae spiral cells are formed in the epicarp; and the
epicarp is prolonged as a wing in _Fraxinus_ (fig. 1) and _Acer_ (fig.
21).
[Illustration:
FIG. 35.--Seed of Pea (_Pisum_) with one cotyledon removed. c,
Remaining cotyledon; ch, chalaza-point at which the nourishing
vessels enter; e, tegmen or inner coat; f, funicle or stalk; g,
plumule of embryo; m, micropyle; pl, placenta; r, radicle of embryo;
t, tigellum or stalk between root and plumule; te, testa.
FIG. 36.--Seed of _Asclepias_, with a cluster of hairs arising from
the edges of the micropyle.]
Sometimes there is an additional covering to the seed, formed after
fertilization, to which the name _arillus_ has been given (fig. 38).
This is seen in the passion-flower, where the covering arises from the
placenta or extremity of the funicle at the base of the ovule and
passes upwards towards the apex, leaving the micropyle uncovered. In
the nutmeg and spindle tree this additional coat is formed from above
downwards, constituting in the former case a laciniated scarlet
covering called _mace_. In such instances it has been called an
_arillode_ (fig. 39). This arillode, after growing downwards, may be
reflected upwards so as to cover the micropyle. The fleshy scarlet
covering formed around the naked seed in the yew is by some considered
of the nature of an aril. On the testa, at various points, there are
produced at times other cellular bodies, to which the name of
_strophioles_, or _caruncles_, has been given, the seeds being
strophiolate or carunculate. These tumours may occur near the base of
the seed, as in _Polygala_, or at the apex, as in Castor-oil plant
(_Ricinus_); or they may occur in the course of the raphe, as in
blood-root (_Sanguinaria_) and _Asarabacca_. The funicles of the
ovules frequently attain a great length in the seed, and in some
magnolias, when the fruit dehisces, they appear as long scarlet cords
suspending the seeds outside. The hilum or umbilicus of the seed is
usually well marked, as a scar of varying size; in the calabar bean
and in some species of Mucuna and Dolichos it extends along a large
portion of the edge of the seed; it frequently exhibits marked
colours, being black in the bean, white in many species of Phaseolus,
&c. The micropyle (fig. 35, m) of the seed may be recognizable by the
naked eye, as in the pea and bean tribe, _Iris_, &c., or it may be
very minute or microscopic. It indicates the true apex of the seed,
and is important as marking the point to which the root of the embryo
is directed. At the micropyle in the bean is observed a small process
of integument, which, when the young plant sprouts, is pushed up like
a lid; it is called the _embryotega_. The chalaza (fig. 38, ch) is
often of a different colour from the rest of the seed. In the orange
(fig. 40) it is of a reddish-brown colour, and is easily recognized at
one end of the seed when the integuments are carefully removed. In
anatropal seeds the raphe forms a distinct ridge along one side of the
seed (fig. 41).
The position of the seed as regards the pericarp resembles that of the
ovule in the ovary, and the same terms are applied--erect, ascending,
pendulous, suspended, curved, &c. These terms have no reference to the
mode in which the fruit is attached to the axis. Thus the seed may be
erect while the fruit itself is pendent, in the ordinary meaning of
that term. The part of the seed next the axis or the ventral suture is
its face, the opposite side being the back. Seeds exhibit great
varieties of form. They may be flattened laterally (_compressed_), or
from above downwards (_depressed_). They may be round, oval,
triangular, polygonal, rolled up like a snail, as in _Physostemon_, or
coiled up like a snake, as in _Ophiocaryon paradoxum_.
[Illustration:
FIG. 37.--Seed of Pine (_Pinus_), with a membranous appendage w to
the testa, called a wing.
FIG. 38.--Young anatropal seed of the white Water-lily (_Nymphaea
alba_), cut vertically. It is attached to the placenta by the
funicle f, cellular prolongations from which form an aril a a. The
vessels of the cord are prolonged to the base of the nucellus n by
means of the raphe r. The base of the nucellus is indicated by the
chalaza ch, while the apex is at the micropyle m. The covering of
the seed is marked i. n is the nucellus or perisperm, enclosing the
embryo-sac es, in which the endosperm is formed. The embryo e, with
its suspensor, is contained in the sac, the radicle pointing to the
micropyle m.
FIG. 39.--Arillode a, or false aril, of the Spindle-tree
(_Euonymus_), arising from the micropyle f.
FIG. 40.--Anatropal seed of the Orange (_Citrus Aurantium_) opened
to show the chalaza c, which forms a brown spot at one end.
FIG. 41.--Entire anatropal seed of the Orange (_Citrus Aurantium_),
with its rugose or wrinkled testa, and the raphe r ramifying in the
thickness of the testa on one side.]
The endosperm formed in the embryo-sac of angiosperms after
fertilization, and found previous to it in gymnosperms, consists of
cells containing nitrogenous and starchy or fatty matter, destined for
the nutriment of the embryo. It occupies the whole cavity of the
embryo-sac, or is formed only at certain portions of it, at the apex,
as in _Rhinanthus_, at the base, as in _Vaccinium_, or in the middle,
as in _Veronica_. As the endosperm increases in size along with the
embryo-sac and the embryo, the substance of the original nucellus of
the ovule is gradually absorbed. Sometimes, however, as in Musaceae,
Cannaceae, Zingiberaceae, no endosperm is formed; the cells of the
original nucellus, becoming filled with food-materials for the embryo,
are not absorbed, but remain surrounding the embryo-sac with the
embryo, and constitute the _perisperm_. Again, in other plants, as
Nymphaeaceae (fig. 38) and Piperaceae, both endosperm and perisperm
are present. It was from observations on cases such as these that old
authors, imagining a resemblance betwixt the plant-ovule and the
animal ovum, applied the name _albumen_ to the outer nutrient mass or
perisperm, and designated the endosperm as _vitellus_. The term
albumen is very generally used as including all the nutrient matter
stored up in the seed, but it would be advisable to discard the name
as implying a definite chemical substance. There is a large class of
plants in which although at first after fertilization a mass of
endosperm is formed, yet, as the embryo increases in size, the
nutrient matter from the endospermic cells passes out from them, and
is absorbed by the cells of the embryo plant. In the mature seed, in
such cases, there is no separate mass of tissue containing nutrient
food-material apart from the embryo itself. Such a seed is said to be
_exalbuminous_, as in Compositae, Cruciferae and most Leguminosae
(e.g. pea, fig. 35). When either endosperm or perisperm or both are
present the seed is said to be _albuminous_.
[Illustration: FIG. 42.--The dicotyledonous embryo of the Pea laid
open. c, c, The two fleshy cotyledons, or seed-lobes, which remain
under ground when the plant sprouts; r, the radicular extremity of the
axis whence the root arises; t, the axis (hypocotyl) bearing the young
stalk and leaves g (plumule), which lie in a depression of the
cotyledons f.]
The albumen varies much in its nature and consistence, and furnishes
important characters. It may be farinaceous or mealy, consisting
chiefly of cells filled with starch, as in cereal grains, where it is
abundant; fleshy or cartilaginous, consisting of thicker cells which
are still soft, as in the coco-nut, and which sometimes contain oil,
as in the oily albumen of _Croton_, _Ricinus_ and poppy; horny, when
the cell-walls are slightly thickened and capable of distension, as in
date and coffee; the cell-walls sometimes become greatly thickened,
filling up the testa as a hard mass, as in vegetable ivory
(_Phytelephas_). The albumen may be uniform throughout, or it may
present a mottled appearance, as in the nutmeg, the seeds of Anonaceae
and some Palms, where it is called _ruminated_. This mottled
appearance is due to a protrusion of a dark lamella of the integument
between folded protuberances of albumen. A cavity is sometimes left in
the centre which is usually filled with fluid, as in the coco-nut. The
relative size of the embryo and of the endosperm varies much. In
Monocotyledons the embryo is usually small, and the endosperm large,
and the same is true in the case of coffee and many other plants
amongst Dicotyledons. The opposite is the case in other plants, as in
the Labiatae, Plumbaginaceae, &c.
The embryo consists of an axis bearing the _cotyledons_ (fig. 42, c),
or the first leaves of the plant. To that part of this axis
immediately beneath the cotyledons the terms _hypocotyl_, _caulicle_
or _tigellum_ (t) have been applied, and continuous backwards with it
is the young root or _radicle_ (r), the descending axis, their point
of union being the collar or neck. The terminal growing bud of the
axis is called the _plumule_ or _gemmule_ (g), and represents the
ascending axis. The radicular extremity points towards the micropyle,
while the cotyledonary extremity is pointed towards the base of the
ovule or the chalaza. Hence, by ascertaining the position of the
micropyle and chalaza, the two extremities of the embryo can in
general be discovered. It is in many cases difficult to recognize the
parts in an embryo; thus in _Cuscuta_, the embryo appears as an
elongated axis without divisions; and in _Caryocar_ the mass of the
embryo is made up by the radicular extremity and hypocotyl, in a
groove of which the cotyledonary extremity lies embedded (fig. 52). In
some monocotyledonous embryos, as in Orchidaceae, the embryo is a
cellular mass showing no parts. In parasitic plants also which form no
chlorophyll, as _Orobanche_, _Monotropa_, &c., the embryo remains
without differentiation, consisting merely of a mass of cells until
the ripening of the seed. When the embryo is surrounded by the
endosperm on all sides except its radicular extremity it is internal
(see figs. 19, 20); when lying outside the endosperm, and only coming
into contact with it at certain points, it is external, as in grasses
(e.g. wheat, fig. 22). When the embryo follows the direction of the
axis of the seed, it is axile or axial (fig. 43); when it is not in
the direction of the axis, it becomes abaxile or abaxial. In
campylotropal seeds the embryo is curved, and in place of being
embedded in endosperm, is frequently external to it, following the
concavity of the seed (fig. 44), and becoming peripherical, with the
chalaza situated in the curvature of the embryo, as in
Caryophyllaceae.
It has been already stated that the radicle of the embryo is directed
to the micropyle, and the cotyledons to the chalaza. In some cases, by
the growth of the integuments, the former is turned round so as not to
correspond with the apex of the nucellus, and then the embryo has the
radicle directed to one side, and is called excentric, as is seen in
Primulaceae, Plantaginaceae and many palms, especially the date. The
position of the embryo in different kinds of seeds varies. In an
orthotropal seed the embryo is inverted or _antitropal_, the radicle
pointing to the apex of the seed, or to the part opposite the hilum.
Again, in an anatropal seed the embryo is erect or _homotropal_ (fig.
43), the radicle being directed to the base of the seed. In curved or
campylotropal seeds the embryo is folded so that its radicular and
cotyledonary extremities are approximated, and it becomes
_amphitropal_ (fig. 44). In this instance the seed may be
exalbuminous, and the embryo may be folded on itself; or albuminous,
the embryo surrounding more or less completely the endosperm and being
peripherical. According to the mode in which the seed is attached to
the pericarp, the radicle may be directed upwards or downwards, or
laterally, as regards the ovary. In an orthotropal seed attached to
the base of the pericarp it is superior, as also in a suspended
anatropal seed. In other anatropal seeds the radicle is inferior. When
the seed is horizontal as regards the pericarp, the radicle is either
centrifugal, when it points to the outer wall of the ovary; or
centripetal, when it points to the axis or inner wall of the ovary.
These characters are of value for purposes of classification, as they
are often constant in large groups of genera.
Plants in which there are two cotyledons produced in the embryo are
_dicotyledonous_. The two cotyledons thus formed are opposite to each
other (figs. 42 and 45), but are not always of the same size. Thus, in
Abronia and other members of the order Nyctaginaceae, one of them is
smaller than the other (often very small), and in _Carapa guianensis_
there appears to be only one, in consequence of the intimate union
which takes place between the two. The union between the cotyledonary
leaves may continue after the young plant begins to germinate. Such
embryos have been called _pseudomonocotyledonous_. The texture of the
cotyledons varies. They may be thick, as in the pea (fig. 42),
exhibiting no traces of venation, with their flat internal surfaces in
contact, and their backs more or less convex; or they may be in the
form of thin and delicate laminae, flattened on both sides, and having
distinct venation, as in _Ricinus_, _Jatropha_, _Euonymus_, &c. The
cotyledons usually form the greater part of the mature embryo, and
this is remarkably well seen in such exalbuminous seeds as the bean
and pea.
[Illustration:
FIG. 43.--Seed of Pansy (_Viola tricolor_) cut vertically. The
embryo pl is axial, in the midst of fleshy endosperm al. The seed is
anatropal, and the embryo is homotropal; the cotyledons co point to
the base of the nucellus or chalaza ch, while the radicle, or the
other extremity of the embryo, points to the micropyle, close to the
hilum h. The hilum or base of the seed, and the chalaza or base of
the nucellus are united by means of the raphe r.
FIG. 44.--Seed of the Red Campion (_Lychnis_), cut vertically,
showing the peripheral embryo, with its two cotyledons and its
radicle. The embryo is curved round the albumen, so that its
cotyledons and radicle both come near the hilum (_amphitropal_).
FIG. 45.--Mature dicotyledonous embryo of the Almond, with one of
the cotyledons removed. r, Radicle; t, young stem or caulicle; c,
one of the cotyledons left; i, line of insertion of the cotyledon
which has been removed; g, plumule.
FIG. 46.--Exalbuminous seed of Wallflower (Cheiranthus) cut
vertically. The radicle r is folded on the edges of the cotyledons c
which are accumbent.
FIG. 47.--Transverse section of the seed of the Wallflower
(_Cheiranthus_), showing the radicle r folded on the edges of the
accumbent cotyledons c.
FIG. 48.--Transverse section of the seed of the Dame's Violet
(_Hesperis_). The radicle r is folded on the back of the cotyledons
c, which are said to be incumbent.]
Cotyledons are usually entire and sessile. But they occasionally
become lobed, as in the walnut and the lime; or petiolate, as in
_Geranium molle_; or auriculate, as in the ash. Like leaves in the
bud, cotyledons may be either applied directly to each other, or may
be folded in various ways. In geranium the cotyledons are twisted and
doubled; in convolvulus they are corrugated; and in the potato and in
_Bunias_, they are spiral,--the same terms being applied as to the
foliage leaves. The radicle and cotyledons are either straight or
variously curved. Thus, in some cruciferous plants, as the wallflower,
the cotyledons are applied by their faces, and the radicle (figs. 46,
47) is folded on their edges, so as to be lateral; the cotyledons are
here _accumbent_. In others, as _Hesperis_, the cotyledons (fig. 48)
are applied to each other by their faces, and the radicle, r, is
folded on their back, so as to be dorsal, and the cotyledons are
_incumbent_. Again, the cotyledons are _conduplicate_ when the radicle
is dorsal, and enclosed between their folds. In other divisions the
radicle is folded in a spiral manner, and the cotyledons follow the
same course.
In many gymnosperms more than two cotyledons are present, and they are
arranged in a whorl. This occurs in Coniferae, especially in the pine,
fir (fig. 49), spruce and larch, in which six, nine, twelve and even
fifteen have been observed. They are linear, and resemble in their
form and mode of development the clustered or fasciculated leaves of
the larch. Plants having numerous cotyledons are termed
_polycotyledonous_. In species of _Streptocarpus_ the cotyledons are
permanent, and act the part of leaves. One of them is frequently
largely developed, while the other is small or abortive.
[Illustration:
FIG. 49.--Polycotylodonous embryo of the Pine (_Pinus_) beginning to
sprout. t, Hypocotyl; r, radicle. The cotyledons c are numerous.
Within the cotyledons the primordial leaves are seen, constituting
the plumule or first bud of the plant.
FIG. 50.--Embryo of a species of Arrow-grass (_Triglochin_), showing
a uniform conical mass, with a slit s near the lower part. The
cotyledon c envelops the young bud, which protrudes at the slit
during germination. The radicle is developed from the lower part of
the axis r.
FIG. 51.--Grain of wheat (_Triticum_) germinating, showing (b) the
cotyledon and (c) the rootlets surrounded by their sheaths
(_coleorrhizae_).
FIG. 52.--Embryo of _Caryocar_. t, Thick hypocotyl, forming nearly
the whole mass, becoming narrowed and curved at its extremity, and
applied to the groove s. In the figure this narrowed portion is
slightly separated from the groove; c, two rudimentary cotyledons.]
In those plants in which there is only a single cotyledon in the
embryo, hence called _monocotyledonous_, the embryo usually has a
cylindrical form more or less rounded at the extremities, or elongated
and fusiform, often oblique. The axis is usually very short compared
with the cotyledon, which in general encloses the plumule by its lower
portion, and exhibits on one side a small slit which indicates the
union of the edges of the vaginal or sheathing portion of the leaf
(fig. 50). In grasses, by the enlargement of the embryo in a
particular direction, the endosperm is pushed on one side, and thus
the embryo comes to lie outside at the base of the endosperm (figs.
22, 51). The lamina of the cotyledon is not developed. Upon the side
of the embryo next the endosperm and enveloping it is a large
shield-shaped body, termed the _scutellum_. This is an outgrowth from
the base of the cotyledon, enveloping more or less the cotyledon and
plumule, in some cases, as in maize, completely investing it; in other
cases, as in rice, merely sending small prolongations over its
anterior face at the apex. By others this scutellum is considered as
the true cotyledon, and the sheathing structure covering the plumule
is regarded as a ligule or axillary stipule (see GRASSES). In many
aquatic monocotyledons (e.g. _Potamogeton_, _Ruppia_ and others) there
is a much-developed hypocotyl, which forms the greater part of the
embryo and acts as a store of nutriment in germination; these are
known as _macropodous_ embryos. A similar case is that of _Caryocar_
among Dicotyledons, where the swollen hypocotyl occupies most of the
embryo (fig. 52). In some grasses, as oats and rice, a projection of
cellular tissue is seen upon the side of the embryo opposite to the
scutellum, that is, on the anterior side. This has been termed the
_epiblast_. It is very large in rice. This by some was considered the
rudimentary second cotyledon; but is now generally regarded as an
outgrowth of the sheath of the true cotyledon. (A. B. R.)
FRUIT AND FLOWER FARMING. The different sorts of fruits and flowers are
dealt with in articles under their own headings, to which reference may
be made; and these give the substantial facts as to their cultivation.
See also the article HORTICULTURE.
TABLE I.--_Extent of Orchards in Great Britain in each Year, 1887 to
1901._
+------+---------++------+---------++------+---------+
| Year.| Acres. || Year.| Acres. || Year.| Acres. |
+------+---------++------+---------++------+---------+
| 1887 | 202,234 || 1892 | 208,950 || 1897 | 224,116 |
| 1888 | 199,178 || 1893 | 211,664 || 1898 | 226,059 |
| 1889 | 199,897 || 1894 | 214,187 || 1899 | 228,603 |
| 1890 | 202,305 || 1895 | 218,428 || 1900 | 232,129 |
| 1891 | 209,996 || 1896 | 221,254 || 1901 | 234,660 |
+------+---------++------+---------++------+---------+
TABLE II.--_Areas under Orchards in England, Wales and Scotland--Acres._
+------+----------+--------+---------+--------------+
| Year.| England. | Wales. |Scotland.|Great Britain.|
+------+----------+--------+---------+--------------+
| 1896 | 215,642 | 3677 | 1935 | 221,254 |
| 1897 | 218,261 | 3707 | 2148 | 224,116 |
| 1898 | 220,220 | 3690 | 2149 | 226,059 |
| 1899 | 222,712 | 3666 | 2225 | 228,603 |
| 1900 | 226,164 | 3695 | 2270 | 232,129 |
| 1901 | 228,580 | 3767 | 2313 | 234,660 |
| 1908 | 244,430 | 3577 | 2290 | 250,297 |
+------+----------+--------+---------+--------------+
GREAT BRITAIN
The extent of the fruit industry may be gathered from the figures for
the acreage of land under cultivation in orchards and small fruit
plantations. The Board of Agriculture returns concerning the orchard
areas of Great Britain showed a continuous expansion year by year from
199,178 acres in 1888 to 234,660 acres in 1901, as will be learnt from
Table I. There was, it is true, an exception in 1892, but the decline in
that year is explained by the circumstance that since 1891 the
agricultural returns have been collected only from holdings of more than
one acre, whereas they were previously obtained from all holdings of a
quarter of an acre or more. As there are many holdings of less than an
acre in extent upon which fruit is grown, and as fruit is largely raised
also in suburban and other gardens which do not come into the returns,
it may be taken for granted that the actual extent of land devoted to
fruit culture exceeds that which is indicated by the official figures.
In the Board of Agriculture returns up to June 1908, 308,000 acres are
stated to be devoted to fruit cultivation of all kinds in Great Britain.
Table II. shows that the expansion of the orchard area of Great Britain
is mainly confined to England, for it has slightly decreased in Wales
and Scotland. The acreage officially returned as under orchards is that
of arable or grass land which is also used for fruit trees of any kind.
Conditions of soil and climate determine the irregular distribution of
orchards in Great Britain. The dozen counties which possess the largest
extent of orchard land all lie in the south or west of the island.
According to the returns for 1908 (excluding small fruit areas) they
were the following:--
+----------+--------++-----------+--------++----------+------+
| County. | Acres. || County. | Acres. || County. |Acres.|
+----------+--------++-----------+--------++----------+------+
| Kent | 32,751 || Worcester | 23,653 || Salop | 4685 |
| Devon | 27,200 || Gloucester| 20,424 || Dorset | 4464 |
| Hereford | 28,316 || Cornwall | 5,415 || Monmouth | 3914 |
| Somerset | 25,279 || Middlesex | 5,300 || Wilts | 3630 |
+----------+--------++-----------+--------++----------+------+
Leaving out of consideration the county of Kent, which grows a greater
variety of fruit than any of the others, the counties of Devon,
Hereford, Somerset, Worcester and Gloucester have an aggregate orchard
area of 124,872 acres. These five counties of the west and south-west of
England--constituting in one continuous area what is essentially the
cider country of Great Britain--embrace therefore rather less than half
of the entire orchard area of the island, while Salop, Monmouth and
Wilts have about 300 less than they had a few years ago. Five English
counties have less than 1000 acres each of orchards, namely, the county
of London, and the northern counties of Cumberland, Westmorland,
Northumberland and Durham. Rutland has just over 100 acres. The largest
orchard areas in Wales are in the two counties adjoining
Hereford--Brecon with 1136 acres and Radnor with 727 acres; at the other
extreme is Anglesey, with a decreasing orchard area of only 22 acres. Of
the Scottish counties, Lanark takes the lead with 1285 acres, Perth,
Stirling and Haddington following with 684 and 129 acres respectively.
Ayr and Midlothian are the only other counties possessing 100 acres or
more of orchards, whilst Kincardine, Orkney and Shetland return no
orchard area, and Banff, Bute, Kinross, Nairn, Peebles, Sutherland and
Wigtown return less than 10 acres each. It may be added that in 1908
Jersey returned 1090 acres of orchards, Guernsey, &c., 144 acres, and
the Isle of Man, 121 acres; the two last-named places showing a decline
as compared with eight years previously.
Outside the cider counties proper of England, the counties in which
orchards for commercial fruit-growing have increased considerably in
recent years include Berks, Buckingham, Cambridge, Essex, Lincoln,
Middlesex, Monmouth, Norfolk, Oxford, Salop, Sussex, Warwick and Wilts.
Apples are the principal fruit grown in the western and south-western
counties, pears also being fairly common. In parts of Gloucestershire,
however, and in the Evesham and Pershore districts of Worcestershire,
plum orchards exist. Plums are almost as largely grown as apples in
Cambridgeshire. Large quantities of apples, plums, damsons, cherries,
and a fair quantity of pears are grown for the market in Kent, whilst
apples, plums and pears predominate in Middlesex. In many counties
damsons are cultivated around fruit plantations to shelter the latter
from the wind.
Of small fruit (currants, gooseberries, strawberries, raspberries, &c.)
no return was made of the acreage previous to 1888, in which year it was
given as 36,724 acres for Great Britain. In 1889 it rose to 41,933
acres.
Later figures are shown in Table III. It will be observed that, owing to
corrections made in the enumeration in 1897, a considerable reduction in
the area is recorded for that year, and presumably the error then
discovered existed in all the preceding returns. The returns for 1907
gave the acreage of small fruit as 82,175 acres, and in 1908 at 84,880
acres--an area more than double that of 1889.
TABLE III.--_Areas of Small Fruit in Great Britain._
+-------+--------+-------+--------+-------+--------+
| Year. | Acres. | Year. | Acres. | Year. | Acres. |
+-------+--------+-------+--------+-------+--------+
| 1890 | 46,234 | 1894 | 68,415 | 1898 | 69,753 |
| 1891 | 58,704 | 1895 | 74,547 | 1899 | 71,526 |
| 1892 | 62,148 | 1896 | 76,245 | 1900 | 73,780 |
| 1893 | 65,487 | 1897 | 69,792 | 1901 | 74,999 |
+-------+--------+-------+--------+-------+--------+
TABLE IV.--_Areas under Small Fruit in England, Wales and
Scotland--Acres._
+-------+----------+--------+-----------+----------------+
| Year. | England. | Wales. | Scotland. | Great Britain. |
+-------+----------+--------+-----------+----------------+
| 1898 | 63,438 | 1044 | 5271 | 69,753 |
| 1899 | 64,867 | 1106 | 5553 | 71,526 |
| 1900 | 66,749 | 1109 | 5922 | 73,780 |
| 1901 | 67,828 | 1092 | 6079 | 74,999 |
| 1908 | 75,750 | 1200 | 7930 | 84,880 |
+-------+----------+--------+-----------+----------------+
There has undoubtedly been a considerable expansion, rather than a
contraction, of small fruit plantations since 1896. The acreage of small
fruit in Great Britain is about one-third that of the orchards. As may
be seen in Table IV., it is mainly confined to England, though Scotland
has over 4000 more acres of small fruit than of orchards. About
one-third of the area of small fruit in England belongs to Kent alone,
that county having returned 24,137 acres in 1908. Cambridge now ranks
next with 6878 acres, followed by Norfolk with 5876 acres,
Worcestershire with 4852 acres, Middlesex with 4163 acres, Hants with
3320 acres and Essex with 2150 acres. It should be remarked that between
1900 and 1908 Cambridgeshire had almost doubled its area of small
fruits, from 3740 to 6878 acres; whilst both Norfolk and Worcestershire
in 1908 had larger areas devoted to small fruits than Middlesex--in
which county there had been a decrease of about 400 acres during the
same period. The largest county area of small fruit in Wales is 806
acres in Denbighshire, and in Scotland 2791 acres in Perthshire, 2259
acres in Lanarkshire, followed by 412 acres in Forfarshire. The only
counties in Great Britain which make no return under the head of small
fruit are Orkney and Shetland; and Sutherland only gives 2-1/2 acres. It
is hardly necessary to say that considerable areas of small fruit, in
kitchen gardens and elsewhere, find no place in the official returns,
which, however, include small fruit grown between and under orchard
trees.
Gooseberries are largely grown in most small fruit districts. Currants
are less widely cultivated, but the red currant is more extensively
grown than the black, the latter having suffered seriously from the
ravages of the black currant mite. Kent is the great centre for
raspberries and for strawberries, though, in addition, the latter fruit
is largely grown in Cambridgeshire (2411 acres), Hampshire (2327 acres),
Norfolk (2067 acres) and Worcestershire (1273 acres). Essex,
Lincolnshire, Cheshire, Cornwall and Middlesex each has more than 500
acres devoted to strawberry cultivation.
The following statement from returns for 1908 shows the area under
different kinds of fruit in 1907 and 1908 in Great Britain, and also
whether there had been an increase or decrease:
+-----------------+---------+---------+-------------+
| | 1907. | 1908. | Increase or |
| | | | Decrease. |
+-----------------+---------+---------+-------------+
| | Acres. | Acres. | Acres. |
| Small Fruit-- | | | |
| Strawberries | 27,827 | 28,815 | + 988 |
| Raspberries | 8,878 | 9,323 | + 445 |
| Currants and | | | |
| Gooseberries | 25,590 | 26,241 | + 651 |
| Other kinds | 19,880 | 20,501 | + 621 |
| +---------+---------+-------------+
| | 82,175 | 84,880 | +2705 |
| +---------+---------+-------------+
| Orchards-- | | | |
| Apples | 172,643 | 172,751 | + 108 |
| Pears | 8,911 | 9,604 | + 693 |
| Cherries | 12,027 | 11,868 | - 159 |
| Plums | 14,901 | 15,683 | + 782 |
| Other kinds | 41,694 | 40,391 | -1303 |
| +---------+---------+-------------+
| | 250,176 | 250,297 | + 121 |
+-----------------+---------+---------+-------------+
It appears from the Board of Agriculture returns that 27,433 acres of
small fruit was grown in orchards, so that the total extent of land
under fruit cultivation in Great Britain at the end of 1908 was about
308,000 acres.
There are no official returns as to the acreage devoted to orchard
cultivation in Ireland. The figures relating to small fruit, moreover,
extend back only to 1899, when the area under this head was returned as
4809 acres, which became 4359 acres in 1900 and 4877 acres in 1901. In
most parts of the country there are districts favourable to the culture
of small fruits, such as strawberries, raspberries, gooseberries and
currants, and of top fruits, such as apples, pears, plums and damsons.
The only localities largely identified with fruit culture as an industry
are the Drogheda district and the Armagh district. In the former all the
kinds named are grown except strawberries, the speciality being
raspberries, which are marketed in Dublin, Belfast and Liverpool. In the
Armagh district, again, all the kinds named are grown, but in this case
strawberries are the speciality, the markets utilized being Richhill,
Belfast, and those in Scotland. In the Drogheda district the grower
bears the cost of picking, packing and shipping, but he cannot estimate
his net returns until his fruit is on the market. Around Armagh the
Scottish system prevails--that is, the fruit is sold while growing, the
buyer being responsible for the picking and marketing.
The amount of fruit imported into the United Kingdom has such an
important bearing on the possibilities of the industry that the
following figures also may be useful:
The quantities of apples, pears, plums, cherries and grapes imported
in the raw condition into the United Kingdom in each year, 1892 to
1901, are shown in Table V. Previous to 1892 apples only were
separately enumerated. Up to 1899 inclusive the quantities were given
in bushels, but in 1900 a change was made to hundred-weights. This
renders the quantities in that and subsequent years not directly
comparable with those in earlier years, but the comparison of the
values, which are also given in the table, continues to hold good. The
figures for 1908 have been added to show the increase that had taken
place. In some years the value of imported apples exceeds the
aggregate value of the pears, plums, cherries and grapes imported. The
extreme values for apples shown in the table are L844,000 in 1893 and
L2,079,000 in 1908. Grapes rank next to apples in point of value, and
over the seventeen years the amount ranged between L394,000 in 1892
and L728,000 in 1908. On the average, the annual outlay on imported
pears is slightly in excess of that on plums. The extremes shown are
L167,000 in 1895 and L515,000 in 1908. In the case of plums, the
smallest outlay tabulated is L166,000 in 1895, whilst the largest is
L498,000 in 1897. The amounts expended upon imported cherries varied
between L96,000 in 1895 and L308,000 in 1900. In 1900 apricots and
peaches, imported raw, previously included with raw plums, were for
the first time separately enumerated, the import into the United
Kingdom for that year amounting to 13,689 cwt., valued at L25,846; in
1901 the quantity was 13,463 cwt. and the value L32,350. The latter
rose in 1908 to L60,000. In 1900, also, currants, gooseberries and
strawberries, hitherto included in unenumerated raw fruit, were
likewise for the first time separately returned. Of raw currants the
import was 64,462 cwt., valued at L87,170 (1908, L121,850); of raw
gooseberries 26,045 cwt., valued at L14,626 (1908, L25,520); and of
raw strawberries, 52,225 cwt., valued at L85,949. In 1907 only 44,000
cwt. of strawberries were imported. In 1901 the quantities and values
were respectively--currants, 70,402 cwt., L75,308; gooseberries,
21,735 cwt., L11,420; strawberries, 38,604 cwt., L51,290. Up to 1899
the imports of tomatoes were included amongst unenumerated raw
vegetables, so that the quantity was not separately ascertainable. For
1900 the import of tomatoes was 833,032 cwt., valued at L792,339,
which is equivalent to a fraction under 2-1/2d. per lb. For 1901 the
quantity was 793,991 cwt., and the value L734,051; for 1906, there
were 1,124,700 cwt., valued at L953,475; for 1907, 1,135,499 cwt.,
valued at L1,020,805; and for 1908, 1,160,283 cwt., valued at
L955,983.
TABLE V.--_Imports of Raw Apples, Pears, Plums, Cherries and Grapes
into the United Kingdom, 1892 to 1901. Quantities in Thousands of
Bushels (thousands of cwt. in 1900 and 1901). Values in Thousands of
Pounds Sterling._
+------+-------------------------------------------------+
| | Quantities. |
| Year.+---------+--------+--------+-----------+---------+
| | Apples. | Pears. | Plums. | Cherries. | Grapes. |
+------+---------+--------+--------+-----------+---------+
| 1892 | 4515 | 637 | 413 | 217 | 762 |
| 1893 | 3460 | 915 | 777 | 346 | 979 |
| 1894 | 4969 | 1310 | 777 | 311 | 833 |
| 1895 | 3292 | 407 | 401 | 196 | 865 |
| 1896 | 6177 | 483 | 560 | 219 | 883 |
| 1897 | 4200 | 1052 | 1044 | 312 | 994 |
| 1898 | 3459 | 492 | 922 | 402 | 1136 |
| 1899 | 3861 | 572 | 558 | 281 | 1158 |
| 1900 | 2129* | 477* | 423* | 243* | 593* |
| 1901 | 1830* | 349* | 264* | 213* | 680* |
+------+---------+--------+--------+-----------+---------+
| Values. |
+------+---------+--------+--------+-----------+---------+
| 1892 | 1354 | 297 | 200 | 135 | 394 |
| 1893 | 844 | 347 | 332 | 195 | 530 |
| 1894 | 1389 | 411 | 302 | 167 | 470 |
| 1895 | 960 | 167 | 166 | 96 | 487 |
| 1896 | 1582 | 207 | 242 | 106 | 443 |
| 1897 | 1187 | 378 | 498 | 178 | 495 |
| 1898 | 1108 | 222 | 435 | 231 | 550 |
| 1899 | 1186 | 266 | 294 | 154 | 588 |
| 1900 | 1225 | 367 | 393 | 308 | 595 |
| 1901 | 1183 | 296 | 244 | 214 | 695 |
| 1908 | 2079 | 515 | 428 | 235 | 728 |
+------+---------+--------+--------+-----------+---------+
* Thousands of cwts.
In 1908 the outlay of the United Kingdom upon imported raw fruits,
such as can easily be produced at home, was L4,195,654, made up as
follows:
Apples L2,079,703 | Plums L428,966
Grapes 728,026 | Currants 121,852
Pears 515,914 | Apricots and peaches 60,141
Cherries 235,523 | Gooseberries 25,529
In addition about L280,000 was spent upon "unenumerated" raw fruit,
and L560,000 on nuts other than almonds "used as fruit," which would
include walnuts and filberts, both produced at home. It is certain,
therefore, that the expenditure on imported fruits, such as are grown
within the limits of the United Kingdom, exceeds four millions
sterling per annum. The remainder of the outlay on imported fruit in
1908, amounting to over L5,000,000, was made up of L2,269,651 for
oranges, L471,713 for lemons, L1,769,249 for bananas, and L560,301 for
almond-nuts; these cannot be grown on an industrial scale in the
British Isles.
It may be interesting to note the source of some of these imported
fruits. The United States and Canada send most of the apples, the
quantity for 1907 being 1,413,000 cwt. and 1,588,000 cwt.
respectively, while Australia contributes 280,000 cwt. Plums come
chiefly from France (200,000 cwt.), followed with 38,000 cwt. from
Germany and 28,000 cwt. from the Netherlands. Pears are imported
chiefly from France (204,000 cwt.) and Belgium (176,000); but the
Netherlands send 52,000 cwt., and the United States 24,000 cwt. The
great bulk of imported tomatoes comes from the Canary Islands, the
quantity in 1907 being 604,692 cwt. The Channel Islands also sent
223,800 cwt., France 115,500 cwt., Spain 169,000 cwt., and Portugal a
long way behind with 11,700 cwt. Most of the strawberries imported
come from France (33,800 cwt.) and the Netherlands (10,300 cwt.).
_Fruit-growing in Kent._--Kent is by far the largest fruit-growing
county in England. For centuries that county has been famous for its
fruit, and appears to have been the centre for the distribution of trees
and grafts throughout the country. The cultivation of fruit land upon
farms in many parts of Kent has always been an important feature in its
agriculture. An excellent description of this noteworthy characteristic
of Kentish farming is contained in a comprehensive paper on the
agriculture of Kent by Mr Charles Whitehead,[1] whose remarks, with
various additions and modifications, are here reproduced.
Where the conditions are favourable, especially in East and Mid Kent,
there is a considerable acreage of fruit land attached to each farm,
planted with cherry, apple, pear, plum and damson trees, and with bush
fruits, or soft fruits as they are sometimes called, including
gooseberries, currants, raspberries, either with or without standard
trees, and strawberries, and filberts and cob-nuts in Mid Kent. This
acreage has largely increased, and will no doubt continue to increase,
as, on the whole, fruit-growing has been profitable and has materially
benefited those fortunate enough to have fruit land on their farms.
There are also cultivators who grow nothing but fruit. These are
principally in the district of East Kent, between Rochester and
Canterbury, and in the district of Mid Kent near London, and they
manage their fruit land, as a rule, better than farmers, as they give
their undivided attention to it and have more technical knowledge. But
there has been great improvement of late in the management of fruit
land, especially of cherry and apple orchards, the grass of which is
fed off by animals having corn or cake, or the land is well manured.
Apple trees are grease-banded and sprayed systematically by advanced
fruit-growers to prevent or check the attacks of destructive insects.
Far more attention is being paid to the selection of varieties of
apples and pears having colour, size, flavour, keeping qualities, and
other attributes to meet the tastes of the public, and to compete with
the beautiful fruit that comes from the United States and Canada.
Of the various kinds of apples at present grown in Kent mention should
be made of Mr Gladstone, Beauty of Bath, Devonshire Quarrenden, Lady
Sudely, Yellow Ingestre and Worcester Pearmain. These are dessert
apples ready to pick in August and September, and are not stored. For
storing, King of the Pippins, Cox's Orange Pippin (the best dessert
apple in existence), Cox's Pomona, Duchess, Favourite, Gascoyne's
Scarlet Seedling, Court Pendu Plat, Baumann's Red Reinette, Allington
Pippin, Duke of Devonshire and Blenheim Orange. Among kitchen apples
for selling straight from the trees the most usually planted are Lord
Grosvenor, Lord Suffield, Keswick Codlin, Early Julian, Eclinville
Seedling, Pott's Seedling, Early Rivers, Grenadier, Golden Spire,
Stirling Castle and Domino. For storing, the cooking sorts favoured
now are Stone's or Loddington, Warner's King, Wellington, Lord Derby,
Queen Caroline, Tower of Glamis, Winter Queening, Lucombe's Seedling,
Bismarck, Bramley's Seedling, Golden Noble and Lane's Prince Albert.
Almost all these will flourish equally as standards, pyramids and
bushes. Among pears are Hessle, Clapp's Favourite, William's Bon
Chretien, Beurre de Capiaumont, Fertility, Beurre Riche, Chissel,
Beurre Clairgeau, Louise Bonne of Jersey, Doyenne du Comice and Vicar
of Winkfield. Among plums, Rivers's Early Prolific, Tsar, Belgian
Purple, Black Diamond, Kentish Bush Plum, Pond's Seedling, Magnum
Bonum and Victoria are mainly cultivated. The damson known as Farleigh
Prolific, or Crittenden's, is most extensively grown throughout the
county, and usually yields large crops, which make good prices. As a
case in point, purchasers were offering to contract for quantities of
this damson at L20 per ton in May of 1899, as the prospects of the
yield were unsatisfactory. On the other hand, in one year recently
when the crop was abnormally abundant, some of the fruit barely paid
the expenses of sending to market. The varieties of cherries most
frequently grown are Governor Wood, Knight's Early Black, Frogmore
Blackheart, Black Eagle, Waterloo, Amberheart, Bigarreau, Napoleon
Bigarreau and Turk. A variety of cherry known as the Kentish cherry,
of a light red colour and fine subacid flavour, is much grown in Kent
for drying and cooking purposes. Another cherry, similar in colour and
quality, which comes rather late, known as the Flemish, is also
extensively cultivated, as well as the very dark red large Morello,
used for making cherry brandy. These three varieties are grown
extensively as pyramids, and the last-named also on walls and sides of
buildings. Sometimes the cherry crop is sold by auction to dealers,
who pick, pack and consign the fruit to market. Large prices are often
made, as much as L80 per acre being not uncommon. The crop on a large
cherry orchard in Mid Kent has been sold for more than L100 per acre.
Where old standard trees have been long neglected and have become
overgrown by mosses and lichens, the attempts made to improve them
seldom succeed. The introduction of bush fruit trees dwarfed by
grafting on the Paradise stock has been of much advantage to fruit
cultivators, as they come into bearing in two or three years, and are
more easily cultivated, pruned, sprayed and picked than standards.
Many plantations of these bush trees have been formed in Kent of
apples, pears and plums. Half standards and pyramids have also been
planted of these fruits, as well as of cherries. Bushes of
gooseberries and currants, and clumps or stools of raspberry canes,
have been planted to a great extent in many parts of the East and Mid
divisions of Kent, but not much in the Weald, where apples are
principally grown. Sometimes fruit bushes are put in alternate rows
with bush of standard trees of apple, pear, plum or damson, or they
are planted by themselves. The distances apart for planting are
generally for cherry and apple trees on grass 30 ft. by 30 ft.; for
standard apples and pear trees from 20 ft. to 24 ft. upon arable land,
with bush fruit, as gooseberries and currants, under them. These are
set 6 ft. by 6 ft. apart, and 5 ft. by 2 ft. for raspberries, and
strawberries 2 ft. 6 in. to 3 ft. by 1 ft. 6 in. to 1 ft. 3 in. apart.
On some fruit farms bush or dwarf trees--apples, pears, plums--are
planted alone, at distances varying from 8 ft. to 10 ft. apart, giving
from 485 to 680 bush trees per acre, nothing being grown between them
except perhaps strawberries or vegetables during the first two or
three years. It is believed that this is the best way of ensuring
fruit of high quality and colour. Another arrangement consists in
putting standard apple or pear trees 30 ft. apart (48 trees per acre),
and setting bush trees of apples or pears 15 ft. apart between them;
these latter come quickly into bearing, and are removed when the
standards are fully grown. Occasionally gooseberry or currant bushes,
or raspberry canes or strawberry plants, are set between the bush
trees, and taken away directly they interfere with the growth of
these. Half standard apple or plum trees are set triangularly 15 ft.
apart, and strawberry plants at a distance of 1-1/2 ft. from plant to
plant and 2-1/2 ft. from row to row. Or currant or gooseberry bushes
are set between the half standards, and strawberry plants between
these.
These systems involve high farming. The manures used are London
manure, where hops are not grown, and bone meal, super-phosphate,
rags, shoddy, wool-waste, fish refuse, nitrate of soda, kainit and
sulphate of ammonia. Where hops are grown the London manure is wanted
for them. Fruit plantations are always dug by hand with the Kent spud.
Fruit land is never ploughed, as in the United States and Canada. The
soil is levelled down with the "Canterbury" hoe, and then the
plantations are kept free from weeds with the ordinary draw or "plate"
hoe. The best fruit farmers spray fruit trees regularly in the early
spring, and continue until the blossoms come out, with quassia and
soft soap and paraffin emulsions, and a very few with Paris green
only, where there is no under fruit, in order to prevent and check the
constant attacks of the various caterpillars and other insect pests.
This is a costly and laborious process, but it pays well, as a rule.
The fallacy that fruit trees on grass land require no manure, and that
the grass may be allowed to grow up to their trunks without any harm,
is exploding, and many fruit farmers are well manuring their grass
orchards and removing the grass for some distance round the stems,
particularly where the trees are young.
Strawberries are produced in enormous quantities in the northern part
of the Mid Kent district round the Crays, and from thence to
Orpington; also near Sandwich, and to some extent near Maidstone.
Raspberry canes have been extensively put in during the last few
years, and in some seasons yield good profits. There is a very great
and growing demand for all soft fruits for jam-making, and prices are
fairly good, taking an average of years, notwithstanding the heavy
importations from France, Belgium, Holland, Spain and Italy. The
extraordinary increase in the national demand for jam and other fruit
preserves has been of great benefit to Kent fruit producers. The
cheapness of duty-free sugar, as compared with sugar paying duty in
the United States and other large fruit-producing countries, afforded
one of the very few advantages possessed by British cultivators, but
the reimposition of the sugar duty in the United Kingdom in 1901 has
modified the position in this respect. Jam factories were established
in several parts of Kent about 1889 or 1890, but most of them
collapsed either from want of capital or from bad management. There
are still a few remaining, principally in connexion with large fruit
farms. One of these is at Swanley, whose energetic owners farm nearly
2000 acres of fruit land in Kent. The fruit grown by them that will
not make satisfactory prices in a fresh raw state is made into jam, or
if time presses it is first made into pulp, and kept until the
opportunity comes for making it into jam. In this factory there are
fifteen steam-jacketed vats in one row, and six others for candied
peel. A season's output on a recent occasion comprised about 3500 tons
of jam, 850 tons of candied peel and 750 gross (108,000 bottles) of
bottled fruit. A great deal of the fruit preserved is purchased,
whilst much of that grown on the farms is sold. A strigging machine is
employed, which does as much work as fifty women in taking currants
off their strigs or stalks. Black currant pulp is stored in casks till
winter, when there is time to convert it into jam. Strawberries cannot
be pulped to advantage, but it is otherwise with raspberries, the pulp
of which is largely made. Apricots for jam are obtained chiefly from
France and Spain. There is another flourishing factory near
Sittingbourne worked on the same lines. It is very advantageous to
fruit farmers to have jam factories in connexion with their farms or
to have them near, as they can thoroughly grade their fruit, and send
only the best to market, thus ensuring a high reputation for its
quality. Carriage is saved, which is a serious charge, though railway
rates from Kent to the great manufacturing towns and to Scotland are
very much less proportionally than those to London, and consequently
Kent growers send increasing quantities to these distant markets,
where prices are better, not being so directly interfered with by
imported fruit, which generally finds its way to London.
Kentish fruit-growers are becoming more particular in picking,
grading, packing and storing fruit, as well as in marketing it. A
larger quantity of fruit is now carefully stored, and sent to selected
markets as it ripens, or when there is an ascertained demand, as it is
found that if it is consigned to market direct from the trees there
must frequently be forced sales and competition with foreign fruit
that is fully matured and in good order. It was customary formerly for
Kentish growers to consign all their fruit to the London markets; now
a good deal of it is sent to Manchester, Birmingham, Liverpool,
Sheffield, Newcastle and other large cities. Some is sent even to
Edinburgh and Glasgow. Many large growers send no fruit to London now.
It is by no means uncommon for growers to sell their fruit crops on
the trees or bushes by auction or private treaty, or to contract to
supply a stipulated quantity of specified fruit, say of currants,
raspberries or strawberries, to jam manufacturers. There is a
considerable quantity of fruit, such as grapes, peaches, nectarines,
grown under glass, and this kind of culture tends to increase.
Filberts and cob-nuts are a special product of Kent, in the
neighbourhood of Maidstone principally, and upon the Ragstone soils,
certain conditions of soil and situation being essential for their
profitable production. A part of the filbert and cob-nut crop is
picked green in September, as they do well for dessert, though their
kernels are not large or firm, and it pays to sell them green, as they
weigh more heavily. One grower in Mid Kent has 100 acres of nuts, and
has grown 100 tons in a good year. The average price of late years has
been about 5d. per lb., which would make the gross return of the 100
acres amount to L4660. Kentish filberts have long been proverbial for
their excellence. Cobs are larger and look better for dessert, though
their flavour is not so fine. They are better croppers, and are now
usually planted. This cultivation is not much extending, as it is very
long before the trees come into full bearing. The London market is
supplied entirely with these nuts from Kent, and there is some demand
in America for them. Filbert and cob trees are most closely pruned.
All the year's growth is cut away except the very finest young wood,
which the trained eye of the tree-cutter sees at a glance is
blossom-bearing. The trees are kept from 5-1/2 to 7 ft. high upon
stems from 1-1/2 to 2 ft. high, and are trained so as to form a cup of
from 7 to 8 ft. in diameter.
There seems no reason to expect any decrease in the acreage of fruit
land in Kent, and if the improvement in the selection of varieties and
in the general management continues it will yet pay. A hundred years
ago every one was grubbing fruit land in order that hops might be
planted, and for this many acres of splendid cherry orchards were
sacrificed. Now the disposition is to grub hop plants and substitute
apples, plums, or small fruit or cherry trees.
_Fruit-growing in other Districts._--The large fruit plantations in
the vicinity of London are to be found mostly in the valley of the
Thames, around such centres as Brentford, Isleworth, Twickenham,
Heston, Hounslow, Cranford and Southall. All varieties of orchard
trees, but mostly apples, pears, and plums and small fruit, are grown
in these districts, the nearness of which to the metropolitan fruit
market at Covent Garden is of course an advantage. Some of the
orchards are old, and are not managed on modern principles. They
contain, moreover, varieties of fruit many of which are out of date
and would not be employed in establishing new plantations. In the
better-managed grounds the antiquated varieties have been removed, and
their places taken by newer and more approved types. In addition to
apples, pears, plums, damsons, cherries and quinces as top fruit,
currants, gooseberries and raspberries are grown as bottom fruit.
Strawberries are extensively grown in some of the localities, and in
favourable seasons outdoor tomatoes are ripened and marketed.
Fruit is extensively grown in Cambridgeshire and adjacent counties in
the east of England. A leading centre is Cottenham, where the Lower
Greensand crops out and furnishes one of the best of soils for
fruit-culture. In Cottenham about a thousand acres are devoted to
fruit, and nearly the same acreage to asparagus, which is, however,
giving place to fruit. Currants, gooseberries and strawberries are the
most largely grown, apples, plums and raspberries following. Of
varieties of plums the Victoria is first in favour, and then Rivers's
Early Prolific, Tsar and Gisborne. London is the chief market, as it
receives about half the fruit sent away, whilst a considerable
quantity goes to Manchester, and some is sent to a neighbouring jam
factory at Histon, where also a moderate acreage of fruit is grown.
Another fruit-growing centre in Cambridgeshire is at Willingham,
where--besides plums, gooseberries and raspberries--outdoor tomatoes
are a feature. Greengages are largely grown near Cambridge. Wisbech is
the centre of an extensive fruit district, situated partly in
Cambridgeshire and partly in Norfolk. Gooseberries, strawberries and
raspberries are largely grown, and as many as 80 tons of the
first-named fruit have been sent away from Wisbech station in a single
day. In the fruit-growing localities of Huntingdonshire apples, plums
and gooseberries are the most extensively grown, but pears,
greengages, cherries, currants, strawberries and raspberries are also
cultivated. As illustrating variations in price, it may be mentioned
that about the year 1880 the lowest price for gooseberries was L10 per
ton, whereas it has since been down to L4. Huntingdonshire fruit is
sent chiefly to Yorkshire, Scotland and South Wales, but railway
freights are high.
Essex affords a good example of successful fruit-farming at Tiptree
Heath, near Kelvedon, where under one management about 260 acres out
of a total of 360 are under fruit. The soil, a stiff loam, grows
strawberries to perfection, and 165 acres are allotted to this fruit.
The other principal crops are 43 acres of raspberries and 30 acres of
black currants, besides which there are small areas of red currants,
gooseberries, plums, damsons, greengages, cherries, apples, quinces
and blackberries. The variety of strawberry known as the Small Scarlet
is a speciality here, and it occupies 55 acres, as it makes the best
of jam. The Paxton, Royal Sovereign and Noble varieties are also
grown. Strawberries stand for six or seven years on this farm, and
begin to yield well when two years old. A jam factory is worked in
conjunction with the fruit farm. Pulp is not made except when there is
a glut of fruit. Perishable fruit intended for whole-fruit preserves
is never held over after it is gathered. The picking of strawberries
begins at 4 A.M., and the first lot is made into jam by 6 A.M.
Hampshire, like Cambridgeshire and Norfolk, are the only counties in
which the area of small fruit exceeds that of orchards. The returns
for 1908 show that Hampshire had 3320 acres of small fruit to 2236
acres of orchards; Cambridge had 6878 acres of small fruit to 5221 of
orchards; and Norfolk had 5876 acres of small fruit against 5188 acres
of orchards. Compared with twenty years previously, the acreage of
small fruit had trebled. This is largely due in Hampshire to the
extension of strawberry culture in the Southampton district, where the
industry is in the hands of many small growers, few of whom cultivate
more than 20 acres each. Sarisbury and Botley are the leading parishes
in which the business is carried on. Most of the strawberry holdings
are from half an acre to 5 acres in extent, a few are from 5 to 10
acres, fewer still from 10 to 20 acres and only half-a-dozen over that
limit. Runners from one-year plants are used for planting, being found
more fruitful than those from older plants. Peat-moss manure from
London stables is much used, but artificial manures are also employed
with good results. Shortly after flowering the plants are bedded down
with straw at the rate of about 25 cwt. per acre. Picking begins some
ten days earlier than in Kent, at a date between 1st June and 15th
June. The first week's gathering is sent mostly to London, but
subsequently the greater part of the fruit goes to the Midlands and to
Scotland and Ireland.
In recent years fruit-growing has much increased in South
Worcestershire, in the vicinity of Evesham and Pershore. Hand-lights
are freely used in the market gardens of this district for the
protection of cucumbers and vegetable marrows, besides which tomatoes
are extensively grown out of doors. At one time the egg plum and the
Worcester damson were the chief fruit crops, apples and cherries
ranking next, pears being grown to only a moderate extent. According
to the 1908 returns, however, apples come first, plums second, pears
third and cherries fourth. In a prolific season a single tree of the
Damascene or Worcester damson will yield from 400 to 500 lb. of
fruit. There is a tendency to grow plum trees in the bush shape, as
they are less liable than standards to injury from wind. The manures
used include soot, fish guano, blood manure and phosphates--basic slag
amongst the last-named. In the Pershore district, where there is a jam
factory, plums are the chief tree fruit, whilst most of the orchard
apples and pears are grown for cider and perry. Gooseberries are a
feature, as are also strawberries, red and black currants and a few
white, but raspberries are little grown. The soil, a strong or medium
loam of fair depth, resting on clay, is so well adapted to plums that
trees live for fifty years. In order to check the ravages of the
winter moth, plum and apple trees are grease-banded at the beginning
of October and again at the end of March. The trees are also sprayed
when necessary with insecticidal solutions. Pruning is done in the
autumn. An approved distance apart at which to grow plum trees is 12
ft. by 12 ft. In the Earl of Coventry's fruit plantation, 40 acres in
extent, at Croome Court, plums and apples are planted alternately, the
bottom fruit being black currants, which are less liable to injury
from birds than are red currants or gooseberries. Details concerning
the methods of cultivation of fruit and flowers in various parts of
England, the varieties commonly grown, the expenditure involved, and
allied matters, will be found in Mr W.E. Bear's papers in the _Journal
of the Royal Agricultural Society_ in 1898 and 1899.
Apart altogether from market gardening and commercial fruit-growing,
it must be borne in mind that an enormous business is done in the
raising of young fruit-trees every year. Hundreds of thousands of
apples, pears, plums, cherries, peaches, nectarines and apricots are
budded or grafted each year on suitable stocks. They are trained in
various ways, and are usually fit for sale the third year. These young
trees replace old ones in private and commercial gardens, and are also
used to establish new plantations in different parts of the kingdom.
_The Woburn Experimental Fruit Farm._--The establishment in 1894 of
the experimental fruit farm at Ridgmont, near Woburn, Beds, has
exercised a healthy influence upon the progress and development of
fruit-farming in England. The farm was founded and carried on by the
public-spirited enterprise of the Duke of Bedford and Mr Spencer U.
Pickering, the latter acting as director. The main object of the
experimental station was "to ascertain facts relative to the culture
of fruit, and to increase our knowledge of, and to improve our
practice in, this industry." The farm is 20 acres in extent, and
occupies a field which up to June 1894 had been used as arable land
for the ordinary rotation of farm crops. The soil is a sandy loam 9 or
10 in. deep, resting on a bed of Oxford Clay. Although it contains a
large proportion of sand, the land would generally be termed very
heavy, and the water often used to stand on it in places for weeks
together in a wet season. The tillage to which the ground was
subjected for the purposes of the fruit farm much improved its
character, and in dry weather it presents as good a tilth as could be
desired. Chemical analyses of the soil from different parts of the
field show such wide differences that it is admitted to be by no means
an ideal one for experimental purposes. Without entering upon further
details, it may be useful to give a summary of the chief results
obtained.
Apples have been grown and treated in a variety of ways, but of the
different methods of treatment careless planting, coupled with
subsequent neglect, has given the most adverse results, the crop of
fruit being not 5% of that from trees grown normally. Of the separate
deleterious items constituting total neglect, by far the most
effective was the growth of weeds on the surface; careless planting,
absence of manure, and the omission of trenching all had comparatively
little influence on the results. A set of trees that had been
carelessly planted and neglected, but subsequently tended in the early
part of 1896, were in the autumn of that year only 10% behind their
normally-treated neighbours, thus demonstrating that the response to
proper attention is prompt. The growth of grass around young apple
trees produced a very striking effect, the injury being much greater
than that due to weeds. It is possible, however, that in wet years the
ill-effects of both grass and weeds would be less than in dry seasons.
Nevertheless, the grass-grown trees, after five years, were scarcely
bigger than when planted, and the actual increase in weight which they
showed during that time was about eighteen times smaller than in the
case of similar trees in tilled ground. It is believed that one of the
main causes of the ill-effects is the large increase in the
evaporation of water from the soil which is known to be produced by
grass, the trees being thereby made to suffer from drought, with
constant deprivation of other nourishment as well. That grass growing
round young apple trees is deleterious was a circumstance known to
many horticulturists, but the extent to which it interferes with the
development of the trees had never before been realized. Thousands of
pounds are annually thrown away in England through want of knowledge
of this fact. Yet trees will flourish in grass under certain
conditions. Whether the dominant factor is the age (or size) of the
tree has been investigated by grassing over trees which have hitherto
been in the open ground, and the results appear to indicate that the
grass is as deleterious to the older trees as it was to the younger
ones. Again, it appears to have been demonstrated that young apple
trees, at all events in certain soils, require but little or no manure
in the early stages of their existence, so that in this case also
large sums must be annually wasted upon manurial dressings which
produce no effects. The experiments have dealt with dwarf trees of
Bramley, Cox and Potts, six trees of each variety constituting one
investigation. Some of the experiments were repeated with Stirling
Castle, and others with standard trees of Bramley, Cox and Lane's
Prince Albert. All were planted in 1894-1895, the dwarfs being then
three years old and the standards four. In each experiment the
"normal" treatment is altered in some one particular, this normal
treatment consisting of planting the trees carefully in trenched
ground, and subsequently keeping the surface clean; cutting back after
planting, pruning moderately in autumn, and shortening the growths
when it appeared necessary in summer; giving in autumn a dressing of
mixed mineral manures, and in February one of nitrate of soda, this
dressing being probably equivalent to one of 12 tons of dung per acre.
In the experiments on branch treatment, the bad effects of omitting to
cut the trees back on planting, or to prune them subsequently, is
evident chiefly in the straggling and bad shape of the resulting
trees, but such trees also are not so vigorous as they should be. The
quantity of fruit borne, however, is in excess of the average. The
check on the vigour and growth of a tree by cutting or injuring its
roots is in marked contrast with the effects of a similar interference
with the branches. Trees which had been root-pruned each year were in
1898 little more than half as big as the normal trees, whilst those
root-pruned every second year were about two-thirds as big as the
normal. The crops borne by these trees were nevertheless heavy in
proportion to the size of the trees. Such frequent root-pruning is
not, of course, a practice which should be adopted. It was found that
trees which had been carefully lifted every other year and replanted
at once experienced no ill-effects from the operation; but in a case
where the trees after being lifted had been left in a shed for three
days before replanting--which would reproduce to a certain extent the
conditions experienced when trees are sent out from a
nursery--material injury was suffered, these trees after four years
being 28% smaller than similar ones which had not been replanted. Sets
of trees planted respectively in November, January and March have, on
the whole, shown nothing in favour of any of these different times for
planting purposes. Some doubt is thrown on the accepted view that
there is a tendency, at any rate with young apple and pear trees, to
fruit in alternate seasons.
Strawberries of eighty-five different varieties have been experimented
with, each variety being represented in 1900 by plants of five
different ages, from one to five years. In 1896 and 1898 the crops of
fruit were about twice as heavy as in 1897 and 1899, but it has not
been found possible to correlate these variations with the
meteorological records of the several seasons. Taking the average of
all the varieties, the relative weights of crop per plant, when these
are compared with the two-year-old plants in the same season, are, for
the five ages of one to five years, 31, 100, 122, 121 and 134,
apparently showing that the bearing power increases rapidly up to two
years, less rapidly up to three years, after which age it remains
practically constant. The relative average size of the berries shows a
deterioration with the age of the plant. The comparative sizes from
plants of one to five years old were 115, 100, 96, 91 and 82
respectively. If the money value of the crop is taken to be directly
dependent on its total weight, and also on the size of the fruits, the
relative values of the crop for the different ages would be 34, 100,
117, 111 and 110, so that, on the Ridgmont ground, strawberry plants
could be profitably retained up to five years and probably longer. As
regards what may be termed the order of merit of different varieties
of strawberries, it appears that even small differences in position
and treatment cause large variations, not only in the features of the
crop generally, but also in the relative behaviour of the different
varieties. The relative cropping power of the varieties under
apparently similar conditions may often be expressed by a number five
or tenfold as great in one case as in the other. A comparison of the
relative behaviour of the same varieties in different seasons is
attended by similar variations. The varying sensitiveness of different
varieties of strawberry plants to small and undefinable differences in
circumstances is indeed one of the most important facts brought to
light in the experiments.
_Fruit Culture in Ireland._--The following figures have been kindly
supplied by the Irish Board of Agriculture, and deal with the acreage
under fruit culture in Ireland up to the end of the year 1907.
1. _Orchard Fruit_-- Statute Acres.
Apples 5829
Pears 224
Plums 223
Damsons 138
Other kinds 129
----
Total 6543
2. _Small Fruit_--
Currants, black 234
Currants, red and white 159
Gooseberries 675
Raspberries 374
Strawberries 994
Mixed fruit 2470
----
Total 4906
It therefore appears that while Ireland grows only about
one-thirty-third the quantity of apples that England does, it is
nevertheless nearly 5000 acres ahead of Scotland and about 2000 acres
ahead of Wales. It grows 41 times fewer pears than England, but still
is ahead of Scotland and a long way ahead of Wales in this fruit.
There are 70 times fewer plums grown in Ireland than in England, and
about the same in Scotland, while Wales does very little indeed. In
small fruit Ireland is a long way behind Scotland in the culture of
strawberries and raspberries, although with currants and gooseberries
it is very close. Considering the climate, and the fact that there
are, according to the latest available returns, over 62,000 holdings
above 1 acre but not exceeding 5 acres (having a total of 224,000
acres), it is possible fruit culture may become more prevalent than it
has been in the past.
_The Flower-growing Industry._--During the last two or three decades of
the 19th century a very marked increase in flower production occurred in
England. Notably was this the case in the neighbourhood of London,
where, within a radius of 15 or 20 m., the fruit crops, which had
largely taken the place of garden vegetables, were themselves ousted in
turn to satisfy the increasing demand for land for flower cultivation.
No flower has entered more largely into the development of the industry
than the narcissus or daffodil, of which there are now some 600
varieties. Comparatively few of these, however, are grown for market
purposes, although all are charming from the amateur point of view. On
some flower farms a dozen or more acres are devoted to narcissi alone,
the production of bulbs for sale as well as of flowers for market being
the object of the growers.
In the London district the country in the Thames valley west of the
metropolis is as largely occupied by flower farms as it is by fruit
farms--in fact, the cultivation of flowers is commonly associated with
that of fruit. In the vicinity of Richmond narcissi are extensively
grown, as they also are more to the west in the Long Ditton district,
and likewise around Twickenham, Isleworth, Hounslow, Feltham and
Hampton. Roses come more into evidence in the neighbourhood of Hounslow,
Cranford, Hillingdon and Uxbridge, and in some gardens daffodils and
roses occupy alternate rows. In this district also such flowers as
herbaceous paeonies, Spanish irises, German irises, Christmas roses,
lilies of the valley, chrysanthemums, foxgloves, hollyhocks,
wallflowers, carnations, &c., are extensively grown in many market
gardens. South of London is the Mitcham country, long noted for its
production of lavender. The incessant growth of the lavender plant upon
the same land, however, has led to the decline of this industry, which
has been largely transferred to districts in the counties of Bedford,
Essex and Hertford. At Mitcham, nevertheless, mixed flowers are very
largely grown for the supply of the metropolis, and one farm alone has
nearly 100 acres under flowers and glass-houses. Chrysanthemums, asters,
Iceland poppies, gaillardias, pansies, bedding calceolarias, zonal
pelargoniums and other plants are cultivated in immense quantities. At
Swanley and Eynsford, in Kent, flowers are extensively cultivated in
association with fruit and vegetables. Narcissi, chrysanthemums,
violets, carnations, campanulas, roses, pansies, irises, sweet peas, and
many other flowers are here raised, and disposed of in the form both of
cut flowers and of plants.
The Scilly Isles are important as providing the main source of supply of
narcissi to the English markets in the early months of the year. This
trade arose almost by accident, for it was about the year 1865 that a
box of narcissi sent to Covent Garden Market, London, realized L1; and
the knowledge of this fact getting abroad, the farmers of the isles
began collecting wild bulbs from the fields in order to cultivate them
and increase their stocks. Some ten years, however, elapsed before the
industry promised to become remunerative. In 1885 a Bulb and Flower
Association was established to promote the industrial growth of flowers.
The exports of flowers in that year reached 65 tons, and they steadily
increased until 1893, when they amounted to 450 tons. A slight decline
followed, but in 1896 the quantity exported was no less than 514 tons.
This would represent upwards of 3-1/2 million bunches of flowers,
chiefly narcissi and anemones. Rather more than 500 acres are devoted to
flower-growing in the isles, by far the greater part of this area being
assigned to narcissi, whilst anemones, gladioli, marguerites, arum
lilies, Spanish irises, pinks and wallflowers are cultivated on a much
smaller scale. The great advantage enjoyed by the Scilly flower-growers
is earliness of production, due to climatic causes; the soil, moreover,
is well suited to flower culture and there is an abundance of sunshine.
The long journey to London is somewhat of a drawback, in regard to both
time and freight, but the earliness of the flowers more than compensates
for this. Open-air narcissi are usually ready at the beginning of
January, and the supply is maintained in different varieties up to the
middle or end of May. The narcissus bulbs are usually planted in
October, 4 in. by 3 in. apart for the smaller sorts and 6 in. by 4 to 6
in. for the larger. A compost of farmyard manure, seaweed, earth and
road scrapings is the usual dressing, but nitrate of soda, guano and
bones are also occasionally employed. A better plan, perhaps, is to
manure heavily the previous crop, frequently potatoes, no direct
manuring then being needed for the bulbs, these not being left in the
ground more than two or three years. The expenses of cultivation are
heavy, the cost of bulbs alone--of which it requires nearly a quarter of
a million of the smaller varieties, or half as many of the largest, to
plant an acre--being considerable. The polyanthus varieties of narcissus
are likely to continue the most remunerative to the flower-growers of
Scilly, as they flourish better in these isles than on the mainland.
In the district around the Wash, in the vicinity of such towns as
Wisbech, Spalding and Boston, the industrial culture of bulbs and
flowers underwent great expansion in the period between 1880 and 1909.
At Wisbech one concern alone has a farm of some 900 acres, devoted
chiefly to flowers and fruit, the soil being a deep fine alluvium. Roses
are grown here, one field containing upwards of 100,000 trees. Nearly 20
acres are devoted to narcissi, which are grown for the bulbs and also,
together with tulips, for cut flowers. Carnations are cultivated both
in the field and in pots. Cut flowers are sent out in large quantities,
neatly and effectively packed, the parcel post being mainly employed as
a means of distribution. In the neighbourhood of Spalding crocuses and
snowdrops are less extensively grown than used to be the case. On one
farm, however, upwards of 20 acres are devoted to narcissi alone, whilst
gladioli, lilies and irises are grown on a smaller scale. Around Boston
narcissi are also extensively grown for the market, both bulbs and cut
blooms being sold. The bulbs are planted 3 in. apart in rows, the latter
being 9 in. apart, and are allowed to stand from two to four years.
The imports of fresh flowers into the United Kingdom were not
separately shown prior to 1900. In that year, however, their value
amounted to L200,585, in 1901 to L225,011, in 1906 to L233,884, in
1907 to L233,641, and in 1908 to L229,802, so that the trade showed a
fairly steady condition. From the monthly totals quoted in Table VI.
it would appear that the trade sinks to its minimum dimensions in the
four months July to October inclusive, and that after September the
business continually expands up to April, subsequent to which
contraction again sets in. About one-half of the trade belongs
practically to the three months of February, March and April.
TABLE VI.--_Values of Fresh Flowers imported into the United Kingdom._
+-----------+----------+----------+----------+
| Month. | 1906. | 1907. | 1908. |
+-----------+----------+----------+----------+
| January | L31,035 | L18,545 | L29,180 |
| February | 34,647 | 25,541 | 30,541 |
| March | 50,232 | 42,611 | 35,185 |
| April | 30,809 | 50,418 | 42,681 |
| May | 22,980 | 21,767 | 23,129 |
| June | 17,641 | 18,358 | 16,904 |
| July | 3,386 | 4,509 | 3,467 |
| August | 1,646 | 1,539 | 1,081 |
| September | 852 | 736 | 953 |
| October | 4,481 | 3,180 | 4,504 |
| November | 17,506 | 15,763 | 15,097 |
| December | 18,669 | 30,674 | 27,080 |
| +----------+----------+----------+
| Total | L233,884 | L233,641 | L229,802 |
+-----------+----------+----------+----------+
_Hothouse Culture of Fruit and Flowers._--The cultivation of fruit and
flowers under glass has increased enormously since about the year 1880,
especially in the neighbourhood of London, where large sums of money
have been sunk in the erection and equipment of hothouses. In the parish
of Cheshunt, Herts, alone there are upwards of 130 acres covered with
glass, and between that place on the north and London on the south
extensive areas of land are similarly utilized. In Middlesex, in the
north, in the districts of Edmonton, Enfield, Ponders End and Finchley,
and in the west from Isleworth to Hampton, Feltham, Hillingdon, Sipson
and Uxbridge, many crops are now cultivated under glass. At Erith,
Swanley, and other places in Kent, as also at Worthing, in Sussex,
glass-house culture has much extended. A careful estimate puts the area
of industrial hothouses in England at about 1200 acres, but it is
probably much more than this. Most of the greenhouses are fixtures, but
in some parts of the kingdom structures that move on rails and wheels
are used, to enable the ground to be prepared in the open for one crop
while another is maturing under glass. The leading products are grapes,
tomatoes and cucumbers, the last-named two being true fruits from the
botanist's point of view, though commercially included with vegetables.
To these may be added on the same ground dwarf or French beans, and
runner or climbing beans. Peaches, nectarines and strawberries are
largely grown under glass, and, in private hothouses--from which the
produce is used mainly for household consumption, and which are not
taken into consideration here--pineapples, figs and other fruit.
Conservative estimates indicate the average annual yield of hothouse
grapes to be about 12 tons per acre and of tomatoes 20 tons. The greater
part of the space in the hothouses is assigned to fruit, but whilst some
houses are devoted exclusively to flowers, in others, where fruit is the
main object, flowers are forced in considerable quantities in winter and
early spring. The flowers grown under glass include tulips, hyacinths,
primulas, cyclamens, spiraeas, mignonettes, fuchsias, calceolarias,
roses, chrysanthemums, daffodils, arum lilies or callas, liliums,
azaleas, eucharises, camellias, stephanotis, tuberoses, bouvardias,
gardenias, heaths or ericas, poinsettias, lilies of the valley, zonal
pelargoniums, tuberous and fibrous rooted begonias, and many others.
There is an increasing demand for foliage hothouse plants, such as
ferns, palms, crotons, aspidistras, araucarias, dracaenas, India-rubber
plants, aralias, grevilleas, &c. Berried plants like solanums and
aucubas also find a ready sale, while the ornamental kinds of asparagus
such as _sprengeri_ and _plumosus_ nanus, are ever in demand for
trailing decorations, as well as myrsiphyilum. Special mention must be
made of the winter or perpetual flowering carnations which are now grown
by hundreds of thousands in all parts of the kingdom for decorative work
during the winter season. The converse of forcing plants into early
blossom is adopted with such an important crop as lily of the valley.
During the summer season the crowns are placed in refrigerators with
about 2 degrees of frost, and quantities are taken out as required every
week and transferred to the greenhouse to develop. Tomatoes are grown
largely in houses exclusively occupied by them, in which case two and
sometimes three crops can be gathered in the year. In the Channel
Islands, where potatoes grown under glass are lifted in April and May,
in order to secure the high prices of the early markets, tomato
seedlings are planted out from boxes into the ground as quickly as the
potatoes are removed, the tomato planter working only a few rows behind
the potato digger. The trade in imported tomatoes is so considerable
that home growers are well justified in their endeavours to meet the
demand more fully with native produce, whether raised under glass or in
the open. Tomatoes were not separately enumerated in the imports
previous to 1900. It has already been stated that in 1900 the raw
tomatoes imported amounted to 833,032 cwt., valued at L792,339, and in
1901 to 793,991 cwt., valued at L734,051. From the monthly quantities
given in Table VII., it would appear that the imports are largest in
June, July and August, about one-half of the year's total arriving
during those three months. It is too early in June and July for
home-grown outdoor tomatoes to enter into competition with the imported
product, but home-grown hothouse tomatoes should be qualified to
challenge this trade.
TABLE VII.--_Quantities of Tomatoes imported into the United Kingdom._
+-----------+-----------+------------+------------+
| Month. | 1906. | 1907. | 1908. |
+-----------+-----------+------------+------------+
| January | 61,940 | 56,022 | 73,409 |
| February | 58,187 | 58,289 | 69,350 |
| March | 106,458 | 98,028 | 86,928 |
| April | 103,273 | 109,057 | 74,917 |
| May | 67,933 | 114,041 | 88,901 |
| June | 62,906 | 144,379 | 127,793 |
| July | 238,362 | 150,907 | 171,978 |
| August | 180,046 | 102,600 | 124,757 |
| September | 114,860 | 101,198 | 119,224 |
| October | 52,678 | 67,860 | 75,722 |
| November | 41,513 | 66,522 | 74,292 |
| December | 36,316 | 66,591 | 73,012 |
| +-----------+------------+------------+
| Total | 1,124,472 | 1,135,494 | 1,160,283 |
| +-----------+------------+------------+
| Value | L953,475 | L1,135,499 | L1,160,283 |
+-----------+-----------+------------+------------+
An important feature of modern flower growing is the production and
cultivation of what are known as "hardy herbaceous perennials." Some
2000 or 3000 different species and varieties of these are now raised in
special nurseries; and during the spring, summer and autumn seasons
magnificent displays are to be seen not only in the markets but at the
exhibitions in London and at the great provincial shows held throughout
the kingdom. The production of many of these perennials is so easy that
amateurs in several instances have taken it up as a business hobby; and
in some cases, chiefly through advertising in the horticultural press,
very lucrative concerns have been established.
Ornamental flowering trees and shrubs constitute another feature of
modern gardening. These are grown and imported by thousands chiefly for
their sprays of blossom or foliage, and for planting in large or small
gardens, public parks, &c., for landscape effect. Indeed there is
scarcely an easily grown plant from the northern or southern temperate
zones that does not now find a place in the nursery or garden, provided
it is sufficiently attractive to sell for its flowers, foliage or
appearance.
_Conditions of the Fruit and Flower growing Industries._--As regards
open-air fruit-growing, the outlook for new ventures is perhaps brighter
than in the hothouse industry, not--as Mr Bear has pointed out--because
the area of fruit land in England is too small, but because the level of
efficiency, from the selection of varieties to the packing and marketing
of the produce, is very much lower in the former than in the latter
branch of enterprise. In other words, whereas the practice of the
majority of hothouse nurserymen is so skilled, so up-to-date, and so
entirely under high pressure that a new competitor, however well
trained, will find it difficult to rise above mediocrity, the converse
is true of open-air fruit-growers. Many, and an increasing proportion,
of the latter are thoroughly efficient in all branches of their
business, and are in possession of plantations of the best market
varieties of fruit, well cultivated, pruned and otherwise managed. But
the extent of fruit plantations completely up to the mark in relation to
varieties and treatment of trees and bushes, and in connexion with which
the packing and marketing of the produce are equally satisfactory, is
small in proportion to the total fruit area of the country. Information
concerning the best treatment of fruit trees has spread widely in recent
years, and old plantations, as a rule, suffer from the neglect or errors
of the past, however skilful their present holders may be. Although the
majority of professional market fruit-growers may be well up to the
standard in skill, there are numerous contributors to the fruit supply
who are either ignorant of the best methods of cultivation and marketing
or careless in their application. The bad condition of the great
majority of farm orchards is notorious, and many landowners, farmers and
amateur gardeners who have planted fruit on a more or less extensive
scale have mismanaged their undertakings. For these reasons new growers
of open-air fruit for market have opportunities of succeeding by means
of superiority to the majority of those with whom they will compete,
provided that they possess the requisite knowledge, energy and capital.
It has been asserted on sound authority that there is no chance of
success for fruit-growers except in districts favourable as regards
soil, climate and nearness to a railway or a good market; and, even
under these conditions, only for men who have had experience in the
industry and are prepared to devote their unremitting attention to it.
Most important is it to a beginner that he should ascertain the
varieties of fruit that flourish best in his particular district.
Certain kinds seem to do well or fairly well in all parts of the
country; others, whilst heavy croppers in some localities, are often
unsatisfactory in others.
As has been intimated, there is probably in England less room for
expansion of fruit culture under glass than in the open. The large
increase of glass-houses in modern times appears to have brought the
supply of hothouse produce, even at greatly reduced prices, at least up
to the level of the demand; and as most nurserymen continue to extend
their expanse of glass, the prospect for new competitors is not a bright
one. Moreover, the vast scale upon which some of the growers conduct the
hothouse industry puts small producers at a great disadvantage, not only
because the extensive producers can grow grapes and other fruit more
economically than small growers--with the possible exception of those
who do all or nearly all their own work--but also, and still more,
because the former have greater advantages in transporting and marketing
their fruit. There has, in recent years, been a much greater fall in the
prices of hothouse than of open-air fruit, especially under the existing
system of distribution, which involves the payment by consumers of 50 to
100% more in prices than growers receive. The best openings for new
nurseries are probably not where they are now to be found in large
groups, and especially not in the neighbourhood of London, but in
suitable spots near the great centres of population in the Midlands and
the North, or big towns elsewhere not already well supplied with
nurseries. By such a selection of a locality the beginner may build up a
retail trade in hothouse fruit, or at least a trade with local
fruiterers and grocers, thus avoiding railway charges and salesmen's
commissions to a great extent, though it may often be advantageous to
send certain kinds of produce to a distant market. Above all, a man who
has no knowledge of the hothouse industry should avoid embarking his
capital in it, trusting himself in the hands of a foreman, as experience
shows that such a venture usually leads to disaster. Some years of
training in different nurseries are desirable for any young man who is
desirous of becoming a grower of hothouse fruits or flowers.
There can be no doubt that flower-growing is greatly extending in
England, and that competition among home growers is becoming more
severe. Foreign supplies of flowers have increased, but not nearly as
greatly in proportion as home supplies, and it seems clear that home
growers have gained ground in relation to their foreign rivals, except
with respect to flowers for the growth of which foreigners have
extraordinary natural advantages. There seems some danger of the home
culture of the narcissus being over-done, and the florists'
chrysanthemum appears to be produced in excess of the demand. Again, in
the production of violets the warm and sunny South of France has an
advantage not possessed by England, whilst Holland, likewise for
climatic reasons, maintains her hold upon the hyacinth and tulip trade.
Whether the production of flowers as a whole is gaining ground upon the
demand or not is a difficult question to answer. It is true that the
prices of flowers have fallen generally; but production, at any rate
under glass, has been cheapened, and if a fair profit can be obtained,
the fall in prices, without which the existing consumption of flowers
would be impossible, does not necessarily imply over-production. There
is some difference of opinion among growers upon this point; but nearly
all agree that profits are now so small that production on a large scale
is necessary to provide a fair income. Industrial flower-growing affords
such a wide scope for the exercise of superior skill, industry and
alertness, that it is not surprising to find some who are engaged in it
doing remarkably well to all appearance, while others are struggling on
and hardly paying their way. That a man with only a little capital,
starting in a small way, has many disadvantages is certain; also, that
his chance of saving money and extending his business quickly is much
smaller than it was. To the casual looker-on, who knows nothing of the
drudgery of the industry, flower-growing seems a delightful method of
getting a living. That it is an entrancing pursuit there is no doubt;
but it is equally true that it is a very arduous one, requiring careful
forethought, ceaseless attention and abundant energy. Fortunately for
those who might be tempted, without any knowledge of the industry, to
embark capital in it, flower-growing, if at all comprehensive in scope,
so obviously requires a varied and extensive technical knowledge,
combined with good commercial ability, that any one can see that a
thorough training is necessary to a man who intends to adopt it as a
business, especially if hothouse flowers are to be produced.
The market for fruit, and more especially for flowers, is a fickle
one, and there is nearly always some uncertainty as to the course of
prices. The perishable nature of soft fruit and cut flowers renders
the markets very sensitive to anything in the nature of a glut, the
occurrence of which is usually attended with disastrous results to
producers. Foreign competition, moreover, has constantly to be faced,
and it is likely to increase rather than diminish. French growers have
a great advantage over the open-air cultivators of England, for the
climate enables them to get their produce into the markets early in
the season, when the highest prices are obtainable. The geographical
advantage which France enjoys in being so near to England is, however,
considerably discounted by the increasing facilities for cold storage
in transit, both by rail and sea. The development of such facilities
permits of the retail sale in England of luscious fruit as fresh and
attractive as when it was gathered beneath the sunny skies of
California. In the case of flowers, fashion is an element not to be
ignored. Flowers much in request in one season may meet with very
little demand in another, and it is difficult for the producer to
anticipate the changes which caprice may dictate. Even for the same
kind of flower the requirements are very uncertain, and the white
blossom which is all the rage in one season may be discarded in favour
of one of another colour in the next. The sale of fresh flowers for
church decoration at Christmas and Easter has reached enormous
dimensions. The irregularity in the date of the festival, however,
causes some inconvenience to growers. If it falls very early the great
bulk of suitable flowers may not be sufficiently forward for sale,
whilst a late Easter may find the season too far advanced. The trade
in cut flowers, therefore, is generally attended by uncertainty, and
often by anxiety. (W. FR.; J. WS.)
UNITED STATES
In the United States horticulture and market gardening have now assumed
immense proportions. In a country of over 3,000,000 sq. m., stretching
from the Atlantic to the Pacific on the one hand, and from the Gulf of
Mexico to the great northern lakes and the Dominion of Canada on the
other, a great variation of climatic conditions is not unnatural. From a
horticultural point of view there are practically two well-defined
regions: (1) that to the east of the Rocky Mountains across to the
Atlantic, where the climate is more like that of eastern Asia than of
western Europe so far as rainfall, temperature and seasonable conditions
are concerned; (2) that to the west of the Rockies, known as the Pacific
coast region, where the climate is somewhat similar to that of western
Europe. It may be added that in the northern states--in Washington,
Montana, North Dakota, Minnesota, Wisconsin, &c.--the winters are often
very severe, while the southern states practically enjoy a temperature
somewhat similar to that of the Riviera. Indeed the range of temperature
between the extreme northern states and the extreme southern may vary as
much as 120 deg. F. The great aim of American gardeners, therefore, has
been to find out or to produce the kinds of fruits, flowers and
vegetables that are likely to flourish in different parts of this
immense country.
_Fruit Culture._--There is probably no country in the world where so
many different kinds of fruit can be grown with advantage to the nation
as in the United States. In the temperate regions apples, pears and
plums are largely grown, and orchards of these are chiefly to be found
in the states of New York, Massachusetts, Pennsylvania, Michigan,
Missouri, Colorado, and also in northern Texas, Arkansas and N.
California. To these may be added cranberries and quinces, which are
chiefly grown in the New England states. The quinces are not a crop of
first-rate importance, but as much as 800,000 bushels of cranberries are
grown each year. The peach orchards are assuming great proportions, and
are chiefly to be found in Georgia and Texas, while grapes are grown
throughout the Republic from east to west in all favourable localities.
Oranges, lemons and citrons are more or less extensively grown in
Florida and California, and in these regions what are known as Japanese
or "Kelsey" plums (forms of _Prunus triflora_) are also grown as
marketable crops. Pomegranates are not yet largely grown, but it is
possible their culture will develop in southern Texas and Louisiana,
where the climate is tempered by the waters of the Gulf of Mexico.
Tomatoes are grown in most parts of the country so easily that there is
frequently a glut; while the strawberry region extends from Florida to
Virginia, Pennsylvania and other states--thus securing a natural
succession from south to north for the various great market centres.
Of the fruits mentioned apples are undoubtedly the most important. Not
only are the American people themselves supplied with fresh fruit, but
immense quantities are exported to Europe--Great Britain alone absorbing
as much as 1,430,000 cwt. in 1908. The varieties originally grown were
of course those taken or introduced from Europe by the early settlers.
Since the middle of the 19th century great changes have been brought
about, and the varieties mostly cultivated now are distinctly American.
They have been raised by crossing and intercrossing the most suitable
European forms with others since imported from Russia. In the extreme
northern states indeed, where it is essential to have apple trees that
will stand the severest winters, the Russian varieties crossed with the
berry crab of eastern Europe (_Pyrus baccata_) have produced a race
eminently suited to that particular region. The individual fruits are
not very large, but the trees are remarkably hardy. Farther south larger
fruited varieties are grown, and among these may be noted Baldwins,
Newton pippins, Spitzenbergs and Rhode Island greening. Apple orchards
are numerous in the State of New York, where it is estimated that over
100,000 acres are devoted to them. In the hilly regions of Missouri,
Arkansas and Colorado there are also great plantations of apples. The
trees, however, are grown on different principles from those in New York
State. In the latter state apple trees with ordinary care live to more
than 100 years of age and produce great crops; in the other states,
however, an apple tree is said to be middle-aged at 20, decrepit at 30
and practically useless at 40 years of age. They possess the advantage,
however, of bearing early and heavily.
Until the introduction of the cold-storage system, about the year 1880,
America could hardly be regarded as a commercial fruit-growing country.
Since then, however, owing to the great improvements made in railway
refrigerating vans and storage houses, immense quantities of fruit can
be despatched in good condition to any part of the world; or they can be
kept at home in safety until such time as the markets of Chicago, New
York, Boston, Baltimore, Philadelphia, &c., are considered favourable
for their reception.
Apple trees are planted at distances varying from 25 ft. to 30 ft. apart
in the middle western states, to 40 ft. to 50 ft. apart in New York
State. Here and there, however, in some of the very best orchards the
trees are planted 60 ft. apart every way. Each tree thus has a chance to
develop to its utmost limits, and as air and light reach it better, a
far larger fruit-bearing surface is secured. Actual experience has shown
that trees planted at 60 ft. apart--about 28 to the acre--produce more
fruit by 43 bushels than trees at 30 ft. apart--i.e. about 48 to the
acre.
Until recent years pruning as known to English and French gardeners was
practically unknown. There was indeed no great necessity for it, as the
trees, not being cramped for space, threw their branches outwards and
upwards, and thus rarely become overcrowded. When practised, however,
the operation could scarcely be called pruning; lopping or trimming
would be more accurate descriptions.
Apple orchards are not immune from insect pests and fungoid diseases,
and an enormous business is now done in spraying machines and various
insecticides. It pays to spray the trees, and figures have been given to
show that orchards that have been sprayed four times have produced an
average income of L211 per acre against L103 per acre from unsprayed
orchards.
The spring frosts are also troublesome, and in the Colorado and other
orchards the process known as "smudging" is now adopted to save the
crops. This consists in placing 20 or 30, or even more, iron or tin pots
to an acre, each pot containing wooden chips soaked in tar (or pitch)
mixed with kerosene. Whenever the thermometer shows 3 or 4 degrees of
frost the smudge-pots are lighted. A dense white smoke then arises and
is diffused throughout the orchards, enveloping the blossoming heads of
the trees in a dense cloud. This prevents the frost from killing the
tender pistils in the blossoms, and when several smudge-pots are alight
at the same time the temperature of the orchard is raised two or three
degrees. This work has generally to be done between 3 and 5 A.M., and
the growers naturally have an anxious time until all danger is over. The
failure to attend to smudging, even on one occasion, may result in the
loss of the entire crop of plums, apples or pears.
Next to apples perhaps peaches are the most important fruit crop. The
industry is chiefly carried on in Georgia, Texas and S. Carolina, and on
a smaller scale in some of the adjoining states. Peaches thus flourish
in regions that are quite unsuitable for apples or pears. In many
orchards in Georgia, where over 3,000,000 acres have been planted, there
are as many as 100,000 peach trees; while some of the large fruit
companies grow as many as 365,000. In one place in West Virginia there
is, however, a peach orchard containing 175,000 trees, and in Missouri
another company has 3 sq. m. devoted to peach culture. As a rule the
crops do well. Sometimes, however, a disease known as the "yellows"
makes sad havoc amongst them, and scarcely a fruit is picked in an
orchard which early in the season gave promise of a magnificent crop.
Plums are an important crop in many states. Besides the European
varieties and those that have been raised by crossing with American
forms, there is now a growing trade done in Japanese plums. The largest
of these is popularly known as "Kelseys," named after John Kelsey, who
raised the first fruit in 1876 from trees brought to California in 1870.
Sometimes the fruits are 3 in. in diameter, and like most of the
Japanese varieties are more heart-shaped and pointed than plums of
European origin. One apparent drawback to the Kelsey plum is its
irregularity in ripening. It has been known in some years to be quite
ripe in June, while in others the fruits are still green in October.
Pears are much grown in such states as Massachusetts, New York,
Pennsylvania, Missouri and California; while bush fruits like currants,
gooseberries and raspberries find large spaces devoted in most of the
middle and northern states. Naturally a good deal of crossing and
intercrossing has taken place amongst the European and American forms of
these fruits, but so far as gooseberries are concerned no great advance
seems to have been made in securing varieties capable of resisting the
devastating gooseberry mildew.
Other fruits of more or less commercial value are oranges, lemons and
citrons, chiefly in Florida. Lemons are practically a necessity to the
American people, owing to the heat of the summers, when cool and
refreshing drinks with an agreeable acidulous taste are in great demand.
The pomelo (grape-fruit) is a kind of lemon with a thicker rind and a
more acid flavour. At one time its culture was confined to Florida, but
of recent years it has found its way into Californian orchards.
Notwithstanding the prevailing mildness of the climate in both
California and Florida, the crops of oranges, lemons, citrons, &c., are
sometimes severely injured by frosts when in blossom.
Other fruits likely to be heard of in the future are the kaki or
persimmon, the loquat, which is already grown in Louisiana, as well as
the pomegranate.
Great aid and encouragement are given by the government to the progress
of American fruit-growing, and by the experiments that are being
constantly carried out and tabulated at Cornell University and by the
U.S.A. department of agriculture.
_Flower Culture._--So far as flowers are concerned there appears to be
little difference between the kinds of plants grown in the United States
and in England, France, Belgium, Germany, Holland, &c. Indeed there is a
great interchange of new varieties of plants between Europe and America,
and modifications in systems of culture are being gradually introduced
from one side of the Atlantic to the other. The building of greenhouses
for commercial purposes is perhaps on a somewhat different scale from
that in England, but there are probably no extensive areas of glass such
as are to be seen north of London from Enfield Highway to Broxburne. Hot
water apparatus differs merely in detail, although most of the boilers
used resemble those on the continent of Europe rather than in England.
Great business is done in bulbs--mostly imported from Holland--stove and
greenhouse plants, hardy perennials, orchids, ferns of the "fancy" and
"dagger" types of Nephrolepis, and in carnations and roses. Amongst the
latter thousands of such varieties as Beauty, Liberty, Killarney,
Richmond and Bride are grown, and realize good prices as a rule in the
markets. Carnations of the winter-flowering or "perpetual" type have
long been grown in America, and enormous prices have been given for
individual plants on certain occasions, rivalling the fancy prices paid
in England for certain orchids. The American system of carnation-growing
has quite captivated English cultivators, and new varieties are being
constantly raised in both countries. Chrysanthemums are another great
feature of American florists, and sometimes during the winter season a
speculative grower will send a living specimen to one of the London
exhibitions in the hope of booking large orders for cuttings of it later
on. Sweet peas, dahlias, lilies of the valley, arum lilies and indeed
every flower that is popular in England is equally popular in America,
and consequently is largely grown.
_Vegetables._--So far as these are concerned, potatoes, cabbages,
cauliflowers, beans of all kinds, cucumbers, tomatoes (already
referred to under fruits), musk-melons, lettuces, radishes, endives,
carrots, &c.; are naturally grown in great quantities, not only in the
open air, but also under glass. The French system of intensive
cultivation as practised on hot beds of manure round Paris is
practically unknown at present. In the southern states there would be
no necessity to practise it, but in the northern ones it is likely to
attract attention. (J. Ws.)
FOOTNOTE:
[1] _Jour. Roy. Agric. Soc._, 1899.
FRUMENTIUS (c. 300-c. 360), the founder of the Abyssinian church,
traditionally identified in Abyssinian literature with Abba Salama or
Father of Peace (but see ETHIOPIA), was a native of Phoenicia. According
to the 4th-century historian Rufinus (x. 9), who gives Aedesius himself
as his authority, a certain Tyrian, Meropius, accompanied by his kinsmen
Frumentius and Aedesius, set out on an expedition to "India," but fell
into the hands of Ethiopians on the shore of the Red Sea and, with his
ship's crew, was put to death. The two young men were taken to the king
at Axum, where they were well treated and in time obtained great
influence. With the help of Christian merchants who visited the country
Frumentius gave Christianity a firm footing, which was strengthened when
in 326 he was consecrated bishop by Athanasius of Alexandria, who in his
_Epistola ad Constantinum_ mentions the consecration, and gives some
details of the history of Frumentius's mission. Later witnesses speak of
his fidelity to the homoousian during the Arian controversies. Aedesius
returned to Tyre, where he was ordained presbyter.
FRUNDSBERG, GEORG VON (1473-1528), German soldier, was born at
Mindelheim on the 24th of September 1473. He fought for the German king
Maximilian I. against the Swiss in 1499, and in the same year was among
the imperial troops sent to assist Ludovico Sforza, duke of Milan,
against the French. Still serving Maximilian, he took part in 1504 in
the war over the succession to the duchy of Bavaria-Landshut, and
afterwards fought in the Netherlands. Convinced of the necessity of a
native body of trained infantry Frundsberg assisted Maximilian to
organize the _Landsknechte_ (q.v.), and subsequently at the head of
bands of these formidable troops he was of great service to the Empire
and the Habsburgs. In 1509 he shared in the war against Venice, winning
fame for himself and his men; and after a short visit to Germany
returned to Italy, where in 1513 and 1514 he gained fresh laurels by his
enterprises against the Venetians and the French. Peace being made, he
returned to Germany, and at the head of the infantry of the Swabian
league assisted to drive Ulrich of Wurttemberg from his duchy in 1519.
At the diet of Worms in 1521 he spoke words of encouragement to Luther,
and when the struggle between France and the Empire was renewed he took
part in the invasion of Picardy, and then proceeding to Italy brought
the greater part of Lombardy under the influence of Charles V. through
his victory at Bicocca in April 1522. He was partly responsible for the
great victory over the French at Pavia in February 1525, and, returning
to Germany, he assisted to suppress the Peasant revolt, using on this
occasion, however, diplomacy as well as force. When the war in Italy was
renewed Frundsberg raised an army at his own expense, and skilfully
surmounting many difficulties, joined the constable de Bourbon near
Piacenza and marched towards Rome. Before he reached the city, however,
his unpaid troops showed signs of mutiny, and their leader, stricken
with illness and unable to pacify them, gave up his command. Returning
to Germany, he died at Mindelheim on the 20th of August 1528. He was a
capable and chivalrous soldier, and a devoted servant of the Habsburgs.
His son Caspar (1500-1536) and his grandson Georg (d. 1586) were both
soldiers of some distinction. With the latter's death the family became
extinct.
See Adam Reissner, _Historia Herrn Georgs und Herrn Kaspars von
Frundsberg_ (Frankfort, 1568). A German translation of this work was
published at Frankfort in 1572. F.W. Barthold, _Georg von Frundsberg_
(Hamburg, 1833); J. Heilmann, _Kriegsgeschichte von Bayern, Franken,
Pfalz und Schwaben_ (Munich, 1868).
FRUSTUM (Latin for a "piece broken off"), a term in geometry for the
part of a solid figure, such as a cone or pyramid, cut off by a plane
parallel to the base, or lying between two parallel planes; and hence in
architecture a name given to the drum of a column.
FRUYTIERS, PHILIP (1627-1666), Flemish painter and engraver, was a pupil
of the Jesuits' college at Antwerp in 1627, and entered the Antwerp gild
of painters without a fee in 1631. He is described in the register of
that institution as "illuminator, painter and engraver." The current
account of his life is "that he worked exclusively in water colours, yet
was so remarkable in this branch of his art for arrangement, drawing,
and especially for force and clearness of colour, as to excite the
admiration of Rubens, whom he portrayed with all his family." The truth
is that he was an artist of the most versatile talents, as may be judged
from the fact that in 1646 he executed an Assumption with figures of
life size, and four smaller pictures in oil, for the church of St
Jacques at Antwerp, for which he received the considerable sum of 1150
florins. Unhappily no undoubted production of his hand has been
preserved. All that we can point to with certainty is a series of etched
plates, chiefly portraits, which are acknowledged to have been
powerfully and skilfully handled. If, however, we search the portfolios
of art collections on the European continent, we sometimes stumble upon
miniatures on vellum, drawn with great talent and coloured with
extraordinary brilliancy. In form they quite recall the works of Rubens,
and these, it may be, are the work of Philip Fruytiers.
FRY, the name of a well-known English Quaker family, originally living
in Wiltshire. About the middle of the 18th century JOSEPH FRY
(1728-1787), a doctor, settled in Bristol, where he acquired a large
practice, but eventually abandoned medicine for commerce. He became
interested in china-making, soap-boiling and type-founding businesses in
Bristol, and in a chemical works at Battersea, all of which ventures
proved very profitable. The type-founding business was subsequently
removed to London and conducted by his son Edmund. Joseph Fry, however,
is best remembered as the founder of the great Bristol firm of J.S. Fry
& Sons, chocolate manufacturers. He purchased the chocolate-making
patent of William Churchman and on it laid the foundations of the
present large business. After his death the Bristol chocolate factory
was carried on with increasing success by his widow and by his son,
JOSEPH STORRS FRY (1767-1835).
In 1795 a new and larger factory was built in Union Street, Bristol,
which still forms the centre of the firm's premises, and in 1798 a
Watt's steam-engine was purchased and the cocoa-beans ground by steam.
On the death of Joseph Storrs Fry his three sons, Joseph (1795-1879),
Francis, and Richard (1807-1878) became partners in the firm, the
control being mainly in the hands of FRANCIS FRY (1803-1886). Francis
Fry was in every way a remarkable character. The development of the
business to its modern enormous proportion was chiefly his work, but
this did not exhaust his activities. He took a principal part in the
introduction of railways to the west of England, and in 1852 drew up a
scheme for a general English railway parcel service. He was an ardent
bibliographer, taking a special interest in early English Bibles, of
which he made in the course of a long life a large and striking
collection, and of the most celebrated of which he published facsimiles
with bibliographical notes. Francis Fry died in 1886, and his son
Francis J. Fry and nephew Joseph Storrs Fry carried on the business,
which in 1896 was for family reasons converted into a private limited
company, Joseph Storrs Fry being chairman and all the directors members
of the Fry family.
FRY, SIR EDWARD (1827- ), English judge, second son of Joseph Fry
(1795-1879), was born at Bristol on the 4th of November 1827, and
educated at University College, London, and London University. He was
called to the bar in 1854 and was made a Q.C. in 1869, practising in the
rolls court and becoming recognized as a leading equity lawyer. In 1877
he was raised to the bench and knighted. As chancery judge he will be
remembered for his careful interpretations and elucidations of the
Judicature Acts, then first coming into operation. In 1883 he was made a
lord justice of appeal, but resigned in 1892; and subsequently his
knowledge of equity and talents for arbitration were utilized by the
British government from time to time in various special directions,
particularly as chairman of many commissions. He was also one of the
British representatives at the Paris North Sea Inquiry Commission
(1905), and was appointed a member of the Hague Permanent Arbitration
Court. He wrote _A Treatise on the Specific Performance of Public
Contracts_ (London, 1858, and many subsequent editions).
FRY, ELIZABETH (1780-1845), English philanthropist, and, after Howard,
the chief promoter of prison reform in Europe, was born in Norwich on
the 21st of May 1780. Her father, John Gurney, afterwards of Earlham
Hall, a wealthy merchant and banker, represented an old family which for
some generations had belonged to the Society of Friends. While still a
girl she gave many indications of the benevolence of disposition,
clearness and independence of judgment, and strength of purpose, for
which she was afterwards so distinguished; but it was not until after
she had entered her eighteenth year that her religion assumed a decided
character, and that she was induced, under the preaching of the American
Quaker, William Savery, to become an earnest and enthusiastic though
never fanatical "Friend." In August 1800 she became the wife of Joseph
Fry, a London merchant.
Amid increasing family cares she was unwearied in her attention to the
poor and the neglected of her neighbourhood; and in 1811 she was
acknowledged by her co-religionists as a "minister," an honour and
responsibility for which she was undoubtedly qualified, not only by
vigour of intelligence and warmth of heart, but also by an altogether
unusual faculty of clear, fluent and persuasive speech. Although she had
made several visits to Newgate prison as early as February 1813, it was
not until nearly four years afterwards that the great public work of her
life may be said to have begun. The association for the Improvement of
the Female Prisoners in Newgate was formed in April 1817. Its aim was
the much-needed establishment of some of what are now regarded as the
first principles of prison discipline, such as entire separation of the
sexes, classification of criminals, female supervision for the women,
and adequate provision for their religious and secular instruction, as
also for their useful employment. The ameliorations effected by this
association, and largely by the personal exertions of Mrs Fry, soon
became obvious, and led to a rapid extension of similar methods to other
places. In 1818 she, along with her brother, visited the prisons of
Scotland and the north of England; and the publication (1819) of the
notes of this tour, as also the cordial recognition of the value of her
work by the House of Commons committee on the prisons of the metropolis,
led to a great increase of her correspondence, which now extended to
Italy, Denmark and Russia, as well as to all parts of the United
Kingdom. Through a visit to Ireland, which she made in 1827, she was led
to direct her attention to other houses of detention besides prisons;
and her observations resulted in many important improvements in the
British hospital system, and in the treatment of the insane. In 1838 she
visited France, and besides conferring with many of the leading prison
officials, she personally visited most of the houses of detention in
Paris, as well as in Rouen, Caen and some other places. In the following
year she obtained an official permission to visit all the prisons in
that country; and her tour, which extended from Boulogne and Abbeville
to Toulouse and Marseilles, resulted in a report which was presented to
the minister of the interior and the prefect of police. Before returning
to England she had included Geneva, Zurich, Stuttgart and
Frankfort-on-Main in her inspection. The summer of 1840 found her
travelling through Belgium, Holland and Prussia on the same mission; and
in 1841 she also visited Copenhagen. In 1842, through failing health,
Mrs Fry was compelled to forgo her plans for a still more widely
extended activity, but had the satisfaction of hearing from almost every
quarter of Europe that the authorities were giving increased practical
effect to her suggestions. In 1844 she was seized with a lingering
illness, of which she died on the 12th of October 1845. She was
survived by a numerous family, the youngest of whom was born in 1822.
Two interesting volumes of _Memoirs, with Extracts from her Journals and
Letters_, edited by two of her daughters, were published in 1847. See
also _Elizabeth Fry_, by G. King Lewis (1910).
FRYXELL, ANDERS (1795-1881), Swedish historian, was born at Hesselskog,
Dalsland, Sweden, on the 7th of February 1795. He was educated at
Upsala, took holy orders in 1820, was made a doctor of philosophy in
1821, and in 1823 began to publish the great work of his life, the
_Stories from Swedish History_. He did not bring this labour to a close
until, fifty-six years later, he published the forty-sixth and crowning
volume of his vast enterprise. Fryxell, as a historian, appealed to
every class by the picturesqueness of his style and the breadth of his
research; he had the gift of awakening to an extraordinary degree the
national sense in his readers. In 1824 he published his _Swedish
Grammar_, which was long without a rival. In 1833 he received the title
of professor, and in 1835 he was appointed to the incumbency of Sunne,
in the diocese of Karlstad, where he resided for the remainder of his
life. In 1840 he was elected to the Swedish Academy in succession to the
poet Wallin (1779-1839). In 1847 Fryxell received from his bishop
permission to withdraw from all the services of the Church, that he
might devote himself without interruption to historical investigation.
Among his numerous minor writings are prominent his _Characteristics of
Sweden between 1592 and 1600_ (1830), his _Origins of the Inaccuracy
with which the History of Sweden in Catholic Times has been Treated_
(1847), and his _Contributions to the Literary History of Sweden_. It is
now beginning to be seen that the abundant labours of Fryxell were
rather of a popular than of a scientific order, and although their
influence during his lifetime was unbounded, it is only fair to later
and exacter historians to admit that they threaten to become obsolete in
more than one direction. On the 21st of March 1881 Anders Fryxell died
at Stockholm, and in 1884 his daughter Eva Fryxell (born 1829) published
from his MS. an interesting _History of My History_, which was really a
literary autobiography and displays the persistency and tirelessness of
his industry. (E. G.)
FUAD PASHA (1815-1869), Turkish statesman, was the son of the
distinguished poet Kecheji-zade Izzet Molla. He was educated at the
medical school and was at first an army surgeon. About 1836 he entered
the civil service as an official of the foreign ministry. He became
secretary of the embassy in London; was employed on special missions in
the principalities and at St Petersburg (1848), and was sent to Egypt as
special commissioner in 1851. In that year he became minister for
foreign affairs, a post to which he was appointed also on four
subsequent occasions and which he held at the time of his death. During
the Crimean War he commanded the troops on the Greek frontier and
distinguished himself by his bravery. He was Turkish delegate at the
Paris conference of 1856; was charged with a mission to Syria in 1860;
grand vizier in 1860 and 1861, and also minister of war. He accompanied
the sultan Abd-ul-Aziz on his journey to Egypt and Europe, when the
freedom of the city of London was conferred on him. He died at Nice
(whither he had been ordered for his health) in 1869. Fuad was renowned
for his boldness and promptness of decision, as well as for his ready
wit and his many bons mots. Generally regarded as the partisan of a
pro-English policy, he rendered most valuable service to his country by
his able management of the foreign relations of Turkey, and not least by
his efficacious settlement of affairs in Syria after the massacres of
1860.
FUCHOW, FU-CHAU, FOOCHOW, a city of China, capital of the province of
Fu-kien, and one of the principal ports open to foreign commerce. In the
local dialect it is called Hokchiu. It is situated on the river Min,
about 35 m. from the sea, in 26 deg. 5' N. and 119 deg. 20' E., 140 m.
N. of Amoy and 280 S. of Hang-chow. The city proper, lying nearly 3 m.
from the north bank of the river, is surrounded by a wall about 30 ft.
high and 12 ft. thick, which makes a circuit of upwards of 5 m. and is
pierced by seven gateways surrounded by tall fantastic watch-towers.
The whole district between the city and the river, the island of
Nantai, and the southern banks of the Min are occupied by extensive
suburbs; and the river itself bears a large floating population.
Communication from bank to bank is afforded by a long stone bridge
supported by forty solid stone piers in its northern section and by nine
in its southern. The most remarkable establishment of Fuchow is the
arsenal situated about 3 m. down the stream at Pagoda Island, where the
sea-going vessels usually anchor. It was founded in 1867, and is
conducted under the direction of French engineers according to European
methods. In 1870 it employed about 1000 workmen besides fifty European
superintendents, and between that date and 1880 it turned out about 20
or 30 small gunboats. In 1884 it was partially destroyed by the French
fleet, and for a number of years the workshops and machinery were
allowed to stand idle and go to decay. On the 1st of August 1895 an
attack was made on the English mission near the city of Ku-chang, 120 m.
west of Fuchow, on which occasion nine missionaries, of whom eight were
ladies, were massacred. The port was opened to European commerce in
1842; and in 1853 the firm of Russell and Co. shipped the first cargoes
of tea from Fuchow to Europe and America. The total trade in foreign
vessels in 1876 was imports to the value of L1,531,617, and exports to
the value of L3,330,489. In 1904 the imports amounted to L1,440,351, and
the exports to L1,034,436. The number of vessels that entered in 1876
was 275, and of these 211 were British, 27 German, 11 Danish and 9
American. While in 1904, 480 vessels entered the port, 216 of which were
British. A large trade is carried on by the native merchants in timber,
paper, woollen and cotton goods, oranges and olives; but the foreign
houses mainly confine themselves to opium and tea. Commercial
intercourse with Australia and New Zealand is on the increase. The
principal imports, besides opium, are shirtings, T-cloths, lead and tin,
medicines, rice, tobacco, and beans and peas. Two steamboat lines afford
regular communication with Hong-Kong twice a month. The town is the seat
of several important missions, of which the first was founded in 1846.
That supported by the American board had in 1876 issued 1,3000,000
copies of Chinese books and tracts.
FUCHS, JOHANN NEPOMUK VON (1774-1856), German chemist and mineralogist,
was born at Mattenzell, near Brennberg in the Bavarian Forest, on the
15th of May 1774. In 1807 he became professor of chemistry and
mineralogy at the university of Landshut, and in 1823 conservator of the
mineralogical collections at Munich, where he was appointed professor of
mineralogy three years later, on the removal thither of the university
of Landshut. He retired in 1852, was ennobled by the king of Bavaria in
1854, and died at Munich on the 5th of March 1856. His name is chiefly
known for his mineralogical observations and for his work on soluble
glass.
His collected works, including _Uber den Einfluss der Chemie und
Mineralogie_ (1824), _Die Naturgeschichte des Mineralreichs_ (1842),
_Uber die Theorien der Erde_ (1844), were published at Munich in 1856.
FUCHS, LEONHARD (1501-1566), German physician and botanist, was born at
Wembdingen in Bavaria on the 17th of January 1501. He attended school at
Heilbronn and Erfurt, and in 1521 graduated at the university of
Ingolstadt. About the same time he espoused the doctrines of the
Reformation. Having in 1524 received his diploma as doctor of medicine,
he practised for two years in Munich. He became in 1526 professor of
medicine at Ingolstadt, and in 1528 physician to the margrave of
Anspach. In Anspach he was the means of saving the lives of many during
the epidemic locally known as the "English sweating-sickness." By the
duke of Wurttemberg he was, in 1535, appointed to the professorship of
medicine at the university of Tubingen, a post held by him till his
death on the 10th of May 1566. Fuchs was an advocate of the Galenic
school of medicine, and published several Latin translations of
treatises by its founder and by Hippocrates. But his most important
publication was _De historia stirpium commentarii insignes_ (Basel,
1542), a work illustrated with more than five hundred excellent outline
illustrations, including figures of the common foxglove and of another
species of the genus _Digitalis_, which was so named by him.
FUCHSIA, so named by Plumier in honour of the botanist Leonhard Fuchs, a
genus of plants of the natural order Onagraceae, characterized by
entire, usually opposite leaves, pendent flowers, a funnel-shaped,
brightly coloured, quadripartite, deciduous calyx, 4 petals, alternating
with the calycine segments, 8, rarely 10, exserted stamens, a long
filiform style, an inferior ovary, and fruit, a fleshy ovoid many-seeded
berry. All the members of the genus, with the exception of the New
Zealand species, _F. excorticata, F. Colensoi_ and _F. procumbens_, are
natives of Central and South America--occurring in the interior of
forests or in damp and shady mountainous situations. The various species
differ not a little in size as well as in other characters; some, as _F.
verrucosa_, being dwarf shrubs; others, as _F. arborescens_ and _F.
apetala_, attaining a height of 12 to 16 ft., and having stems several
inches in diameter. Plumier, in his _Nova plantarum Americanarum genera_
(p. 14, tab. 14, Paris, 1703), gave a description of a species of
fuchsia, the first known, under the name of _Fuchsia triphylla, flore
coccineo_, and a somewhat conventional outline figure of the same plant
was published at Amsterdam in 1757 by Burmann. In the _Histoire des
plantes medicinales_ of the South American traveller Feuillee (p. 64,
pl. XLVII.), written in 1709-1711, and published by him with his
_Journal_, Paris, 1725, the name _Thilco_ is applied to a species of
fuchsia from Chile, which is described, though not evidently so figured,
as having a pentamerous calyx. The _F. coccinea_ of Alton (fig.) (see
J.D. Hooker, in _Journal Linnean Soc_., Botany, vol. x. p. 458, 1867),
the first species of fuchsia cultivated in England, where it was long
confined to the greenhouse, was brought from South America by Captain
Firth in 1788 and placed in Kew Gardens. Of this species Mr Lee, a
nurseryman at Hammersmith, soon afterwards obtained an example, and
procured from it by means of cuttings several hundred plants, which he
sold at a guinea each. In 1823 _F. macrostemma_ and _F. gracilis_, and
during the next two or three years several other species, were
introduced into England; but it was not until about 1837, or soon after
florists had acquired _F. fulgens_, that varieties of interest began to
make their appearance. The numerous hybrid forms now existing are the
result chiefly of the intercrossing of that or other long-flowered with
globose-flowered plants. _F. Venus-victrix_, raised by Mr Gulliver,
gardener to the Rev. S. Marriott of Horsemonden, Kent, and sold in 1822
to Messrs Cripps, was the earliest white-sepalled fuchsia. The first
fuchsia with a white corolla was produced about 1853 by Mr Storey. In
some varieties the blossoms are variegated, and in others they are
double. There appears to be very little limit to the number of forms to
be obtained by careful cultivation and selection. To hybridize, the
flower as soon as it opens is emasculated, and it is then fertilized
with pollen from some different flower.
[Illustration: _Fuchsia coccinea_. 1, Flower cut open after removal of
sepals; 2, fruit; 3, floral diagram.]
Ripe seed is sown either in autumn or about February or March in light,
rich, well-drained mould, and is thinly covered with sandy soil and
watered. A temperature of 70 deg. to 75 deg. Fahr. has been found
suitable for raising. The seedlings are pricked off into shallow pots or
pans, and when 3 in. in height are transferred to 3-in. pots, and are
then treated the same as plants from cuttings. Fuchsias may be grafted
as readily as camellias, preferably by the splice or whip method, the
apex of a young shoot being employed as a scion; but the easiest and
most usual method of propagation is by cuttings. The most expeditious
way to procure these is to put plants in heat in January, and to take
their shoots when 3 in. in length. For summer flowering in England they
are best made about the end of August, and should be selected from the
shortest-jointed young wood. They root readily in a compost of loam and
silver-sand if kept close and sprinkled for a short time. In from two to
three weeks they may be put into 3-in. pots containing a compost of
equal parts of rich loam, silver-sand and leaf-mould. They are
subsequently moved from the frame or bed, first to a warm and shady, and
then to a more airy part of the greenhouse. In January a little
artificial heat may be given, to be gradually increased as the days
lengthen. The side-shoots are generally pruned when they have made three
or four joints, and for bushy plants the leader is stopped soon after
the first potting. Care is taken to keep the plants as near the glass as
possible, and shaded from bright sunshine, also to provide them
plentifully with water, except at the time of shifting, when the roots
should be tolerably dry. For the second potting a suitable soil is a
mixture of well-rotted cow-dung or old hotbed mould with leaf-mould and
sandy peat, and to promote drainage a little peat-moss may be placed
immediately over the crocks in the lower part of the pot. Weak liquid
manure greatly promotes the advance of the plants, and should be
regularly supplied twice or thrice a week during the flowering season.
After this, water is gradually withheld from them, and they may be
placed in the open air to ripen their wood.
Among the more hardy or half-hardy plants for inside borders are
varieties of the Chilean species, _F. macrostemma_ (or _F.
magellanica_), a shrub 6 to 12 ft. high with a scarlet calyx, such as
_F. m. globosa, F. m. gracilis_; one of the most graceful and hardy of
these, a hybrid _F. riccartoni_, was raised at Riccarton, near
Edinburgh, in 1830. For inside culture may be mentioned _F. boliviana_
(Bolivia), 2 to 4 ft. high, with rich crimson flowers with a
trumpet-shaped tube; _F. corymbiflora_ (Peru), 4 to 6 ft. high, with
scarlet flowers nearly 2 in. long in long terminal clusters; F. fulgens
(Mexico), 4 to 6 ft., with drooping apical clusters of scarlet flowers;
_F. microphylla_ (Central America), with small leaves and small scarlet
funnel-shaped flowers, the petals deep red; _F. procumbens_ (New
Zealand), a pretty little creeper, the small flowers of which are
succeeded by oval magenta-crimson berries which remain on for months;
and _F. splendens_ (Mexico), 6 ft. high, with very showy scarlet and
green flowers. But these cannot compare in beauty or freedom of blossom
with the numerous varieties raised by gardeners. The nectar of fuchsia
flowers has been shown to contain nearly 78% of cane sugar, the
remainder being fruit sugar. The berries of some fuchsias are subacid or
sweet and edible. From certain species a dye is obtainable. The
so-called "native fuchsias" of southern and eastern Australia are plants
of the genus _Correa_, natural order Rutaceae.
FUCHSINE, or MAGENTA, a red dye-stuff consisting of a mixture of the
hydrochlorides or acetates of pararosaniline and rosaniline. It was
obtained in 1856 by J. Natanson (_Ann_., 1856, 98, p. 297) by the action
of ethylene chloride on aniline, and by A.W. Hofmann in 1858 from
aniline and carbon tetrachloride. It is prepared by oxidizing "aniline
for red" (a mixture of aniline and ortho- and para-toluidine) with
arsenic acid (H. Medlock, _Dingler's Poly. Jour_., 1860, 158, p. 146);
by heating aniline for red with nitrobenzene, concentrated hydrochloric
acid and iron (Coupier, _Ber_., 1873, 6, p. 423); or by condensing
formaldehyde with aniline and ortho-toluidine and oxidizing the mixture.
It forms small crystals, showing a brilliant green reflex, and is
soluble in water and alcohol with formation of a deep red solution. It
dyes silk, wool and leather direct, and cotton after mordanting with
tannin and tartar emetic (see DYEING). An aqueous solution of fuchsine
is decolorized on the addition of sulphurous acid, the easily soluble
fuchsine sulphurous acid being formed. This solution is frequently used
as a test reagent for the detection of aldehydes, giving, in most cases,
a red coloration on the addition of a small quantity of the aldehyde.
The constitution of the fuchsine bases (pararosaniline and rosaniline)
was determined by E. and O. Fischer in 1878 (_Ann_., 1878, 194, p.
242); A.W. Hofmann having previously shown that oxidation of pure
aniline alone or of pure toluidine yielded no fuchsine, whilst
oxidation of a mixture of aniline and para-toluidine gave rise to the
fine red dye-stuff para-fuchsine (pararosaniline hydrochloride)
CH3.C6H4NH2 + 2C6H5NH2 + 3O = HO.C(C6H4NH2)3 + 2H2O.
Colour base (pararosaniline).
HO.C(C6H4NH2)3.HCl = H2O + (H2N.C6H4)2C : C6H4 : NH2Cl.
Pararosaniline hydrochloride.
A. Rosenstiehl (_Jahres_., 1869, p. 693) found also that different
rosanilines were obtained according to whether ortho- or
para-toluidine was oxidized with aniline; and he gave the name
rosaniline to the one obtained from aniline and ortho-toluidine,
reserving the term pararosaniline for the other. E. and O. Fischer
showed that these compounds were derivatives of triphenylmethane and
tolyldiphenylmethane respectively. Pararosaniline was reduced to the
corresponding leuco compound (paraleucaniline), from which by
diazotization and boiling with alcohol, the parent hydrocarbon was
obtained
(H2N.C5H4)2C : C6H4:NH2Cl --> HC(C6H4NH2.HCl)3 --> HC(C6H4N2Cl3)
Pararosaniline hydrochloride. Paraleucaniline.
--> HC(C6H5)3.
Triphenylmethane.
The reverse series of operations was also carried out by the Fischers,
triphenylmethane being nitrated, and the nitro compound then reduced
to triaminotriphenylmethane or paraleucaniline, which on careful
oxidation is converted into the dye-stuff. A similar series of
reactions was carried out with rosaniline, which was shown to be the
corresponding derivative of tolyldiphenylmethane.
The free pararosaniline, C19H19N3O, and rosaniline, C20H21N3O, may be
obtained by precipitating solutions of their salts with a caustic
alkali, colourless precipitates being obtained, which crystallize from
hot water in the form of needles or plates. The position of the amino
groups in pararosaniline was determined by the work of H. Caro and C.
Graebe (_Ber_., 1878, II, p. 1348) and of E. and O. Fischer (_Ber._,
1880, 13, p. 2204) as follows: Nitrous acid converts pararosaniline
into aurin, which when superheated with water yields
para-dioxybenzophenone. As the hydroxyl groups in aurin correspond to
the amino groups in pararosaniline, two of these in the latter
compound must be in the para position. The third is also in the para
position; for if benzaldehyde be condensed with aniline, condensation
occurs in the para position, for the compound formed may be converted
into para-dioxybenzophenone,
C6H5CHO --> C6H5CH(C6H4NH2)2 --> C6H5CH(C6H4OH)2 --> CO(C6H4OH)2;
but if para-nitrobenzaldehyde be used in the above reaction and the
resulting nitro compound NO2.C6H4.CH(C6H4NH2)2 be reduced, then
pararosaniline is the final product, and consequently the third amino
group occupies the para position. Many derivatives of pararosaniline
and rosaniline are known, in which the hydrogen atoms of the amino
groups are replaced by alkyl groups; this has the effect of producing
a blue or violet shade, which becomes deeper as the number of groups
increases (see DYEING).
FUCINO, LAGO DI [Lat. _Lacus Fucinus_], a lake bed of the Abruzzi,
Italy, in the province of Aquila, 2 m. E. of the town of Avezzano. The
lake was 37 m. in circumference and 65 ft. deep. From the lack of an
outlet, the level of the lake was subject to great variations, often
fraught with disastrous consequences. As early as A.D. 52 the emperor
Claudius, realizing a project of Julius Caesar, constructed a tunnel
3-1/2 m. long, with 40 shafts at intervals, by which the surplus waters
found an outlet to the Liris (or Garigliano). No less than 30,000
workmen were employed for eleven years in driving this tunnel. In the
following reign the tunnel was allowed to fall into disrepair, but was
repaired by Trajan. When, however, it finally went out of use is
uncertain. The various attempts made to reopen it from 1240 onwards were
unsuccessful. By 1852 the lake had gradually risen until it was 30 ft.
above its original level, and had become a source of danger to the
surrounding countryside. A company undertook to drain it on condition of
becoming proprietors of the site when dry; in 1854, however, the rights
and privileges were purchased by Prince Giulio Torlonia (d. 1886), the
great Roman banker, who carried on the work at his own expense until, in
1876, the lake was finally drained at the cost of some L1,700,000. The
reclaimed area is 12-1/2 m. long, 7 m. broad, and is cultivated by
families from the Torlonia estates. The outlet by which it was drained
is 4 m. long and 24 sq. yds. in section.
See A. Brisse and L. de Rotron, _Le Dessechement du lac Fucin, execute
par S.E. le Prince A. Torlonia_ (Rome, 1876). (T. As.)
FUEL (O. Fr. _feuaile_, popular Lat. _focalia_, from _focus_, hearth,
fire), a term applicable to all substances that can be usefully employed
for the production of heat by combustion. Any element or combination of
elements susceptible of oxidation may under appropriate conditions be
made to burn; but only those that ignite at a moderate initial
temperature and burn with comparative rapidity, and, what is practically
of more importance, are obtainable in quantity at moderate prices, can
fairly be regarded as fuels. The elementary substances that can be so
classed are primarily hydrogen, carbon and sulphur, while others finding
more special applications are silicon, phosphorus, and the more readily
oxidizable metals, such as iron, manganese, aluminium and magnesium.
More important, however, than the elements are the carbohydrates or
compounds of carbon, oxygen and hydrogen, which form the bulk of the
natural fuels, wood, peat and coal, as well as of their liquid and
gaseous derivatives--coal-gas, coal-tar, pitch, oil, &c., which have
high values as fuel. Carbon in the elementary form has its nearest
representative in the carbonized fuels, charcoal from wood and coke from
coal.
_Solid Fuels._
Wood.
Wood may be considered as having the following average composition when
in the air-dried state: Carbon, 39.6; hydrogen, 4.8; oxygen, 34.8; ash,
1.0; water, 20%. When it is freshly felled, the water may be from 18 to
50%. Air-dried or even green wood ignites readily when a considerable
surface is exposed to the kindling flame, but in large masses with
regular or smooth surfaces it is often difficult to get it to burn. When
previously torrefied or scorched by heating to a temperature of about
200 deg., at which incipient charring is set up, it is exceedingly
inflammable. The ends of imperfectly charred boughs from the charcoal
heaps in this condition are used in Paris and other large towns in
France for kindling purposes, under the name of _fumerons_. The
inflammability, however, varies with the density,--the so-called hard
woods, oak, beech and maple, taking fire less readily than the softer,
and, more especially, the coniferous varieties rich in resin. The
calorific power of absolutely dry woods may as an average be taken at
about 4000 units, and when air-dried, i.e. containing 25% of water, at
2800 to 3000 units. Their evaporative values, i.e. the quantities of
water evaporated by unit weight, are 3.68 and 4.44.
Wood being essentially a flaming fuel is admirably adapted for use with
heat-receiving surfaces of large extent, such as locomotive and marine
boilers, and is also very clean in use. The absence of all cohesion in
the cinders or unburnt carbonized residue causes a large amount of
ignited particles to be projected from the chimney, when a rapid draught
is used, unless special spark-catchers of wire gauze or some analogous
contrivance are used. When burnt in open fireplaces the volatile
products given off in the apartment on the first heating have an acrid
penetrating odour, which is, however, very generally considered to be
agreeable. Owing to the large amount of water present, no very high
temperatures can be obtained by the direct combustion of wood, and to
produce these for metallurgical purposes it is necessary to convert it
previously either into charcoal or into inflammable gas.
Peat.
Peat includes a great number of substances of very unequal fuel value,
the most recently formed spongy light brown kind approximating in
composition to wood, while the dense pitchy brown compact substance,
obtained from the bottom of bogs of ancient formation, may be compared
with lignite or even in some instances with coal. Unlike wood, however,
it contains incombustible matter in variable but large quantity, from 5
to 15% or even more. Much of this, when the amount is large, is often
due to sand mechanically intermixed; when air-dried the proportion of
water is from 8 to 20%. When these constituents are deducted the average
composition may be stated to be--carbon, 52 to 66; hydrogen, 4.7 to
7.4; oxygen, 28 to 39; and nitrogen, 1.5 to 3%. Average air-dried peat
may be taken as having a calorific value of 3000 to 3500 units, and when
dried at 100 deg. C., and with a minimum of ash (4 to 5%), at about 5200
units, or from a quarter to one-third more than that of an equal weight
of wood. The lighter and more spongy varieties of peat when air-dried
are exceedingly inflammable, firing at a temperature of 200 deg. C.; the
denser pulpy kinds ignite less readily when in the natural state, and
often require a still higher temperature when prepared by pulping and
compression or partial carbonization. Most kinds burn with a red smoky
flame, developing a very strong odour, which, however, has its admirers
in the same way that wood smoke has. This arises from the destructive
distillation of imperfectly carbonized organic matter. The ash, like
that of wood, is light and powdery, except when much sand is present,
when it is of a denser character.
Peat is principally found in high latitudes, on exposed high tablelands
and treeless areas in more temperate climates, and in the valleys of
slow-flowing rivers,--as in Ireland, the west of Scotland, the tableland
of Bavaria, the North German plain, and parts of the valleys of the
Somme, Oise and a few other rivers in northern France. A principal
objection to its use is its extreme bulk, which for equal evaporative
effect is from 8 to 18 times that of coal. Various methods have been
proposed, and adopted more or less successfully, for the purpose of
increasing the density of raw peat by compression, either with or
without pulping; the latter process gives the heaviest products, but the
improvement is scarcely sufficient to compensate for the cost.
Lignite.
Lignite or brown coal is of intermediate character between peat and coal
proper. The best kinds are undistinguishable in quality from
free-burning coals, and the lowest earthy kinds are not equal to average
peat. When freshly raised, the proportion of water may be from 45 to 50%
and even more, which is reduced from 28 to 20% by exposure to dry air.
Most varieties, however, when fully dried, break up into powder, which
considerably diminishes their utility as fuel, as they cannot be
consolidated by coking. Lignite dust may, however, be compacted into
serviceable blocks for burning, by pressure in machines similar to those
used for brickmaking, either in the wet state as raised from the mines
or when kiln-dried at 200 deg. C. This method was adopted to a very large
extent in Prussian Saxony. The calorific value varies between 3500 and
5000 units, and the evaporative factor from 2.16 when freshly raised to
5.84 for the best kinds of lignite when perfectly dried.
Other natural fuels.
Of the other natural fuels, apart from coal (q.v.), the most important
is so-called vegetable refuse, such as cotton stalks, brushwood, straw,
and the woody residue of sugar-cane after the extraction of the
saccharine juice known as megasse or cane trash. These are extensively
used in countries where wood and coal are scarce, usually for providing
steam in the manufactures where they arise, e.g. straw for thrashing,
cotton stalks for ploughing, irrigating, or working presses, and cane
trash for boiling down sugar or driving the cane mill. According to J.
Head (_Proc. Inst. of Civil Engineers_, vol. xlviii. p. 75), the
evaporative values of 1 lb. of these different articles when burnt in a
tubular boiler are--coal, 8 lb.; dry peat, 4 lb.; dry wood, 3.58-3.52
lb.; cotton stalks or megasse, 3.2-2.7 lb.; straw, 2.46-2.30 lb.
Owing to the siliceous nature of the ash of straw, it is desirable to
have a means of clearing the grate bars from slags and clinkers at short
intervals, and to use a steam jet to clear the tubes from similar
deposits.
The common fuel of India and Egypt is derived from the dung of camels
and oxen, moulded into thin cakes, and dried in the sun. It has a very
low heating power, and in burning gives off acrid ammoniacal smoke and
vapour.
Somewhat similar are the tan cakes made from spent tanners' bark, which
are used to some extent in eastern France and in Germany. They are made
by moulding the spent bark into cakes, which are then slowly dried by
exposure to the air. Their effect is about equivalent to 80 and 30% of
equal weights of wood and coal respectively.
Sulphur, phosphorus and silicon, the other principal combustible
elements, are only of limited application as fuels. The first is used in
the liquidation of sulphur-bearing rocks. The ore is piled into large
heaps, which are ignited at the bottom, a certain proportion, from
one-fourth to one-third, of the sulphur content being sacrificed, in
order to raise the mass to a sufficient temperature to allow the
remainder to melt and run down to the collecting basin. Another
application is in the so-called "pyritic smelting," where ores of copper
(q.v.) containing iron pyrites, FeS2, are smelted with appropriate
fluxes in a hot blast, without preliminary roasting, the sulphur and
iron of the pyrites giving sufficient heat by oxidation to liquefy both
slag and metal. Phosphorus, which is of value from its low igniting
point, receives its only application in the manufacture of lucifer
matches. The high temperature produced by burning phosphorus is in part
due to the product of combustion (phosphoric acid) being solid, and
therefore there is less heat absorbed than would be the case with a
gaseous product. The same effect is observed in a still more striking
manner with silicon, which in the only special case of its application
to the production of heat, namely, in the Bessemer process of
steel-making, gives rise to an enormous increase of temperature in the
metal, sufficient indeed to keep the iron melted. The absolute calorific
value of silicon is lower than that of carbon, but the product of
combustion (silica) being non-volatile at all furnace temperatures, the
whole of the heat developed is available for heating the molten iron,
instead of a considerable part being consumed in the work of
volatilization, as is the case with carbonic oxide, which burns to waste
in the air.
Calorific power.
_Assay and Valuation of Carbonaceous Fuels._--The utility or value of
a fuel depends upon two principal factors, namely, its calorific power
and its calorific intensity or pyrometric effect, that is, the
sensible temperature of the products of combustion. The first of these
is constant for any particular product of combustion independently of
the method by which the burning is effected, whether by oxygen, air or
a reducible metallic oxide. It is most conveniently determined in the
laboratory by measuring the heat evolved during the combustion of a
given weight of the fuel. The method of Lewis Thompson is one of the
most useful. The calorimeter consists of a copper cylinder in which a
weighed quantity of coal intimately mixed with 10-12 parts of a
mixture of 3 parts of potassium chlorate and 1 of potassium nitrate is
deflagrated under a copper case like a diving-bell, placed at the
bottom of a deep glass jar filled with a known weight of water. The
mixture is fired by a fuse of lamp-cotton previously soaked in a nitre
solution and dried. The gases produced by the combustion rising
through the water are cooled, with a corresponding increase of
temperature in the latter, so that the difference between the
temperature observed before and after the experiment measures the heat
evolved. The instrument is so constructed that 30 grains (2 grammes)
of coal are burnt in 29,010 grains of water, or in the proportion of 1
to 937, these numbers being selected that the observed rise of
temperature in Fahrenheit degrees corresponds to the required
evaporative value in pounds, subject only to a correction for the
amount of heat absorbed by the mass of the instrument, for which a
special coefficient is required and must be experimentally determined.
The ordinary bomb calorimeter is also used. An approximate method is
based upon the reduction of lead oxide by the carbon and hydrogen of
the coal, the amount of lead reduced affording a measure of the oxygen
expended, whence the heating power may be calculated, 1 part of pure
carbon being capable of producing 34-1/2 times its weight of lead. The
operation is performed by mixing the weighed sample with a large
excess of litharge in a crucible, and exposing it to a bright red heat
for a short time. After cooling, the crucible is broken and the
reduced button of lead is cleaned and weighed. The results obtained by
this method are less accurate with coals containing much disposable
hydrogen and iron pyrites than with those approximating to anthracite,
as the heat equivalent of the hydrogen in excess of that required to
form water with the oxygen of the coal is calculated as carbon, while
it is really about four times as great. Sulphur in iron pyrites also
acts as a reducing agent upon litharge, and increases the apparent
effect in a similar manner.
The evaporative power of a coal found by the above methods, and also
by calculating the separate calorific factors of the components as
determined by the chemical analysis, is always considerably above that
obtained by actual combustion under a steam boiler, as in the latter
case numerous sources of loss, such as imperfect combustion of gases,
loss of unburnt coal in cinders, &c., come into play, which cannot be
allowed for in laboratory experiments. It is usual, therefore, to
determine the value of a coal by the combustion of a weighed quantity
in the furnace of a boiler, and measuring the amount of water
evaporated by the heat developed.
In a research upon the heating power and other properties of coal for
naval use, carried out by the German admiralty, the results tabulated
below were obtained with coals from different localities.
+-----------------------+-----------+------------+-----------+------------------+
| | Slag left | Ashes in | Soot in | Water evaporated |
| | in Grate. | Ashpit. | Flues. | by 1 lb. of Coal.|
+-----------------------+-----------+------------+-----------+------------------+
| Westphalian gas coals | 0.33-6.42 | 2.83- 6.53 | 0.32-0.46 | 6.60-7.45 lb. |
| Do. bituminous coals | 0.98-9.10 | 1.97- 9.63 | 0.24-0.88 | 7.30-8.66 |
| Do. dry coals | 1.93-5.70 | 4.37-10.63 | 0.24-0.48 | 7.03-8.51 |
| Silesian coals | 0.92-1.30 | 3.15- 3.50 | 0.24-0.30 | 6.73-7.10 |
| Welsh steam coals | 1.20-4.07 | 4.07 | 0.32 | 8.41 |
| Newcastle coals | 1.92 | 2.57 | 0.35 | 7.28 |
+-----------------------+-----------+------------+-----------+------------------+
The heats of combustion of elements and compounds will be found in
most of the larger works on physical and chemical constants; a
convenient series is given in the _Annuaire du Bureau des Longitudes_,
appearing in alternate years. The following figures for the principal
fuel elements are taken from the issue for 1908; they are expressed in
gramme "calories" or heat units, signifying the weight of water in
grammes that can be raised 1 deg. C. in temperature by the combustion
of 1 gramme of the substance, when it is oxidized to the condition
shown in the second column:
+----------------+---------------------------------+-----------+
| Element. | Product of Combustion. | Calories. |
+----------------+---------------------------------+-----------+
| Hydrogen \ | Water, H2O, condensed to liquid | 34,500 |
| / | " as vapour | 29,650 |
| Carbon-- | | |
| Diamond | Carbon Dioxide, CO2 | 7,868 |
| Graphite | " " | 7,900 |
| Amorphous | " " | 8,133 |
| Silicon-- | | |
| Amorphous | Silicon Dioxide, SiO2 | 6,414 |
| Crystallized | " " | 6,570 |
| Phosphorus | Phosphoric pentoxide, P2O5 | 5,958 |
| Sulphur | Sulphur dioxide, SO2, gaseous | 2,165 |
+----------------+---------------------------------+-----------+
The results may also be expressed in terms of the atomic equivalent of
the combustible by multiplying the above values by the atomic weight
of the substance, 12 for carbon, 28 for silicon, &c.
In all fuels containing hydrogen the calorific value as found by the
calorimeter is higher than that obtainable under working conditions by
an amount equal to the latent heat of volatilization of water which
reappears as heat when the vapour is condensed, though under ordinary
conditions of use the vapour passes away uncondensed. This gives rise
to the distinction of higher and lower calorific values for such
substances, the latter being those generally used in practice. The
differences for the more important compound gaseous fuels are as
follows:--
Calorific Value.
Higher. Lower.
Acetylene, C2H2 11,920 11,500
Ethylene, C2H4 11,880 11,120
Methane, CH4 13,240 11,910
Carbon monoxide, CO 2,440 2,440
Caloric intensity.
The calorific intensity or pyrometric effect of any particular fuel
depends upon so many variable elements that it cannot be determined
except by actual experiment. The older method was to multiply the
weight of the products of combustion by their specific heats, but this
gave untrustworthy results as a rule, on account of two
circumstances--the great increase in specific heat at high
temperatures in compound gases such as water and carbon dioxide, and
their instability when heated to 1800 deg. or 2000 deg. At such
temperatures dissociation to a notable extent takes place, especially
with the latter substance, which is also readily reduced to carbon
monoxide when brought in contact with carbon at a red heat--a change
which is attended with a large heat absorption. This effect is higher
with soft kinds of carbon, such as charcoal or soft coke, than with
dense coke, gas retort carbon or graphite. These latter substances,
therefore, are used when an intense local heat is required, as for
example, in the Deville furnace, to which air is supplied under
pressure. Such a method is, however, only of very special application,
the ordinary method being to supply air to the fire in excess of that
required to burn the fuel to prevent the reduction of the carbon
dioxide. The volume of flame, however, is increased by inert gas, and
there is a proportionate diminution of the heating effect. Under the
most favourable conditions, when the air employed has been previously
raised to a high temperature and pressure, the highest attainable
flame temperature from carbonaceous fuel seems to be about 2100
deg.-2300 deg. C.; this is realized in the bright spots or "eyes" of
the tuyeres of blast furnaces.
Very much higher temperatures may be reached when the products of
combustion are not volatile, and the operation can be effected by
using the fuel and oxidizing agent in the proportions exactly
required for perfect combustion and intimately mixed. These
conditions are met in the "Thermit" process of Goldschmidt, where
finely divided aluminium is oxidized by the oxide of some similar
metal, such as iron, manganese or chromium, the reaction being started
by a primer of magnesium and barium peroxide. The reaction is so
rapidly effected that there is an enormous rise in temperature,
estimated to be 5400 deg. F. (3000 deg. C.), which is sufficient to
melt the most refractory metals, such as chromium. The slag consists
of alumina which crystallizes in the forms of corundum and ruby, and
is utilized as an abrasive under the name of corubin.
The chemical examination includes the determination of (1) moisture,
(2) ash, (3) coke, (4) volatile matter, (5) fixed carbon in coke, (6)
sulphur, (7) chlorine, (8) phosphorus. Moisture is determined by
noting the loss in weight when a sample is heated at 100 deg. for
about one hour. The ash is determined by heating a sample in a muffle
furnace until all the combustible matter has been burnt off. The ash,
which generally contains silica, oxides of the alkaline earths, ferric
oxide (which gives the ash a red colour), sulphur, &c., is analysed by
the ordinary gravimetric methods. The determination of coke is very
important on account of the conclusions concerning the nature of the
coal which it permits to be drawn. A sample is finely powdered and
placed in a covered porcelain crucible, which is surrounded by an
outer one, the space between them being packed with small coke. The
crucibles are heated in a wind furnace for 1 to 1-1/2 hours, then
allowed to cool, the inner crucible removed, and the coke weighed. The
coke may be (1) pulverulent, (2) slightly fritted, (3) spongy and
swelled, (4) compact. Pulverulent cokes indicate a non-caking
bituminous coal, rich in oxygen if the amount be below 60%, but if the
amount be very much less it generally indicates a lignite; if the
amount be above 80% it indicates an anthracite containing little
oxygen or hydrogen. A fritted coke indicates a slightly coking coal,
while the spongy appearance points to a highly coking coal which has
been partly fused in the furnace. A compact coke is yielded by good
coking coals, and is usually large in amount. The volatile matters are
determined as the loss of weight on coking less the amount of
moisture. The "fixed carbon" is the carbon retained in the coke, which
contains in addition the ash already determined. The fixed carbon is
therefore the difference between the coke and the ash, and may be
determined from these figures; or it may be determined directly by
burning off the coke in a muffle and noting the loss in weight.
Sulphur may be present as (1) organic sulphur, (2) as iron pyrites or
other sulphides, (3) as the sulphates of calcium, aluminium and other
metals; but the amount is generally so small that only the total
sulphur is determined. This is effected by heating a mixture of the
fuel with lime and sodium carbonate in a porcelain dish to redness in
a muffle until all the carbonaceous matter has been burnt off. The
residue, which contains the sulphur as calcium sulphate, is
transferred to a beaker containing water to which a little bromine has
been added. Hydrochloric acid is carefully added, the liquid filtered
and the residue washed. To the filtrate ammonia is added, and then
barium chloride, which precipitates the sulphur as barium sulphate.
Sulphur existing in the form of sulphates may be removed by washing a
sample with boiling water and determining the sulphuric acid in the
solution. The washed sample is then fused in the usual way to
determine the proportion of sulphur existing as iron pyrites. The
distinction between sulphur present as sulphate and sulphide is of
importance in the examination of coals intended for iron smelting, as
the sulphates of the earthy metals are reduced by the gases of the
furnace to sulphides, which pass into the slag without affecting the
quality of the iron produced, while the sulphur of the metallic
sulphides in the ash acts prejudicially upon the metal. Coals for
gas-making should contain little sulphur, as the gases produced in the
combustion are noxious and have very corrosive properties. Chlorine is
rarely determined, but when present in quantity it corrodes copper and
brass boiler tubes, with which consequently chlorine-bearing coals
cannot be used. The element is determined by fusing with soda lime in
a muffle, dissolving the residue in water and precipitating with
silver nitrate. Phosphorus is determined in the ash by fusing it with
a mixture of sodium and potassium carbonates, extracting the residue
with hydrochloric acid, and twice evaporating to dryness with the same
acid. The residue is dissolved in hydrochloric acid, a few drops of
ferric chloride added, and then ammonia in excess. The precipitate of
ferric phosphate is then treated as in the ordinary estimation of
phosphates. If it be necessary to determine the absolute amount of
carbon and hydrogen in a fuel, the dried sample is treated with copper
oxide as in the ordinary estimation of these elements in organic
compounds. (H. B.)
_Liquid Fuel._
Vegetable oil is not used for fuel except for laboratory purposes,
partly because its constituent parts are less adaptable for combustion
under the conditions necessary for steam-raising, but chiefly because of
the commercial difficulty of producing it with sufficient economy to
compete with mineral fuel either solid or liquid.
The use of petroleum as fuel had long been recognized as a scientific
possibility, and some attempts had been made to adopt it in practice
upon a commercial scale, but the insufficiency, and still more the
irregularity, of the supplies prevented it from coming into practical
use to any important extent until about 1898, when discoveries of oil
specially adapted by chemical composition for fuel purposes changed the
aspect of the situation. These discoveries of special oil were made
first in Borneo and later in Texas, and experience in treating the oils
from both localities has shown that while not less adapted to produce
kerosene or illuminating oil, they are better adapted to produce fuel
oil than either the Russian or the Pennsylvanian products. Texas oil did
not hold its place in the market for long, because the influx of water
into the wells lowered their yield, but discoveries of fuel oil in
Mexico have come later and will help to maintain the balance of the
world's supply, although this is still a mere fraction of the assured
supply of coal.
With regard to the chemical properties of petroleum, it is not necessary
to say more in the present place than that the lighter and more volatile
constituents, known commercially as naphtha and benzene, must be removed
by distillation in order to leave a residue composed principally of
hydrocarbons which, while containing the necessary carbon for
combustion, shall be sufficiently free from volatile qualities to avoid
premature ignition and consequent danger of explosion. Attempts have
been made to use crude oil for fuel purposes, and these have had some
success in the neighbourhood of the oil wells and under boilers of
unusually good ventilation both as regards their chimneys and the
surroundings of their stokeholds; but for reasons both of commerce and
of safety it is not desirable to use crude oil where some distillation
is possible. The more complete the process of distillation, and the
consequent removal of the volatile constituents, the higher the
flash-point, and the more turgid and viscous is the fuel resulting; and
if the process is carried to an extreme, the residue or fuel becomes
difficult to ignite by the ordinary process of spraying or atomizing
mechanically at the moment immediately preceding combustion. The
proportions which have been found to work efficiently in practice are as
follows:--
Carbon 88.00 %
Hydrogen 10.75 %
Oxygen 1.25 %
---------
Total 100
The standards of safety for liquid fuel as determined by flash-point are
not yet finally settled, and are changing from time to time. The British
admiralty require a flash-point of 270 deg. F., and to this high
standard, and the consequent viscosity of the fuel used by vessels in
the British fleet, may partly be attributed the low rate of combustion
that was at first found possible in them. The German admiralty have
fixed a flash-point of 187 deg. F., and have used oil of this standard
with perfect safety, and at the same time with much higher measure of
evaporative duty than has been attained in British war-vessels. In the
British mercantile marine Lloyd's Register has permitted fuel with a
flash-point as low as 150 deg. F. as a minimum, and no harm has
resulted. The British Board of Trade, the department of the government
which controls the safety of passenger vessels, has fixed a higher
standard upon the basis of a minimum of 185 deg. In the case of
locomotives the flash-point as a standard of safety is of less
importance than in the case of stationary or marine boilers, because the
storage is more open, and the ventilation, both of the storage tanks and
the boilers during combustion, much more perfect than in any other class
of steam-boilers.
The process of refining by distillation is also necessary to reduce two
impurities which greatly retard storage and combustion, i.e. water and
sulphur. Water is found in all crude petroleum as it issues from the
wells, and sulphur exists in important quantities in oil from the Texas
wells. Its removal was at first found very expensive, but there no
longer exists difficulty in this respect, and large quantities of
petroleum fuel practically free from sulphur are now regularly exported
from Texas to New York and to Europe.
Water mixed with fuel is in intimate mechanical relation, and frequently
so remains in considerable quantities even after the process of
distillation. It is in fact so thoroughly mixed as to form an emulsion.
The effect of feeding such a mixture into a furnace is extremely
injurious, because the water must be decomposed chemically into its
constituents, hydrogen and oxygen, thus absorbing a large quantity of
heat which would otherwise be utilized for evaporation. Water also
directly delays combustion by producing from the jet a long, dull, red
flame instead of a short bright, white flame, and the process of
combustion, which should take place by vaporization of the oil near the
furnace mouth, is postponed and transferred to the upper part of the
combustion-box, the tubes, and even the base of the chimney, producing
loss of heat and injury to the boiler structure. The most effective
means of ridding the fuel of this dangerous impurity is by heat and
settlement. The coefficients of expansion of water and oil by heat are
substantially different, and a moderate rise of temperature therefore
separates the particles and precipitates the water, which is easily
drawn off--leaving the oil available for use. The heating and
precipitation are usually performed upon a patented system of settling
tanks and heating apparatus known as the Flannery-Boyd system, which has
proved itself indispensable for the successful use at sea of petroleum
fuel containing any large proportion of water.
Progress of liquid fuel.
The laboratory and mechanical use of petroleum for fuel has already been
referred to, but it was not until the year 1870 that petroleum was
applied upon a wider and commercial scale. In the course of distillation
of Russian crude petroleum for the production of kerosene or lamp oil,
large quantities of refuse were produced--known by the Russian name of
_astatki_--and these were found an incumbrance and useless for any
commercial purpose. To a Russian oil-refiner gifted with mechanical
instinct and the genius for invention occurred the idea of utilizing the
waste product as fuel by spraying or atomizing it with steam, so that,
the thick and sluggish fluid being broken up into particles, the air
necessary for combustion could have free access to it. The earliest
apparatus for this purpose was a simple piece of gas-tube, into which
the thick oil was fed; by another connexion steam at high pressure was
admitted to an inner and smaller tube, and, the end of the tube nearest
to the furnace being open, the pressure of the steam blew the oil into
the furnace, and by its velocity broke it up into spray. The apparatus
worked with success from the first. Experience pointed out the proper
proportionate sizes for the inlets of steam and oil, the proper pressure
for the steam, and the proportionate sizes for the orifices of admission
to the furnaces, as well as the sizes of air-openings and best
arrangements of fire-bricks in the furnaces themselves; and what had
been a waste product now became a by-product of great value. Practically
all the steam power in South Russia, both for factories and navigation
of the inland seas and rivers, is now raised from _astatki_ fuel.
In the Far East, including Burma and parts of China and Japan, the use
of liquid fuel spread rapidly during the years 1899, 1900 and 1901,
owing entirely to the development of the Borneo oil-fields by the
enterprise of Sir Marcus Samuel and the large British corporation known
as the Shell Transport and Trading Company, of which he is the head.
This corporation has since amalgamated with the Royal Dutch Petroleum
Company controlling the extensive wells in Dutch Borneo, and together
they supply large quantities of liquid fuel for use in the Far East. In
the United States of America liquid fuel is not only used for
practically the whole of the manufacturing and locomotive purposes of
the state of Texas, but factories in New York, and a still larger number
in California, are now discarding the use of coal and adopting
petroleum, because it is more economical in its consumption and also
more easily handled in transit, and saves nearly all the labour of
stoking. So far the supplies for China and Japan have been exported from
Borneo, but the discoveries of new oil-fields in California, of a
character specially adapted for fuel, have encouraged the belief that it
may be possible to supply Chile and Peru and other South American
countries, where coal is extremely expensive, with Californian fuel; and
it has also found its way across the Pacific to Japan. There are
believed to be large deposits in West Africa, but in the meantime the
only sources of supply to those parts of Africa where manufacture is
progressing, i.e. South Africa and Egypt, are the oil-fields of Borneo
and Texas, from which the import has well begun, from Texas to
Alexandria via the Mediterranean, and from Borneo to Cape Town via
Singapore.
In England, notwithstanding the fact that there exist the finest
coal-fields in the world, there has been a surprising development of the
use of petroleum as fuel. The Great Eastern railway adapted 120
locomotive engines to its use, and these ran with regularity and success
both on express passenger and goods trains until the increase in price
due to short supply compelled a return to coal fuel. The London,
Brighton & South Coast railway also began the adaptation of some of
their locomotive engines, but discontinued the use of liquid fuel from
the same cause. Several large firms of contractors and cement
manufacturers, chiefly on the banks of the Thames, made the same
adaptations which proved mechanically successful, but were not continued
when the price of liquid fuel increased with the increased demand.
[Illustration: FIG. 1.--Holden Burner.]
Economy of liquid fuel.
The chief factors of economy are the greater calorific value of oil than
coal (about 16 lb. of water per lb. of oil fuel evaporated from a
temperature of 212 deg. F.), not only in laboratory practice, but in
actual use on a large scale, and the saving of labour both in transit
from the source of supply to the place of use and in the act of stoking
the furnaces. The use of cranes, hand labour with shovels, wagons and
locomotives, horses and carts, is unavoidable for the transit of coal;
and labour to trim the coal, to stoke it when under combustion, and to
handle the residual ashes, are all indispensable to steam-raising by
coal. On the other hand, a system of pipes and pumps, and a limited
quantity of skilled labour to manage them, is all that is necessary for
the transit and combustion of petroleum fuel; and it is certain that
even in England will be found places which, from topographical and other
circumstances, will use petroleum more economically than coal as fuel
for manufacturing purposes under reasonable conditions of price for the
fuel.
[Illustration: FIG. 2.--Rusden and Eeles Burner.]
The theoretical calorific value of oil fuel is more nearly realized in
practice than the theoretical calorific value of coal, because the
facilities for complete combustion, due to the artificial admixture of
the air by the atomizing process, are greater in the case of oil than
coal, and for this reason, among others, the practical evaporative
results are proportionately higher with liquid fuel. In some cases the
work done in a steam-engine by 2 tons of coal has been performed by 1
ton of oil fuel, but in others the proportions have been as 3 to 2, and
these latter can be safely relied on in practice as a minimum. This
saving, combined with the savings of labour and transit already
explained, will in the near future make the use of liquid fuel
compulsory, except in places so near to coal-fields that the cost of
coal becomes sufficiently low to counterbalance the savings in weight of
fuel consumed and in labour in handling it. In some locomotives on the
Great Eastern railway the consumption of oil and coal for the same
development of horse-power was as 17 lb. oil is to 35 lb. coal; all,
however, did not realize so high a result.
Liquid fuel in locomotives.
The mechanical apparatus for applying petroleum to steam-raising in
locomotives is very simple. The space in the tender usually occupied by
coal is closed up by steel-plating closely riveted and tested, so as to
form a storage tank. From this tank a feed-pipe is led to a burner of
the combined steam-and-oil type already indicated, and this burner is so
arranged as to enter a short distance inside the furnace mouth. The
ordinary fire-bars are covered with a thin layer of coal, which starts
the ignition in the first place, and the whole apparatus is ready for
work. The burner best adapted for locomotive practice is the Holden
Burner (fig. 1), which was used on the Great Eastern railway. The
steam-pipe is connected at A, the oil-pipe at B, and the hand-wheels C
and D are for the adjustment of the internal orifices according to the
rate of combustion required. The nozzle E is directed towards the
furnace, and the external ring FF, supplied by the small pipe G and the
by-pass valve H, projects a series of steam jets into the furnace,
independent of the injections of atomized fuel, and so induces an
artificial inrush of air for the promotion of combustion. This type of
burner has also been tried on stationary boilers and on board ship. It
works well, although the great consumption of steam by the supplementary
ring is a difficulty at sea, where the water lost by the consumption of
steam cannot easily be made up.
Liquid fuel at sea.
Although the application of the new fuel for land and locomotive boilers
has already been large, the practice at sea has been far more extensive.
The reason is chiefly to be found in the fact that although the sources
of supply are at a distance from Great Britain, yet they are in
countries to whose neighbourhood British steamships regularly trade, and
in which British naval squadrons are regularly stationed, so that the
advantages of adopting liquid fuel have been more immediate and the
economy more direct. The certainty of continuous supply of the fuel and
the wide distribution of storage stations have so altered the conditions
that the general adoption of the new fuel for marine purposes becomes a
matter of urgency for the statesman, the merchant and the engineer. None
of these can afford to neglect the new conditions, lest they be noted
and acted upon by their competitors. Storage for supply now exists at a
number of sea ports: London, Barrow, Southampton, Amsterdam, Copenhagen,
New Orleans, Savannah, New York, Philadelphia, Singapore, Hong Kong,
Madras, Colombo, Suez, Hamburg, Port Arthur, Rangoon, Calcutta, Bombay,
Alexandria, Bangkok, Saigon, Penang, Batavia, Surabaya, Amoy, Swatow,
Fuchow, Shanghai, Hankow, Sydney, Melbourne, Adelaide, Zanzibar,
Mombasa, Yokohama, Kobe and Nagasaki; also in South African and South
American ports.
[Illustration: FIG. 3.--Storage of Liquid Fuel on Oil-carrying Steamers
(Flannery-Boyd System).]
The British admiralty have undertaken experiments with liquid fuel at
sea, and at the same time investigations of the possibility of supply
from sources within the regions of the British empire. There is an
enormous supply of shale under the north-eastern counties of England,
but no oil that can be pumped--still less oil with a pressure above it
so as to "gush" like the wells in America--and the only sources of
liquid supply under the British flag appear to be in Burma and Trinidad.
The Borneo fields are not under British control, although developed
entirely by British capital. The Italian admiralty have fitted several
large warships with boiler apparatus to burn petroleum. The German
admiralty are regularly using liquid fuel on the China station. The
Dutch navy have fitted coal fuel and liquid fuel furnaces in
combination, so that the smaller powers required may be developed by
coal alone, and the larger powers by supplementing coal fuel with oil
fuel. The speeds of some vessels of the destroyer type have by this
means been accelerated nearly two knots.
[Illustration: FIG. 4.--Installation on ss. "Trochas."]
[Illustration: FIG. 5.--Details of Furnace, Meyer System.]
[Illustration: FIG. 6.--Details of Exterior Elongation of Furnace, Meyer
System.]
Advantages in warships.
The questions which govern the use of fuel in warships are more largely
those of strategy and fighting efficiency than economy of evaporation.
Indeed, the cost of constructing and maintaining in fighting efficiency
a modern warship is so great that the utmost use strategically must be
obtained from the vessel, and in this comparison the cost of fuel is
relatively so small an item that its increase or decrease may be
considered almost a negligible quantity. The desideratum in a warship is
to obtain the greatest fighting efficiency based on the thickest
armour, the heaviest and most numerous guns, the highest maximum speed,
and, last and not least, the greatest range of effective action based
upon the maximum supplies of fuel, provisions and other consumable
stores that the ship can carry. Now, if by changing the type of fuel it
be possible to reduce its weight by 30%, and to abolish the stokers, who
are usually more than half the ship's company, the weight saved will be
represented not merely by the fuel, but by the consumable stores
otherwise necessary for the stokers. Conversely, the radius of effective
action of the ship will be doubled as regards consumable stores if the
crew be halved, and will be increased by 50% if the same weight of fuel
be carried in the form of liquid instead of coal. In space the gain by
using oil fuel is still greater, and 36 cubic feet of oil as stored are
equal in practical calorific value to 67 cubic feet of coal according to
the allowance usual for ship's bunkering. On the other hand, coal has
been relied upon, when placed in the side bunkers of unarmoured ships,
as a protection against shot and shell, and this advantage, if it really
exists, could not be claimed in regard to liquid fuel.
Recent experiments in coaling warships at sea have not been very
successful, as the least bad weather has prevented the safe transmission
of coal bags from the collier to the ship. The same difficulty does not
exist for oil fuel, which has been pumped through flexible tubing from
one ship to the other even in comparatively rough weather.
Smokelessness, so important a feature of sea strategy, has not always
been attained by liquid fuel, but where the combustion is complete, by
reason of suitable furnace arrangements and careful management, there is
no smoke. The great drawback, however, to the use of liquid fuel in fast
small vessels is the confined space allotted to the boilers, such
confinement being unavoidable in view of the high power concentrated in
a small hull. The British admiralty's experiments, however, have gone
far to solve the problem, and the quantity of oil which can be consumed
by forced draught in confined boilers now more nearly equals the
quantity of coal consumed under similar conditions. All recent vessels
built for the British navy are so constructed that the spaces between
their double bottoms are oil-tight and capable of storing liquid fuel in
the tanks so formed. Most recent battleships and cruisers have also
liquid fuel furnace fittings, and in 1910 it already appeared probable
that the use of oil fuel in warships would rapidly develop.
In view of recent accusations of insufficiency of coal storage in
foreign naval depots, by reason of the allegation that coal so stored
quickly perishes, it is interesting to note that liquid fuel may be
stored in tanks for an indefinite time without any deterioration
whatever.
Advantages in merchant ships.
In the case of merchant steamers large progress has also been made. The
Shell Transport and Trading Company have twenty-one vessels successfully
navigating in all parts of the world and using liquid fuel. The
Hamburg-American Steamship Company have four large vessels similarly
fitted for oil fuel, which, however, differ in furnace arrangements, as
will be hereafter described, although using coal when the fluctuation of
the market renders that the more economical fuel. One of the large
American transatlantic lines is adopting liquid fuel, and French,
German, Danish and American mercantile vessels are also beginning to use
it in considerable amounts.
In the case of very large passenger steamers, such as those of 20 knots
and upwards in the Atlantic trade, the saving in cost of fuel is
trifling compared with the advantage arising from the greater weight and
space available for freight. Adopting a basis of 3 to 2 as between coal
consumption and oil consumption, there is an increase of 1000 tons of
dead weight cargo in even a medium-sized Atlantic steamer, and a
collateral gain of about 100,000 cub. ft. of measurement cargo, by
reason of the ordinary bunkers being left quite free, and the oil being
stored in the double bottom spaces hitherto unutilized except for the
purpose of water ballast. The cleanliness and saving of time from
bunkering by the use of oil fuel is also an important factor in
passenger ships, whilst considerable additional speed is obtainable. The
cost of the installation, however, is very considerable, as it includes
not only burners and pipes for the furnaces, but also the construction
of oil-tight tanks, with pumps and numerous valves and pipe connexions.
[Illustration: FIG. 7.--Furnace on ss. "Ferdinand Laeisz." A, it is
proposed to do away with this ring of brickwork as being useless; B, it
is proposed to fill this space up, thus continuing lining of furnace to
combustion chamber, and also to fit protection bricks in way of saddle
plate.]
[Illustration: FIG. 8.--Fuel Tanks, &c., of ss. "Murex."]
[Illustration: FIG. 9.--Furnace Gear of ss. "Murex."]
Fig. 2 shows a burner of Rusden and Eeles' patent as generally used on
board ships for the purpose of injecting the oil. A is a movable cap
holding the packing B, which renders the annular spindle M oil and
steam tight. E is the outer casing containing the steam jacket from
which the steam, after being fed through the steam-supply pipe G,
passes into the annular space surrounding the spindle P. It will be
seen that if the spindle P be travelled inwards by turning the handle
N, the orifice at the nozzle RR will be opened so as to allow the
steam to flow out radially. If at the same time the annular spindle M
be drawn inwards by revolving the handle L, the oil which passes
through the supply pipe F will also have emission at RR, and, coming
in contact with the outflowing steam, will be pulverized and sprayed
into the furnace. Fig. 3 is a profile and plan of a steamer adapted
for carrying oil in bulk, and showing all the storage arrangements for
handling liquid fuel. Fig. 4 shows the interior arrangement of the
boiler furnace of the steamship "Trocas." A is broken fire-brick
resting on the ordinary fire-bars, B is a brick bridge, C a casing of
fire-brick intended to protect the riveted seam immediately above it
from the direct impact of the flame, and D is a lining of fire-brick
at the back of the combustion-box, also intended to protect the
plating from the direct impact of the petroleum flame. The arrangement
of the furnace on the Meyer system is shown in fig. 5, where E is an
annular projection built at the mouth of the furnace, and BB are
spiral passages for heating the air before it passes into the furnace.
Fig. 6 shows the rings CC and details of the casting which forms the
projection or exterior elongation of the furnace. The brickwork
arrangement adopted for the double-ended boilers on the
Hamburg-American Steamship Company's "Ferdinand Laeisz" is represented
in fig. 7. The whole furnace is lined with fire-brick, and the burner
is mounted upon a circular disk plate which covers the mouth of the
furnace. The oil is injected not by steam pulverization, but by
pressure due to a steam-pump. The oil is heated to about 60 deg. C.
before entering the pump, and further heated to 90 deg. C. after
leaving the pump. It is then filtered, and passes to the furnace
injector C at about 30-lb. pressure; and its passage through this
injector and the spiral passages of which it consists pulverizes the
oil into spray, in which form it readily ignites on reaching the
interior of the furnace. The injector is on the Korting principle,
that is, it atomizes by fracture of the liquid oil arising from its
own momentum under pressure. The advantage of this system as compared
with the steam-jet system is the saving of fresh water, the
abstraction of which is so injurious to the boiler by the formation of
scale.
[Illustration: FIG. 10.--Section through Furnace of ss. "Murex."]
The general arrangement of the fuel tanks and filling pipes on the ss.
"Murex" is shown in fig. 8; and fig. 9 represents the furnace gear of
the same vessel, A being the steam-pipe, B the oil-pipe, C the
injector, D the swivel upon which the injector is hung so that it may
be swung clear of the furnace, E the fire-door, and F the handle for
adjusting the injector. In fig. 10, which represents a section of the
furnace, H is a fire-brick pier and K a fire-brick baffling bridge.
It is found in practice that to leave out the fire-bars ordinarily
used for coal produces a better result with liquid fuel than the
alternative system of keeping them in place and protecting them by a
layer of broken fire-brick.
Boilers fitted upon all the above systems have been run for thousands
of miles without trouble. In new construction it is desirable to give
larger combustion chambers and longer and narrower boiler tubes than
in the case of boilers intended for the combustion of coal alone.
(F. F.*)
_Gaseous Fuel._
Strictly speaking, much, and sometimes even most, of the heating
effected by solid or liquid fuel is actually performed by the gases
given off during the combustion. We speak, however, of gaseous fuel only
in those cases where we supply a combustible gas from the outset, or
where we produce from ordinary solid (or liquid) fuel in one place a
stream of combustible gas which is burned in another place, more or less
distant from that where it has been generated.
The various descriptions of gaseous fuel employed in practice may be
classified under the following heads:
I. Natural Gas.
II. Combustible Gases obtained as by-products in various technical
operations.
III. Coal Gas (Illuminating Gas).
IV. Combustible Gases obtained by the partial combustion of coal, &c.
I. _Natural Gas._--From time immemorial it has been known that in some
parts of the Caucasus and of China large quantities of gases issue from
the soil, sometimes under water, which can be lighted and burn with a
luminous flame. The "eternal fires" of Baku belong to this class. In
coal-mines frequently similar streams of gas issue from the coal; these
are called "blowers," and when they are of somewhat regular occurrence
are sometimes conducted away in pipes and used for underground lighting.
As a regular source of heating power, however, natural gas is employed
only in some parts of the United States, especially in Pennsylvania,
Kansas, Ohio and West Virginia, where it always occurs in the
neighbourhood of coal and petroleum fields. The first public mention of
it was made in 1775, but it was not till 1821 that it was turned to use
at Fredonia, N.Y. In Pennsylvania natural gas was discovered in 1859,
but at first very little use was made of it. Its industrial employment
dates only from 1874, and became of great importance about ten years
later. Nobody ever doubted that the gas found in these localities was an
accumulation of many ages and that, being tapped by thousands of
bore-holes, it must rapidly come to an end. This assumption was
strengthened by the fact that the "gas-wells," which at first gave out
the gas at a pressure of 700 or 800, sometimes even of 1400 lb. per sq.
in., gradually showed a more and more diminishing pressure and many of
them ceased to work altogether. About the year 1890 the belief was
fairly general that the stock of natural gas would soon be entirely
exhausted. Indeed, the value of the annual production of natural gas in
the United States, computed as its equivalent of coal, was then
estimated at twenty-one million dollars, in 1895 at twelve millions, in
1899 at eleven and a half millions. But the output rose again to a value
of twenty-seven millions in 1901, and to fifty million dollars in 1907.
Mostly the gas, derived from upwards of 10,000 gas-wells, is now
artificially compressed to a pressure of 300 or 400 lb. per sq. in. by
means of steam-power or gas motors, fed by the gas itself, and is
conveyed over great distances in iron pipes, from 9 or 10 to 36 in. in
diameter. In 1904 nearly 30,000 m. of pipe lines were in operation. In
1907 the quantity of natural gas consumed in the United States (nearly
half of which was in Pennsylvania) was 400,000 million cub. ft., or
nearly 3 cub. m. Canada (Ontario) also produces some natural gas,
reaching a maximum of about $746,000 in 1907.
The principal constituent of natural gas is always methane, CH4, of
which it contains from 68.4 to 94.0% by volume. Those gases which
contain less methane contain all the more hydrogen, viz. 2.9 to 29.8%.
There is also some ethylene, ethane and carbon monoxide, rarely
exceeding 2 or 3%. The quantity of incombustible gases--oxygen, carbon
dioxide, nitrogen--ranges from mere traces to about 5%. The density is
from 0.45 to 0.55. The heating power of 1000 cub. ft. of natural gas is
equal to from 80 to 120 lb., on the average 100 lb., of good coal, but
it is really worth much more than this proportion would indicate, as it
burns completely, without smoke or ashes, and without requiring any
manual labour. It is employed for all domestic and for most industrial
purposes.
The origin of natural gas is not properly understood, even now. The most
natural assumption is, of course, that its formation is connected with
that of the petroleum always found in the same neighbourhood, the latter
principally consisting of the higher-boiling aliphatic hydrocarbons of
the methane series. But whence do they both come? Some bring them into
connexion with the formation of coal, others with the decomposition of
animal remains, others with that of _diatomaceae_, &c., and even an
inorganic origin of both petroleum and natural gas has been assumed by
chemists of the rank of D.I. Mendeleeff and H. Moissan.
II. _Gases obtained as By-products._--There are two important cases in
which gaseous by-products are utilized as fuel; both are intimately
connected with the manufacture of iron, but in a very different way, and
the gases are of very different composition.
(a) _Blast-furnace Gases._--The gases issuing from the mouths of
blast-furnaces (see IRON AND STEEL) were first utilized in 1837 by Faber
du Faur, at Wasseralfingen. Their use became more extensive after 1860,
and practically universal after 1870. The volume of gas given off per
ton of iron made is about 158,000 cub. ft. Its percentage composition by
volume is:
Carbon monoxide 21.6 to 29.0, mostly about 26%
Hydrogen 1.8 " 6.3, " " 3%
Methane 0.1 " 0.8, " " 0.5%
Carbon dioxide 6 " 12, " " 9.5%
Nitrogen 51 " 60, " " 56 %
Steam 5 " 12, " " 5 %
-----
100 %
There is always a large amount of mechanically suspended flue-dust in
this gas. It is practically equal to a poor producer-gas (see below),
and is everywhere used, first for heating the blast in Cowper stoves or
similar apparatus, and secondly for raising all the steam required for
the operation of the blast-furnace, that is, for driving the
blowing-engines, hoisting the materials, &c. Where the iron ore is
roasted previously to being fed into the furnace, this can also be done
by this gas, but in some cases the waste in using it is so great that
there is not enough left for the last purpose. The calorific power of
this gas per cubic foot is from 80 to 120 B.Th.U.
Since about 1900 a great advance has been made in this field. Instead of
burning the blast-furnace gas under steam boilers and employing the
steam for producing mechanical energy, the gas is directly burned in
gas-motors on the explosion principle. Thus upwards of three times the
mechanical energy is obtained in comparison with the indirect way
through the steam boiler. After all the power required for the
operations of the blast-furnace has been supplied, there is a surplus of
from 10 to 20 h.p. for each ton of pig-iron made, which may be applied
to any other purpose.
(b) _Coke-oven Gases._--Where the coking of coal is performed in the old
beehive ovens or similar apparatus the gas issuing at the mouth of the
ovens is lost. The attempts at utilizing the gases in such cases have
not been very successful. It is quite different where coke is
manufactured in the same way as illuminating gas, viz. by the
destructive distillation of coal in closed apparatus (retorts), heated
from the outside. This industry, which is described in detail in G.
Lunge's _Coal-Tar and Ammonia_ (4th ed., 1909), originated in France,
but has spread far more in Germany, where more than half of the coke
produced is made by it; in the United Kingdom and the United States its
progress has been much slower, but there also it has long been
recognized as the only proper method. The output of coke is increased by
about 15% in comparison with the beehive ovens, as the heat required for
the process of distillation is not produced by burning part of the coal
itself (as in the beehive ovens), but by burning part of the gas. The
quality of the coke for iron-making is quite as good as that of beehive
coke, although it differs from it in appearance. Moreover, the gases can
be made to yield their ammonia, their tar, and even their benzene
vapours, the value of which products sometimes exceeds that of the coke
itself. And after all this there is still an excess of gas available for
any other purpose.
As the principle of distilling the coal is just the same, whether the
object is the manufacture of coal gas proper or of coke as the main
product, although there is much difference in the details of the
manufacture, it follows that the quality of the gas is very similar in
both cases, so far as its heating value is concerned. Of course this
heating value is less where the benzene has been extracted from
coke-oven gas, since this compound is the richest heat-producer in the
gas. This is, however, of minor importance in the present case, as there
is only about 1% benzene in these gases.
The composition of coke-oven gases, after the extraction of the ammonia
and tar, is about 53% hydrogen, 36% methane, 6% carbon monoxide, 2%
ethylene and benzene, 0.5% sulphuretted hydrogen, 1.5% carbon dioxide,
1% nitrogen.
III. _Coal Gas (Illuminating Gas)._--Although ordinary coal gas is
primarily manufactured for illuminating purposes, it is also extensively
used for cooking, frequently also for heating domestic rooms, baths,
&c., and to some extent also for industrial operations on a small scale,
where cleanliness and exact regulation of the work are of particular
importance. In chemical laboratories it is preferred to every other kind
of fuel wherever it is available. The manufacture of coal gas being
described elsewhere in this work (see GAS, S _Manufacture_), we need
here only point out that it is obtained by heating bituminous coal in
fireclay retorts and purifying the products of this destructive
distillation by cooling, washing and other operations. The residual gas,
the ordinary composition of which is given in the table below, amounts
to about 10,000 cub. ft. for a ton of coal, and represents about 21% of
its original heating value, 56.5% being left in the coke, 5.5% in the
tar and 17% being lost. As we must deduct from the coke that quantity
which is required for the heating of the retorts, and which, even when
good gas producers are employed, amounts to 12% of the weight of the
coal, or 10% of its heat value, the total loss of heat rises to 27%.
Taking, further, into account the cost of labour, the wear and tear, and
the capital interest on the plant, coal gas must always be an expensive
fuel in comparison with coal itself, and cannot be thought of as a
general substitute for the latter. But in many cases the greater expense
of the coal gas is more than compensated by its easy distribution, the
facility and cleanliness of its application, the general freedom from
the mechanical loss, unavoidable in the case of coal fires, the
prevention of black smoke and so forth. The following table shows the
average composition of coal gas by volume and weight, together with the
heat developed by its single constituents, the latter being expressed in
kilogram-calories per cub. metre (0.252 kilogram-calories = 1 British
heat unit; 1 cub. metre = 35.3 cub. ft.; therefore 0.1123 calories per
cub. metre = 1 British heat unit per cub. foot).
+----------------------+----------+----------+------------+--------------+------------+
| | | | Heat-value | Heat-value | Heat-value |
| Constituents. | Volume | Weight | per Cubic | per Quantity | per cent. |
| | per cent.| per cent.| Metre | contained in | of Total. |
| | | | Calories. | 1 Cub. Met. | |
+----------------------+----------+----------+-----------+---------------+------------+
| Hydrogen, H2 | 47 | 7.4 | 2,582 | 1213 | 22.8 |
| Methane, CH4 | 34 | 42.8 | 8,524 | 2898 | 54.5 |
| Carbon monoxide, CO | 9 | 19.9 | 3,043 | 273 | 5.1 |
| Benzene vapour, C6H6 | 1.2 | 7.4 | 33,815 | 405 | 7.7 |
| Ethylene, C2H4 | 3.8 | 8.4 | 13,960 | 530 | 9.9 |
| Carbon dioxide, CO2 | 2.5 | 8.6 | .. | .. | .. |
| Nitrogen, N2 | 2.5 | 5.5 | .. | .. | .. |
| +----------+----------+------------+--------------+------------+
| Total | 100.0 | 100.0 | .. | 5319 | 100.0 |
+----------------------+----------+----------+------------+--------------+------------+
One cubic metre of such gas weighs 568 grammes. _Rich gas_, or gas made
by the destructive distillation of certain bituminous schists, of oil,
&c., contains much more of the heavy hydrocarbons, and its heat-value is
therefore much higher than the above. The carburetted water gas, very
generally made in America, and sometimes employed in England for mixing
with coal gas, is of varying composition; its heat-value is generally
rather less than that of coal gas (see below).
IV. _Combustible Gases produced by the Partial Combustion of Coal,
&c._--These form by far the most important kind of gaseous fuel. When
coal is submitted to destructive distillation to produce the
illuminating gas described in the preceding paragraph, only a
comparatively small proportion of the heating value of the coal (say, a
sixth or at most a fifth part) is obtained in the shape of gaseous fuel,
by far the greater proportion remaining behind in the shape of coke.
An entirely different class of gaseous fuels comprises those produced by
the incomplete combustion of the total carbon contained in the raw
material, where the result is a mixture of gases which, being capable of
combining with more oxygen, can be burnt and employed for heating
purposes. Apart from some descriptions of waste gases belonging to this
class (of which the most notable are those from blast-furnaces), we must
distinguish two ways of producing such gaseous fuels entirely different
in principle, though sometimes combined in one operation. The incomplete
combustion of carbon may be brought about by means of atmospheric
oxygen, by means of water, or by a simultaneous combination of these two
actions. In the first case the chemical reaction is
C + O= CO (a);
the nitrogen accompanying the oxygen in the atmospheric air necessarily
remains mixed with carbon monoxide, and the resulting gases, which
always contain some carbon dioxide, some products of the destructive
distillation of the coal, &c., are known as _producer gas_ or _Siemens
gas_. In the second case the chemical reaction is mainly
C + H2O = CO + H2 (b);
that is to say, the carbon is converted into monoxide and the hydrogen
is set free. As both of these substances can combine with oxygen, and as
there is no atmospheric nitrogen to deal with, the resulting gas (_water
gas_) is, apart from a few impurities, entirely combustible. Another
kind of water gas is formed by the reaction
C + 2H2O = CO2 + 2H2 (c),
but this reaction, which converts all the carbon into the incombustible
form of CO2, is considered as an unwelcome, although never entirely
avoidable, concomitant of (b).
The reaction by which water gas is produced being endothermic (as we
shall see), this gas cannot be obtained except by introducing the
balance of energy in another manner. This might be done by heating the
apparatus from without, but as this method would be uneconomical, the
process is carried out by alternating the endothermic production of
water gas with the exothermic combustion of carbon by atmospheric air.
Pure water gas is not, therefore, made by a continuous process, but
alternates with the production of other gases, combustible or not. But
instead of constantly interrupting the process in this way, a continuous
operation may be secured by simultaneously carrying on both the
reactions (a) and (b) in such proportions that the heat generated by (a)
at least equals the heat absorbed by (b). For this purpose the apparatus
is fed at the same time with atmospheric air and with a certain quantity
of steam, preferably in a superheated state. Gaseous mixtures of this
kind have been made, more or less intentionally, for a long time past.
One of the best known of them, intended less for the purpose of serving
as ordinary fuel than for that of driving machinery, is the Dowson gas.
An advantage common to all kinds of gaseous fuel, which indeed forms the
principal reason why it is intentionally produced from solid fuel, in
spite of inevitable losses in the course of the operation, is the
following. The combustion of solid fuel (coal, &c.) cannot be carried on
with the theoretically necessary quantity of atmospheric air, but
requires a considerable excess of the latter, at least 50%, sometimes
100% and more. This is best seen from the analyses of smoke gases. If
all the oxygen of the air were converted into CO2 and H2O, the amount of
CO2 in the smoke gases should be in the case of pure carbon nearly 21
volumes %, as carbon dioxide occupies the same volume as oxygen; while
ordinary coal, where the hydrogen takes up a certain quantity of oxygen
as well, should show about 18.5% CO2. But the best smoke gases of steam
boilers show only 12 or 13%, much more frequently only 10% CO2, and
gases from reverberatory furnaces often show less than 5%. This means
that the volume of the smoke gases escaping into the air is from 1-1/2
to 2 times (in the case of high-temperature operations often 4 times)
greater than the theoretical minimum; and as these gases always carry
off a considerable quantity of heat, the loss of heat is all the greater
the less complete is the utilization of the oxygen and the higher the
temperature of the operation. This explains why, in the case of the
best-constructed steam-boiler fires provided with heat economizers,
where the smoke gases are deprived of most of their heat, the proportion
of the heat value of the fuel actually utilized may rise to 70 or even
75%, while in some metallurgical operations, in glass-making and similar
cases, it may be below 5%.
One way of overcoming this difficulty to a certain extent is to reduce
the solid fuel to a very fine powder, which can be intimately mixed with
the air so that the consumption of the latter is only very slightly in
excess of the theoretical quantity; but this process, which has been
only recently introduced on a somewhat extended scale, involves much
additional expense and trouble, and cannot as yet be considered a real
success. Generally, too, it is far less easily applied than gaseous
fuel. The latter can be readily and intimately mixed with the exact
quantity of air that is required and distributed in any suitable way,
and much of the waste heat can be utilized for a preliminary heating of
the air and the gas to be burned by means of "recuperators."
We shall now describe the principal classes of gaseous fuel, produced by
the partial combustion of coal.
A. _Producer Gas, Siemens Gas._--As we have seen above, this gas is made
by the incomplete combustion of fuel. The materials generally employed
for its production are anthracite, coke or other fuels which are not
liable to cake during the operation, and thus stop the draught or
otherwise disturb the process, but by special measures also bituminous
coal, lignite, peat and other fuel may be utilized for gas producers.
The fuel is arranged in a deep layer, generally from 4 ft. up to 10 ft.,
and the air is introduced from below, either by natural draught or by
means of a blast, and either by a grate or only by a slit in the wall of
the "gas producer." Even if the primary action taking place at the
entrance of the air consisted in the complete combustion of the carbon
to dioxide, CO2, the latter, in rising through the high column of
incandescent fuel, must be reduced to monoxide: CO2 + C = 2CO. But as
the temperature in the producer rises rather high, and as in ordinary
circumstances the action of oxygen on carbon above 1000 deg. C. consists
almost entirely in the direct formation of CO, we may regard this
compound as primarily formed in the hotter parts of the gas-producer. It
is true that ordinary producer gas always contains more or less CO2, but
this may be formed higher up by air entering through leakages in the
apparatus. If we ignore the hydrogen contained in the fuel, the
theoretical composition of producer gas would be 33.3% CO and 66.7% N,
both by volume and weight. Its weight per cubic metre is 1.251 grammes,
and its heat value 1013 calories per cubic metre, or less than one-fifth
of the heat-value of coal gas. Practically, however, producer gas
contains a small percentage of gases, increasing its heat-value, like
hydrogen, methane, &c., but on the other hand it is never free from
carbon dioxide to the extent of from 2 to 8%. Its heat-value may
therefore range between 800 and 1100 calories per cubic metre. Even when
taking as the basis of our calculation a theoretical gas of 33.3% CO, we
find that there is a great loss of heat-value in the manufacture of this
gas. Thermochemistry teaches us that the reaction C + O develops 29.5%
of the heat produced by the complete oxidation of C to CO2, thus leaving
only 70.5% for the stage CO + O = CO2. If, therefore, the gas given off
in the producer is allowed to cool down to ordinary temperature, nearly
30% of the heat-value of the coal is lost by radiation. If, however, the
gas producer is built in close proximity to the place where the
combustion takes place, so that the gas does not lose very much of its
heat, the loss is correspondingly less. Even then there is no reason why
this mode of burning the fuel, i.e. first with "primary air" in the
producer (C + O = CO), then with "secondary air" in the furnace (CO + O
= CO2), should be preferred to the direct complete burning of the fuel
on a grate, unless the above-mentioned advantage is secured, viz.
reduction of the smoke gases to a minimum by confining the supply of air
as nearly as possible to that required for the formation of CO2, which
is only possible by producing an intimate mixture of the producer gas
with the secondary air. The advantage in question is not very great
where the heat of the smoke gases can be very fully utilized, e.g. in
well-constructed steam boilers, salt-pans and the like, and as a matter
of fact gas producers have not found much use in such cases. But a very
great advantage is attained in high-temperature operations, where the
smoke gases escape very hot, and where it is on that account
all-important to confine their quantity to a minimum.
It is precisely in these cases that another requirement frequently comes
in, viz. the production at a given point of a higher temperature than is
easily attained by ordinary fires. Gas-firing lends itself very well to
this end, as it is easily combined with a preliminary heating up of the
air, and even of the gas itself, by means of "recuperators." The
original and best-known form of these, due to Siemens Brothers, consists
of two brick chambers filled with loosely stacked fire-bricks in such
manner that any gases passed through the chambers must seek their way
through the interstices left between the bricks, by which means a
thorough interchange of temperature takes place. The smoke gases,
instead of escaping directly into the atmosphere, are made to pass
through one of these chambers, giving up part of their heat to the
brickwork. After a certain time the draught is changed by means of
valves, the smoke gases are passed through another chamber, and the cold
air intended to feed the combustion is made to pass through the first
chamber, where it takes up heat from the white-hot bricks, and is thus
heated up to a bright red heat until the chamber is cooled down too far,
when the draughts are again reversed. Sometimes the producer gas itself
is heated up in this manner (especially when it has been cooled down by
travelling a long distance); in that case four recuperator chambers must
be provided instead of two. Another class of recuperators is not founded
on the alternating system, but acts continuously; the smoke gases travel
always in the same direction in flues contiguous to other flues or pipes
in which the air flows in the opposite direction, an interchange of heat
taking place through the walls of the flues or pipes. Here the surface
of contact must be made very large if a good effect is to be produced.
In both cases not merely is a saving effected of all the calories which
are abstracted by the cold air from the recuperator, but as less fuel
has to be burned to get a given effect, the quantity of smoke gas is
reduced. For details and other producer gases, see GAS, II. _For Fuel
and Power._
Gas-firing in the manner just described can be brought about by very
simple means, viz. by lowering the fire-grate of an ordinary fire-place
to at least 4 ft. below the fire-bridge, and by introducing the air
partly below the grate and partly behind the fire-place, at or near the
point where the greatest heat is required. Usually, however, more
elaborate apparatus is employed, some of which we shall describe below.
Gas-firing has now become universal in some of the most important
industries and nearly so in others. The present extension of
steel-making and other branches of metallurgy is intimately connected
with this system, as is the modern method of glass-making, of heating
coal gas retorts and so forth.
The composition of producer gas differs considerably, principally
according to the material from which it is made. Analyses of ordinary
producer gas (not such as falls under the heading of "semi-water gas,"
see _sub_ C) by volume show 22 to 33% CO, 1 to 7% CO2, 0.5 to 2% H2, 0.5
to 3% hydrocarbons, and 64 to 68% N2.
B. _Water Gas._--The reaction of steam on highly heated carbonaceous
matter was first observed by Felice Fontana in 1780. This was four years
before Henry Cavendish isolated hydrogen from water, and thirteen years
before William Murdoch made illuminating gas by the distillation of
coal, so that it was no wonder that Fontana's laboratory work was soon
forgotten. Nor had the use of carburetted water gas, as introduced by
Donovan in 1830 for illuminating purposes, more than a very short life.
More important is the fact that during nine years the illumination of
the town of Narbonne was carried on by incandescent platinum wire,
heated by water gas, where also internally heated generators were for
the first time regularly employed. The Narbonne process was abandoned in
1865, and for some time no real progress was made in this field in
Europe. But in America, T.S.C. Lowe, Strong, Tessie du Motay and others
took up the matter, the first permanent success being obtained by the
introduction (1873) of Lowe's system at Phoenixville, Pa. In the United
States the abundance of anthracite, as well as of petroleum naphtha,
adapted for carburetting the gas, secures a great commercial advantage
to this kind of illuminant over coal gas, so that now three-fourths of
all American gas-works employ carburetted water gas. In Europe the
progress of this industry was naturally much less rapid, but here also
since 1882, when the apparatus of Lowe and Dwight was introduced in the
town of Essen, great improvements have been worked out, principally by
E. Blass, and by these improvements water gas obtained a firm footing
also for certain heating purposes. The American process for making
carburetted water gas, as an auxiliary to ordinary coal gas, was first
introduced by the London Gas Light and Coke Company on a large scale in
1890.
Water gas in its original state is called "blue gas," because it burns
with a blue, non-luminous flame, which produces a very high temperature.
According to the equation C + H2O = CO + H2, this gas consists
theoretically of equal volumes of carbon monoxide and hydrogen. We shall
presently see why it is impossible to avoid the presence of a little
carbon dioxide and other gases, but we shall for the moment treat of
water gas as if it were composed according to the above equation. The
reaction C + H2O = CO + H2 is endothermic, that is, its thermal value is
negative. One gram-molecule of carbon produces 97 great calories (1
great calorie or kilogram-calorie = 1000 gram-calories) when burning to
CO2, and this is of course the maximum effect obtainable from this
source. If the same gram-molecule of carbon is used for making water
gas, that is, CO + H2, the heat produced by the combustion of the
product is 68.4 + 57.6 = 126 great calories, an apparent surplus of 29
calories, which cannot be got out of nothing. This is made evident by
another consideration. In the above reaction C is not burned to CO2, but
to CO, a reaction which produces 28.6 calories per gram-molecule. But as
the oxygen is furnished from water, which must first be decomposed by
the expenditure of energy, we must introduce this amount, 68.5 calories
in the case of liquid water, or 57.6 calories in the case of steam, as a
negative quantity, and the difference, viz. + 28.6 - 57.6 = 29 great
calories, represents the amount of heat to be expended from another
source in order to bring about the reaction of one gram-molecule of
carbon on one gram-molecule of H2O in the shape of steam. This explains
why steam directed upon incandescent coal will produce water gas only
for a very short time: even a large mass of coal will quickly be cooled
down so much that at first a gas of different composition is formed and
soon the process will cease altogether. We can avoid this result by
carrying on the process in a retort heated from without by an ordinary
coal fire, and all the early water gas apparatus was constructed in this
way; but such a method is very uneconomical, and was long ago replaced
by a process first patented by J. and T.N. Kirkham in 1854, and very
much improved by successive inventors. This process consists in
conducting the operation in an upright brick shaft, charged with
anthracite, coke or other suitable fuel. This shaft resembles an
ordinary gas producer, but it differs in being worked, not in a
continuous manner, which, as shown above, would be impossible, but by
alternately blowing air and steam through the coal for periods of a few
minutes each. During the first phase, when carbon is burned by
atmospheric oxygen, and thereby heat is produced, this heat, or rather
that part of it which is not carried away by radiation and by the
products of combustion on leaving the apparatus, is employed in raising
the temperature of the remaining mass of fuel, and is thus available for
the second phase, in which the reaction (b) C + H2O = CO + H2 goes on
with the abstraction of a corresponding amount of heat from the
incandescent fuel, so that the latter rapidly cools down, and the
process must be reversed by blowing in air and so forth. The formation
of exactly equal volumes of carbon monoxide and hydrogen goes on only at
temperatures over 1200 deg. C., that is, for a very few minutes. Even at
1100 deg. C. a little CO2 can be proved to exist in the gas, and at 900
deg. its proportion becomes too high to allow the process to go on.
About 650 deg. C. the CO has fallen to a minimum, and the reaction is
now essentially (c) C + 2H2O = CO2 + 2H2; soon after the temperature of
the mass will have fallen to such a low point that the steam passes
through it without any perceptible action. The gas produced by reaction
(c) contains only two-thirds of combustible matter, and is on that
account less valuable than proper water gas formed by reaction (b);
moreover, it requires the generation of twice the amount of steam, and
its presence is all the less desirable since it must soon lead to a
total cessation of the process. In ordinary circumstances it is evident
that the more steam is blown in during a unit of time, the sooner
reaction (c) will set in; on the other hand, the more heat has been
accumulated in the producer the longer can the blowing-in of steam be
continued.
The process of making water gas consequently comprises two alternating
operations, viz. first "blowing-up" by means of a current of air, by
which the heat of the mass of fuel is raised to about 1200 deg. C.; and,
secondly "steaming," by injecting a current of (preferably superheated)
steam until the temperature of the fuel had fallen to about 900 deg. C.,
and too much carbon dioxide appears in the product. During the steaming
the gas is carried off by a special conduit into a scrubber, where the
dust mechanically carried away in the current is washed out, and the gas
is at the same time cooled down nearly to the ordinary temperature. It
is generally stored in a gas-holder, from which it is conducted away as
required. It is never quite free from nitrogen, as the producer at the
beginning of steaming contains much of this gas, together with CO or
CO2. The proportion of hydrogen may exceed 50%, in consequence of
reaction (c) setting in at the close of the steaming. Ordinary "blue"
water gas, if, as usual, made from coke or anthracite, contains 48-52%
H2, 40-41% CO, 1-5% CO2, 4-5% N2, and traces of hydrocarbons, especially
methane. If made from bituminous coal, it contains more of the latter.
If "carburetted" (a process which increases its volume 50% and more) by
the vapours from superheated petroleum naphtha, the proportion of CO
ranges about 25%, with about as much methane, and from 10 to 15% of
"illuminants" (heavy hydrocarbons). The latter, of course, greatly
enhance the fuel-value of the gas. Pure water gas would possess the
following fuel-value per cubic metre:
0.5 cub. met. H2 = 1291 calories
0.5 " " CO = 1522 "
----
2813 "
Ordinary "blue" water gas has a fuel-value of at least 2500 calories.
Carburetted water gas, which varies very much in its percentage of
hydrocarbons, sometimes reaches nearly the heat-value of coal gas, but
such gas is only in exceptional cases used for heating purposes.
We must now turn to the "blowing-up" stage of the process. Until
recently it was assumed that during this stage the combustion of carbon
cannot be carried on beyond the formation of carbon monoxide, for as the
gas-producer must necessarily contain a deep layer of fuel (generally
about 6 to 10 ft.), any CO2 formed at first would be reduced to CO; and
it was further assumed that hardly any CO2 would be formed from the
outset, as the temperature of the apparatus is too high for this
reaction to take place. But as the combustion of C to CO produces only
about 30% of the heat produced when C is burned into CO2, the quantity
of fuel consumed for "blowing-up" is very large, and in fact
considerably exceeds that consumed in "steaming." There is, of course, a
further loss by radiation and minor sources, and the result is that 1
kilogram of carbon yields only about 1.2 cub. met. of water gas. Each
period of blowing-up generally occupies from 8 to 12 minutes, that of
steaming only 4 or 5 minutes. This low yield of water gas until quite
recently appeared to be unavoidable, and the only question seemed to be
whether and to what extent the gas formed during blowing-up, which is in
fact identical with ordinary producer gas (Siemens gas), could be
utilized. In America, where the water gas is mostly employed for
illuminating purposes, at least part of the blowing-up gas is utilized
for heating the apparatus in which the naphtha is volatilized and the
vapours are "fixed" by superheating. This process, however, never
utilizes anything like the whole of the blowing-up gas, nor can this be
effected by raising and superheating the steam necessary for the second
operation; indeed, the employment of this gas for raising steam is not
very easy, owing to the irregularities of and constant interruptions in
the supply. In some systems the gas made during the blowing-up stage is
passed through chambers, loosely filled with bricks, like Siemens
recuperators, where it is burned by "secondary" air: the heat thus
imparted to the brickwork is utilized by passing through the
recuperator, and thus superheating, the steam required for the next
steaming operation. In many cases, principally where no carburetting is
practised, the blowing-up gas is simply burned at the mouth of the
producer, and is thus altogether lost; and in no case can it be utilized
without great waste. A very important improvement in this respect was
effected by C. Dellwik and E. Fleischer. They found that the view that
it is unavoidable to burn the carbon to monoxide during the blowing-up
holds good only for the pressure of blast formerly applied. This did not
much exceed that which is required for overcoming the frictional
resistance within the producer. If, however, the pressure is
considerably increased, and the height of the column of fuel reduced,
both of these conditions being strictly regulated in accordance with the
result desired, it is easy to attain a combustion of the carbon to
dioxide, with only traces of monoxide, in spite of the high temperature.
Evidently the excess of oxygen coming into contact with each particle of
carbon in a given unit of time produces other conditions of chemical
equilibrium than those existing at lower pressures. At any rate,
experience has shown that by this process, in which the full heat-value
of carbon is utilized during the blowing-up stage, the time of
heating-up can be reduced from 10 to 1-1/2 or 2 minutes, and the
steaming can be prolonged from 4 or 5 to 8 or 10 minutes, with the
result that twice the quantity of water gas is obtained, viz. upwards of
2 cub. metres from 1 kilogram of carbon.
The application of water gas as a fuel mainly depends upon the high
temperatures which it is possible to attain by its aid, and these are
principally due to the circumstance that it forms a much smaller flame
than coal gas, not to speak of Siemens gas, which contains at most 33%
of combustible matter against 90% or more in water gas. The latter
circumstance also allows the gas to be conducted and distributed in
pipes of moderate dimensions. Its application, apart from its use as an
illuminant (with which we are not concerned here), was formerly retarded
by its high cost in comparison with Siemens gas and other sources of
heat, but as this state of affairs has been changed by the modern
improvements, its use is rapidly extending, especially for metallurgical
purposes.
C. _Mixed Gas (Semi-Water Gas)._--This class is sometimes called Dowson
gas, irrespective of its method of production, although it was made and
extensively used a long time before J.E. Dowson constructed his
apparatus for generating such a gas principally for driving gas-engines.
By a combination of the processes for generating Siemens gas and water
gas, it is produced by injecting into a gas-producer at the same time a
certain quantity of air and a corresponding quantity of steam, the
latter never exceeding the amount which can be decomposed by the
heat-absorbing reaction, C + H2O = CO + H2, at the expense of the heat
generated by the action of the air in the reaction C + O = CO. Such gas
used to be frequently obtained in an accidental way by introducing
liquid water or steam into an ordinary gas-producer for the purpose of
facilitating its working by avoiding an excessive temperature, such as
might cause the rapid destruction of the brickwork and the fusion of the
ashes of the fuel into troublesome cakes. It was soon found that by
proceeding in this way a certain advantage could be gained in regard to
the consumption of fuel, as the heat abstracted by the steam from the
brickwork and the fuel itself was usefully employed for decomposing
water, its energy thus reappearing in the shape of a combustible gas. It
is hardly necessary to mention explicitly that the total heat obtained
by any such process from a given quantity of carbon (or hydrogen) can in
no case exceed that which is generated by direct combustion; some
inventors, however, whether inadvertently or intentionally, have
actually represented this to be possible, in manifest violation of the
law of the conservation of energy.
Roughly speaking, this gas may be said to be produced by the combination
of the reactions, described _sub_ A and B, to the joint reaction: 2C + O
+ H2O = 2CO + H2. The decomposition of H2O (applied in the shape of
steam) absorbs 57.6 gram calories, the formation of 2CO produces 59 gram
calories; hence there is a small positive excess of 1.4 calories at
disposal. This in reality would not be sufficient to cover the loss by
radiation, &c.; hence rather more free oxygen (i.e. atmospheric air)
must be employed than is represented by the above equation. All this
free oxygen is, of course, accompanied by nearly four times its volume
of nitrogen.
The mixed gas thus obtained differs very much in composition, but is
always much richer in hydrogen (of which it contains sometimes as much
as 20%) and poorer in carbon monoxide (sometimes down to 20%) than
Siemens gas; generally it contains more of CO2 than the latter. The
proportion of nitrogen is always less, about 50%. It is therefore a more
concentrated fuel than Siemens gas, and better adapted to the driving of
gas-engines. It scarcely costs more to make than ordinary Siemens gas,
except where the steam is generated and superheated in special
apparatus, as is done in the Dowson producer, which, on the other hand,
yields a correspondingly better gas. As is natural, its properties are
some way between those of Siemens gas and of water gas; but they
approach more nearly the former, both as to costs and as to fuel-value,
and also as to the temperatures reached in combustion. This is easily
understood if we consider that gas of just the same description can be
obtained by mixing one volume of real water gas with the four volumes of
Siemens gas made during the blowing-up stage--an operation which is
certainly too expensive for practical use.
A modification of this gas is the _Mond gas_, which is made, according
to Mond's patent, by means of such an excess of steam that most of the
nitrogen of the coke is converted into ammonia (Grouven's reaction). Of
course much of this steam passes on undecomposed, and the quantity of
the gas is greatly increased by the reaction C + 2H2O = CO2 + 2H2; hence
the fuel-value of this gas is less than that of semi-water gas made in
other ways. Against this loss must be set the gain of ammonia which is
recovered by means of an arrangement of coolers and scrubbers, and,
except at very low prices of ammonia, the profit thus made is probably
more than sufficient to cover the extra cost. But as the process
requires very large and expensive plant, and its profits would vanish in
the case of the value of ammonia becoming much lower (a result which
would very probably follow if it were somewhat generally introduced), it
cannot be expected to supplant the other descriptions of gaseous fuel to
more than a limited extent.
Semi-water gas is especially adapted for the purpose of driving
gas-engines on the explosive principle (gas-motors). Ordinary
producer-gas is too poor for this purpose in respect of heating power;
moreover, owing to the prevalence of carbon monoxide, it does not light
quickly enough. These defects are sufficiently overcome in semi-water
gas by the larger proportion of hydrogen contained in it. For the
purpose in question the gas should be purified from tar and ashes, and
should also be cooled down before entering the gas-engine. The Dowson
apparatus and others are constructed on this principle.
_Air Gas._--By forcing air over or through volatile inflammable liquids
a gaseous mixture can be obtained which burns with a bright flame and
which can be used for illumination. Its employment for heating purposes
is quite exceptional, e.g. in chemical laboratories, and we abstain,
therefore, from describing any of the numerous appliances, some of them
bearing very fanciful names, which have been devised for its
manufacture. (G. L.)
FUENTE OVEJUNA [_Fuenteovejuna_], a town of Spain, in the province of
Cordova; near the sources of the river Guadiato, and on the Fuente del
Arco-Belmez-Cordova railway. Pop. (1900) 11,777. Fuente Ovejuna is built
on a hill, in a well-irrigated district, which, besides producing an
abundance of wheat, wine, fruit and honey, also contains argentiferous
lead mines and stone quarries. Cattle-breeding is an important local
industry, and leather, preserved meat, soap and flour are manufactured.
The parish church formerly belonged to the knights of Calatrava (c.
1163-1486).
FUENTERRABIA (formerly sometimes written _Fontarabia_; Lat. _Fons
Rapidus_), a town of northern Spain, in the province of Guipuzcoa; on
the San Sebastian-Bayonne railway; near the Bay of Biscay and on the
French frontier. Pop. (1870) about 750; (1900) 4345. Fuenterrabia stands
on the slope of a hill on the left bank of the river Bidassoa, and near
the point where its estuary begins. Towards the close of the 19th
century the town became popular as a summer resort for visitors from the
interior of Spain, and, in consequence, its appearance underwent many
changes and much of its early prosperity returned. Hotels and villas
were built in the new part of the town that sprang up outside the
picturesque walled fortress, and there is quite a contrast between the
part inside the heavy, half-ruined ramparts, with its narrow, steep
streets and curious gable-roofed houses, its fine old church and castle
and its massive town hall, and the new suburbs and fishermen's quarter
facing the estuary of the Bidassoa. Many industries flourish on the
outskirts of the town, including rope and net manufactures, flour mills,
saw mills, mining railways, paper mills.
Fuenterrabia formerly possessed considerable strategic importance, and
it has frequently been taken and retaken in wars between France and
Spain. The rout of Charlemagne in 778, which has been associated with
Fontarabia, by Milton (_Paradise Lost_, i. 587), is generally understood
to have taken place not here but at Roncesvalles (q.v.), which is nearly
40 m. E.S.E. Unsuccessful attempts to seize Fuenterrabia were made by
the French troops in 1476 and again in 1503. In a subsequent campaign
(1521) these were more successful, but the fortress was retaken in 1524.
The prince of Conde sustained a severe repulse under its walls in 1638,
and it was on this occasion that the town received from Philip IV. the
rank of city (_muy noble, muy leal, y muy valerosa ciudad_, "most noble,
most loyal, and most valiant city"), a privilege which involved some
measure of autonomy. After a severe siege, Fuenterrabia surrendered to
the duke of Berwick and his French troops in 1719; and in 1794 it again
fell into the hands of the French, who so dismantled it that it has
never since been reckoned by the Spaniards among their fortified places.
It was by the ford opposite Fuenterrabia that the duke of Wellington, on
the 8th of October 1813, successfully forced a passage into France in
the face of an opposing army commanded by Marshal Soult. Severe fighting
also took place here during the Carlist War in 1837.
FUERO, a Spanish term, derived from the Latin _forum_. The Castillan use
of the word in the sense of a right, privilege or charter is most
probably to be traced to the Roman _conventus juridici_, otherwise known
as _jurisdictiones_ or _fora_, which in Pliny's time were already
numerous in the Iberian peninsula. In each of these provincial _fora_
the Roman magistrate, as is well known, was accustomed to pay all
possible deference to the previously established common law of the
district; and it was the privilege of every free subject to demand that
he should be judged in accordance with the customs and usages of his
proper forum. This was especially true in the case of the inhabitants of
those towns which were in possession of the _jus italicum_. It is not,
indeed, demonstrable, but there are many presumptions, besides some
fragments of direct evidence, which make it more than probable that the
old administrative arrangements both of the provinces and of the towns,
but especially of the latter, remained practically undisturbed at the
period of the Gothic occupation of Spain.[1] The Theodosian Code and the
Breviary of Alaric alike seem to imply a continuance of the municipal
system which had been established by the Romans; nor does the later Lex
Visigothorum, though avowedly designed in some points to supersede the
Roman law, appear to have contemplated any marked interference with the
former _fora_, which were still to a large extent left to be regulated
in the administration of justice by unwritten, immemorial, local custom.
Little is known of the condition of the subject populations of the
peninsula during the Arab occupation; but we are informed that the
Christians were, sometimes at least, judged according to their own laws
in separate tribunals presided over by Christian judges;[2] and the mere
fact of the preservation of the name _alcalde_, an official whose
functions corresponded so closely to those of the _judex_ or _defensor
civitatis_, is fitted to suggest that the old municipal _fora_, if much
impaired, were not even then in all cases wholly destroyed. At all
events when the word _forum_[3] begins to appear for the first time in
documents of the 10th century in the sense of a liberty or privilege,
it is generally implied that the thing so named is nothing new. The
earliest extant written fuero is probably that which was granted to the
province and town of Leon by Alphonso V. in 1020. It emanated from the
king in a general council of the kingdom of Leon and Castile, and
consisted of two separate parts; in the first 19 chapters were contained
a series of statutes which were to be valid for the kingdom at large,
while the rest of the document was simply a municipal charter.[4] But in
neither portion does it in any sense mark a new legislative departure,
unless in so far as it marks the beginning of the era of written
charters for towns. The "fuero general" does not profess to supersede
the _consuetudines antiquorum jurium_ or Chindaswint's codification of
these in the Lex Visigothorum; the "fuero municipal" is really for the
most part but a resuscitation of usages formerly established, a
recognition and definition of liberties and privileges that had long
before been conceded or taken for granted. The right of the burgesses to
self-government and self-taxation is acknowledged and confirmed, they,
on the other hand, being held bound to a constitutional obedience and
subjection to the sovereign, particularly to the payment of definite
imperial taxes, and the rendering of a certain amount of military
service (as the ancient municipia had been). Almost contemporaneous with
this fuero of Leon was that granted to Najera (Naxera) by Sancho el
Mayor of Navarre (_ob._ 1035), and confirmed, in 1076, by Alphonso
VI.[5] Traces of others of perhaps even an earlier date are occasionally
to be met with. In the fuero of Cardena, for example, granted by
Ferdinand I. in 1039, reference is made to a previous forum Burgense
(Burgos), which, however, has not been preserved, if, indeed, it ever
had been reduced to writing at all. The phraseology of that of Sepulveda
(1076) in like manner points back to an indefinitely remote
antiquity.[6] Among the later fueros of the 11th century, the most
important are those of Jaca (1064) and of Logrono (1095). The former of
these, which was distinguished by the unusual largeness of its
concessions, and by the careful minuteness of its details, rapidly
extended to many places in the neighbourhood, while the latter charter
was given also to Miranda by Alphonso VI., and was further extended in
1181 by Sancho el Sabio of Navarre to Vitoria, thus constituting one of
the earliest written _fora_ of the "Provincias Vascongadas." In the
course of the 12th and 13th centuries the number of such documents
increased very rapidly; that of Toledo especially, granted to the
Mozarabic population in 1101, but greatly enlarged and extended by
Alphonso VII. (1118) and succeeding sovereigns, was used as a basis for
many other Castilian fueros. Latterly the word fuero came to be used in
Castile in a wider sense than before, as meaning a general code of laws;
thus about the time of Saint Ferdinand the old Lex Visigothorum, then
translated for the first time into the vernacular, was called the Fuero
Juzgo, a name which was soon retranslated into the barbarous Latin of
the period as Forum Judicum;[7] and among the compilations of Alphonso
the Learned in like manner were an _Espejo de Fueros_ and also the
_Fuero de las leyes_, better known perhaps as the _Fuero Real_. The
famous code known as the _Ordenamiento Real de Alcala_, or _Fuero Viejo
de Castilla_, dates from a still later period. As the power of the
Spanish crown was gradually concentrated and consolidated, royal
pragmaticas began to take the place of constitutional laws; the local
fueros of the various districts slowly yielded before the superior force
of imperialism; and only those of Navarre and the Basque provinces (see
BASQUES) have had sufficient vitality to enable them to survive to
comparatively modern times. While actually owning the lordship of the
Castilian crown since about the middle of the 14th century, these
provinces rigidly insisted upon compliance with their consuetudinary
law, and especially with that which provided that the _senor_, before
assuming the government, should personally appear before the assembly
and swear to maintain the ancient constitutions. Each of the provinces
mentioned had distinct sets of fueros, codified at different periods,
and varying considerably as to details; the main features, however, were
the same in all. Their rights, after having been recognized by
successive Spanish sovereigns from Ferdinand the Catholic to Ferdinand
VII., were, at the death of the latter in 1833, set aside by the
government of Castanos. The result was a civil war, which terminated in
a renewed acknowledgment of the fueros by Isabel II. (1839). The
provisional government of 1868 also promised to respect them, and
similar pledges were given by the governments which succeeded. In
consequence, however, of the Carlist rising of 1873-1876, the Basque
fueros were finally extinguished in 1876. The history of the _Foraes_ of
the Portuguese towns, and of the _Fors du Bearn_, is precisely analogous
to that of the fueros of Castile.
Among the numerous works that more or less expressly deal with this
subject, that of Marina (_Ensayo historico-critico sobre la antigua
legislacion y principales cuerpos legales de los reynos de Leon y
Castilla_) still continues to hold a high place. Reference may also be
made to Colmeiro's _Curso de derecho politico segun la historia de
Leon y de Castilla_ (Madrid, 1873); to Schafer's _Geschichte von
Spanien_, ii. 418-428, iii. 293 seq.; and to Hallam's _Middle Ages_,
c. iv.
FOOTNOTES:
[1] The nature of the evidence may be gathered from Savigny, _Gesch.
d. rom. Rechts_. See especially i. pp. 154, 259 seq.
[2] Compare Lembke u. Schafer, _Geschichte von Spanien_, i. 314; ii.
117.
[3] Or rather _forus_. See Ducange, s.v.
[4] Cap. xx. begins: "Constituimus etiam ut Legionensis civitas, quae
depopulata fuit a Sarracenis in diebus patris mei Veremundi regis,
repopulatur _per hos foros subscriptos_."
[5] "Mando et concedo et confirmo ut ista civitas cum sua plebe et
cum omnibus suis pertinentiis sub tali lege et sub tali foro maneat
per saecula cuncta. Amen. Isti sunt fueros quae habuerunt in Naxera
in diebus Sanctii regis et Gartiani regis."
[6] "Ego Aldefonsus rex et uxor mea Agnes confirmamus ad
Septempublica suo foro quod habuit in tempore antiquo de avolo meo et
in tempore comitum Ferrando Gonzalez et comite Garcia Ferdinandez et
comite Domno Santio."
[7] This Latin is later even than that of Ferdinand, whose words are:
"Statuo et mando quod Liber Judicum, quo ego misi Cordubam,
translatetur in vulgarem et vocetur forum de Corduba ... et quod per
saecula cuncta sit pro foro et nullus sit ausus istud forum aliter
appellare nisi forum de Corduba, et jubeo et mando quod omnis morator
et populator ... veniet ad judicium et ad forum de Corduba."
FUERTEVENTURA, an island in the Atlantic Ocean, forming part of the
Spanish archipelago of the Canary Islands (q.v.). Pop. (1900) 11,669;
area 665 sq. m. Fuerteventura lies between Lanzarote and Grand Canary.
It has a length of 52 m., and an average width of 12 m. Though less
mountainous than the other islands, its aspect is barren. There are only
two springs of fresh water, and these are confined to one valley. Lava
streams and other signs of volcanic action abound, but there has been no
igneous activity since the Spaniards took possession. At each extremity
of the island are high mountains, which send off branches along the
coast so as to enclose a large arid plain. The highest peak reaches 2500
ft. In external appearance, climate and productions, Fuerteventura
greatly resembles Lanzarote. An interval of three years without rain has
been known. Oliva (pop. 1900, 2464) is the largest town. A smaller place
in the centre of the island named Betancuria (586) is the administrative
capital. Cabras (1000) on the eastern coast is the chief port.
Dromedaries are bred here.
FUGGER, the name of a famous German family of merchants and bankers. The
founder of the family was Johann Fugger, a weaver at Graben, near
Augsburg, whose son, Johann, settled in Augsburg probably in 1367. The
younger Johann added the business of a merchant to that of a weaver, and
through his marriage with Clara Widolph became a citizen of Augsburg.
After a successful career he died in 1408, leaving two sons, Andreas and
Jakob, who greatly extended the business which they inherited from their
father. Andreas, called the "rich Fugger," had several sons, among them
being Lukas, who was very prominent in the municipal politics of
Augsburg and who was very wealthy until he was ruined by the repudiation
by the town of Louvain of a great debt owing to him, and Jakob, who was
granted the right to bear arms in 1452, and who founded the family of
Fugger vom Reh--so called from the first arms of the Fuggers, a roe
(_Reh_) or on a field azure--which became extinct on the death of his
great-grandson, Ulrich, in 1583. Johann Fugger's son, Jakob, died in
1469, and three of his seven sons, Ulrich (1441-1510), Georg (1453-1506)
and Jakob (1459-1525), men of great resource and industry, inherited the
family business and added enormously to the family wealth. In 1473
Ulrich obtained from the emperor Frederick III. the right to bear arms
for himself and his brothers, and about the same time he began to act
as the banker of the Habsburgs, a connexion destined to bring fame and
fortune to his house. Under the lead of Jakob, who had been trained for
business in Venice, the Fuggers were interested in silver mines in Tirol
and copper mines in Hungary, while their trade in spices, wool and silk
extended to almost all parts of Europe. Their wealth enabled them to
make large loans to the German king, Maximilian I., who pledged to them
the county of Kirchberg, the lordship of Weissenhorn and other lands,
and bestowed various privileges upon them. Jakob built the castle of
Fuggerau in Tirol, and erected the Fuggerei at Augsburg, a collection of
106 dwellings, which were let at low rents to poor people and which
still exist. Jakob Fugger and his two nephews, Ulrich (d. 1525) and
Hieronymus (d. 1536), the sons of Ulrich, died without direct heirs, and
the family was continued by Georg's sons, Raimund (1489-1535) and Anton
(1493-1560), under whom the Fuggers attained the summit of their wealth
and influence.
Jakob Fugger's florins had contributed largely to the election of
Charles V. to the imperial throne in 1519, and his nephews and heirs
maintained close and friendly relations with the great emperor. In
addition to lending him large sums of money, they farmed his valuable
quicksilver mines at Almaden, his silver mines at Guadalcanal, the great
estates of the military orders which had passed into his hands, and
other parts of his revenue as king of Spain; receiving in return several
tokens of the emperor's favour. In 1530 Raimund and Anton were granted
the imperial dignity of counts of Kirchberg and Weissenhorn, and
obtained full possession of these mortgaged properties; in 1534 they
were given the right of coining money; and in 1541 received rights of
jurisdiction over their lands. During the diet of Augsburg in 1530
Charles V. was the guest of Anton Fugger at his house in the Weinmarkt,
and the story relates how the merchant astonished the emperor by
lighting a fire of cinnamon with an imperial bond for money due to him.
This incident forms the subject of a picture by Carl Becker which is in
the National Gallery at Berlin. Continuing their mercantile career, the
Fuggers brought the new world within the sphere of their operations, and
also carried on an extensive and lucrative business in farming
indulgences. Moreover, both brothers found time to acquire landed
property, and were munificent patrons of literature and art. When Anton
died he is said to have been worth 6,000,000 florins, besides a vast
amount of property in Europe, Asia and America; and before this time the
total wealth of the family had been estimated at 63,000,000 florins. The
Fuggers were devotedly attached to the Roman Catholic Church, which
benefited from their liberality. Jakob had been made a count palatine
(_Pfalzgraf_) and had received other marks of favour from Pope Leo X.,
and several members of the family had entered the church; one, Raimund's
son, Sigmund, becoming bishop of Regensburg.
In addition to the bishop, three of Raimund Fugger's sons attained some
degree of celebrity. Johann Jakob (1516-1575), was the author of
_Wahrhaftigen Beschreibung des osterreichischen und habsburgischen
Nahmens_, which was largely used by S. von Bircken in his _Spiegel der
Ehren des Erzhauses Osterreich_ (Nuremberg, 1668), and of a _Geheim
Ernbuch des Fuggerischen Geschlechtes_. He was also a patron of art, and
a distinguished counsellor of Duke Albert IV. of Bavaria. After the
death of his son Konstantin, in 1627, this branch of the family was
divided into three lines, which became extinct in 1738, 1795 and 1846
respectively. Another of Raimund's sons was Ulrich (1526-1584), who,
after serving Pope Paul III. at Rome, became a Protestant. Hated on this
account by the other members of his family, he took refuge in the
Rhenish Palatinate; greatly interested in the Greek classics, he
occupied himself in collecting valuable manuscripts, which he bequeathed
to the university of Heidelberg. Raimund's other son was Georg (d.
1579), who inherited the countships of Kirchberg and Weissenhorn, and
founded a branch of the family which still exists, its present head
being Georg, Count Fugger of Kirchberg and Weissenhorn (b. 1850).
Anton Fugger left three sons, Marcus (1529-1597), Johann (d. 1598) and
Jakob (d. 1598), all of whom left male issue. Marcus was the author of
a book on horse-breeding, _Wie und wo man ein Gestut von guten edeln
Kriegsrossen aufrichten soll_ (1578), and of a German translation of the
_Historia ecclesiastica_ of Nicephorus Callistus. He founded the
Nordendorf branch of the family, which became extinct on the death of
his grandson, Nicolaus, in 1676. Another grandson of Marcus was Franz
Fugger (1612-1664), who served under Wallenstein during the Thirty
Years' War, and was afterwards governor of Ingolstadt. He was killed at
the battle of St Gotthard on the 1st of August 1664.
Johann Fugger had three sons, Christoph (d. 1615) and Marcus (d. 1614),
who founded the families of Fugger-Glott and Fugger-Kirchheim
respectively, and Jakob, bishop of Constance from 1604 until his death
in 1626. Christoph's son, Otto Heinrich (1592-1644), was a soldier of
some distinction and a knight of the order of the Golden Fleece. He was
one of the most active of the Bavarian generals during the Thirty Years'
War, and acted as governor of Augsburg, where his rule aroused much
discontent. The family of Kirchheim died out in 1672. That of Glott was
divided into several branches by the sons of Otto Heinrich and of his
brother Johann Ernst (d. 1628). These lines, however, have gradually
become extinct except the eldest line, represented in 1909 by Karl
Ernst, Count Fugger of Glott (b. 1859). Anton Fugger's third son Jakob,
the founder of the family of Wellenburg, had two sons who left issue,
but in 1777 the possessions of this branch of the family were again
united by Anselm Joseph (d. 1793), Count Fugger of Babenhausen. In 1803
Anselm's son, Anselm Maria (d. 1821), was made a prince of the Holy
Roman Empire, the title of Prince Fugger of Babenhausen being borne by
his direct descendant Karl (b. 1861). On the fall of the empire in 1806
the lands of the Fuggers, which were held directly of the empire, were
mediatized under Bavaria and Wurttemberg. The heads of the three
existing branches of the Fuggers are all hereditary members of the
Bavarian Upper House.
Augsburg has many interesting mementoes of the Fuggers, including the
family burial-chapel in the church of St Anna; the Fugger chapel in the
church of St Ulrich and St Afra; the Fuggerhaus, still in the possession
of one branch of the family; and a statue of Johann Jakob Fugger.
In 1593 a collection of portraits of the Fuggers, engraved by
Dominique Custos of Antwerp, was issued at Augsburg. Editions with 127
portraits appeared in 1618 and 1620, the former accompanied by a
genealogy in Latin, the latter by one in German. Another edition of
this _Pinacotheca Fuggerorum_, published at Vienna in 1754, includes
139 portraits. See _Chronik der Familie Fugger vom Jahre 1599_, edited
by C. Meyer (Munich, 1902); A. Geiger, _Jakob Fugger, 1459-1525_
(Regensburg, 1895); A. Schulte, _Die Fugger in Rom, 1495-1523_
(Leipzig, 1904); R. Ehrenberg, _Das Zeitalter der Fugger_ (Jena,
1896); K. Habler, _Die Geschichte der Fuggerschen Handlung in Spanien_
(Weimar, 1897); A. Stauber, _Das Haus Fugger_ (Augsburg, 1900); and M.
Jansen, _Die Anfange der Fugger_ (Leipzig, 1907).
FUGITIVE SLAVE LAWS, a term applied in the United States to the Statutes
passed by Congress in 1793 and 1850 to provide for the return of negro
slaves who escaped from one state into another or into a public
territory. A fugitive slave clause was inserted in the Articles of
Confederation of the New England Confederation of 1643, providing for
the return of the fugitive upon the certificate of one magistrate in the
jurisdiction out of which the said servant fled--no trial by jury being
provided for. This seems to have been the only instance of an
inter-colonial provision for the return of fugitive slaves; there were,
indeed, not infrequent escapes by slaves from one colony to another, but
it was not until after the growth of anti-slavery sentiment and the
acquisition of western territory, that it became necessary to adopt a
uniform method for the return of fugitive slaves. Such provision was
made in the Ordinance of 1787 (for the Northwest Territory), which in
Article VI. provided that in the case of "any person escaping into the
same [the Northwest Territory] from whom labor or service is lawfully
claimed in any one of the original states, such fugitive may be lawfully
reclaimed and conveyed to the person claiming his or her labor or
service as aforesaid." An agreement of the sort was necessary to
persuade the slave-holding states to union, and in the Federal
Constitution, Article IV., Section II., it is provided that "no person
held to service or labor in one state, under the laws thereof, escaping
into another, shall, in consequence of any law or regulation therein, be
discharged from such service or labor, but shall be delivered up on
claim of the party to whom such service or labour may be due."
The first specific legislation on the subject was enacted on the 12th of
February 1793, and like the Ordinance for the Northwest Territory and
the section of the Constitution quoted above, did not contain the word
"slave"; by its provisions any Federal district or circuit judge or any
state magistrate was authorized to decide finally and without a jury
trial the status of an alleged fugitive. The measure soon met with
strong opposition in the northern states, and Personal Liberty Laws were
passed to hamper officials in the execution of the law; Indiana in 1824
and Connecticut in 1828 providing jury trial for fugitives who appealed
from an original decision against them. In 1840 New York and Vermont
extended the right of trial by jury to fugitives and provided them with
attorneys. As early as the first decade of the 19th century individual
dissatisfaction with the law of 1793 had taken the form of systematic
assistance rendered to negroes escaping from the South to Canada or New
England--the so-called "Underground Railroad."[1] The decision of the
Supreme Court of the United States in the case of _Prigg_ v.
_Pennsylvania_ in 1842 (16 Peters 539), that state authorities could not
be forced to act in fugitive slave cases, but that national authorities
must carry out the national law, was followed by legislation in
Massachusetts (1843), Vermont (1843), Pennsylvania (1847) and Rhode
Island (1848), forbidding state officials to help enforce the law and
refusing the use of state gaols for fugitive slaves. The demand from the
South for more effective Federal legislation was voiced in the second
fugitive slave law, drafted by Senator J.M. Mason of Virginia, and
enacted on the 18th of September 1850 as a part of the Compromise
Measures of that year. Special commissioners were to have concurrent
jurisdiction with the U.S. circuit and district courts and the inferior
courts of Territories in enforcing the law; fugitives could not testify
in their own behalf; no trial by jury was provided; penalties were
imposed upon marshals who refused to enforce the law or from whom a
fugitive should escape, and upon individuals who aided negroes to
escape; the marshal might raise a _posse comitatus_; a fee of $10 was
paid to the commissioner when his decision favoured the claimant and
only $5 when it favoured the fugitive; and both the fact of the escape
and the identity of the fugitive were to be determined on purely _ex
parte_ testimony. The severity of this measure led to gross abuses and
defeated its purpose; the number of abolitionists increased, the
operations of the Underground Railroad became more efficient, and new
Personal Liberty Laws were enacted in Vermont (1850), Connecticut
(1854), Rhode Island (1854), Massachusetts (1855), Michigan (1855),
Maine (1855 and 1857), Kansas (1858) and Wisconsin (1858). These
Personal Liberty Laws forbade justices and judges to take cognizance of
claims, extended the _habeas corpus_ act and the privilege of jury trial
to fugitives, and punished false testimony severely. The supreme court
of Wisconsin went so far (1859) as to declare the Fugitive Slave Law
unconstitutional. These state laws were one of the grievances officially
referred to by South Carolina (in Dec. 1860) as justifying her secession
from the Union. Attempts to carry into effect the law of 1850 aroused
much bitterness. The arrests of Sims and of Shadrach in Boston in 1851;
of "Jerry" M'Henry, in Syracuse, New York, in the same year; of Anthony
Burns in 1854, in Boston; and of the two Garner families in 1856, in
Cincinnati, with other cases arising under the Fugitive Slave Law of
1850, probably had as much to do with bringing on the Civil War as did
the controversy over slavery in the Territories.
With the beginning of the Civil War the legal status of the slave was
changed by his master's being in arms. General B.F. Butler, in May 1861,
declared negro slaves contraband of war. A confiscation bill was passed
in August 1861 discharging from his service or labour any slave employed
in aiding or promoting any insurrection against the government of the
United States. By an act of the 17th of July 1862 any slave of a
disloyal master who was in territory occupied by northern troops was
declared _ipso facto_ free. But for some time the Fugitive Slave Law was
considered still to hold in the case of fugitives from masters in the
border states who were loyal to the Union government, and it was not
until the 28th of June 1864 that the Act of 1850 was repealed.
See J.F. Rhodes, _History of the United States from the Compromise of
1850_, vols. i. and ii. (New York, 1893); and M.G. M'Dougall,
_Fugitive Slaves, 1619-1865_ (Boston, 1891).
FOOTNOTE:
[1] The precise amount of organization in the Underground Railroad
cannot be definitely ascertained because of the exaggerated use of
the figure of railroading in the documents of the "presidents" of the
road, Robert Purvis and Levi Coffin, and of its many "conductors,"
and their discussion of the "packages" and "freight" shipped by them.
The system reached from Kentucky and Virginia across Ohio, and from
Maryland across Pennsylvania and New York, to New England and Canada,
and as early as 1817 a group of anti-slavery men in southern Ohio had
helped to Canada as many as 1000 slaves. The Quakers of Pennsylvania
possibly began the work of the mysterious Underground Railroad; the
best known of them was Thomas Garrett (1789-1871), a native of
Pennsylvania, who, in 1822, removed to Wilmington, Delaware, where he
was convicted in 1848 on four counts under the Fugitive Slave Law and
was fined $8000; he is said to have helped 2700 slaves to freedom.
The most picturesque figure of the Underground Railroad was Harriet
Tubman (c. 1820), called by her friend, John Brown, "General" Tubman,
and by her fellow negroes "Moses." She made about a score of trips
into the South, bringing out with her 300 negroes altogether. At one
time a reward of $40,000 was offered for her capture. She was a
mystic, with remarkable clairvoyant powers, and did great service as
a nurse, a spy and a scout in the Civil War. Levi Coffin (1798-1877),
a native of North Carolina (whose cousin, Vestal Coffin, had
established before 1819 a "station" of the Underground near what is
now Guilford College, North Carolina), in 1826 settled in Wayne
County, Ohio; his home at New Garden (now Fountain City) was the
meeting point of three "lines" from Kentucky; and in 1847 he removed
to Cincinnati, where his labours in bringing slaves out of the South
were even more successful. It has been argued that the Underground
Railroad delayed the final decision of the slavery question, inasmuch
as it was a "safety valve"; for, without it, the more intelligent and
capable of the negro slaves would, it is asserted, have become the
leaders of insurrections in the South, and would not have been
removed from the places where they could have done most damage.
Consult William Still, _The Underground Railroad_ (Philadelphia,
1872), a collection of anecdotes by a negro agent of the Pennsylvania
Anti-Slavery Society, and of the Philadelphia branch of the Railroad;
and the important and scholarly work of Wilbur H. Siebert, _The
Underground Railroad from Slavery to Freedom_ (New York, 1898).
FUGLEMAN (from the Ger. _Flugelmann_, the man on the _Flugel_ or wing),
properly a military term for a soldier who is selected to act as
"guide," and posted generally on the flanks with the duty of directing
the march in the required line, or of giving the time, &c., to the
remainder of the unit, which conforms to his movements, in any military
exercise. The word is then applied to a ringleader or one who takes the
lead in any movement or concerted movement.
FUGUE (Lat. _fuga_, flight), in music, the mutual "pursuit" of voices or
parts. It was, up to the end of the 16th century, if not later, the name
applied to two art-forms. (A) _Fuga ligata_ was the exact reproduction
by one or more voices of the statement of a leading part. The
reproducing voice (_comes_) was seldom if ever written out, for all
differences between it and the _dux_ were rigidly systematic; e.g. it
was an exact inversion, or exactly twice as slow, or to be sung
backwards, &c. &c. Hence, a rule or _canon_ was given, often in
enigmatic form, by which the _comes_ was deduced from the _dux_: and so
the term _canon_ became the appropriate name for the form itself, and is
still retained. (B) A composition in which the canonic style was
cultivated without canonic restriction was, in the 16th century, called
_fuga ricercata_ or simply a _ricercare_, a term which is still used by
Bach as a title for the fugues in _Das musikalische Opfer_.
The whole conception of fugue, rightly understood, is one of the most
important in music, and the reasons why some contrapuntal compositions
are called fugues, while others are not, are so trivial, technically as
well as aesthetically, that we have preferred to treat the subject
separately under the general heading of CONTRAPUNTAL FORMS, reserving
only technical terms for definition here.
(i.) If in the beginning or "exposition" the material with which the
opening voice accompanies the answer is faithfully reproduced as the
accompaniment to subsequent entries of the subject, it is called a
_countersubject_ (see COUNTERPOINT, under sub-heading _Double
Counterpoint_). Obviously the process may be carried further, the first
countersubject going on to a second when the subject enters in the third
part and so on. The term is also applied to new subjects appearing later
in the fugue in combination (immediate or destined) with the original
subject. Cherubini, holding the doctrine that a fugue cannot have more
than one subject, insists on applying the term to the less prominent of
the subjects of what are commonly called double fugues, i.e. fugues
which begin with two parts and two subjects simultaneously, and so also
with _triple_ and _quadruple fugues_.
(ii.) _Episodes_ are passages separating the entries of the subject.[1]
Episodes are usually developed from the material of the subject and
countersubjects; they are very rarely independent, but then
conspicuously so.
(iii.) _Stretto_, the overlapping of subject and answer, is a resource
the possibilities of which may be exemplified by the setting of the
words _omnes generationes_ in Bach's _Magnificat_ (see BACH).
(iv.) The distinction between _real_ and _tonal_ fugue, which is still
sometimes treated as a thing of great historical and technical
importance, is really a mere detail resulting from the fact that a
violent oscillation between the keys of tonic and dominant is no part of
the function of a fugal exposition, so that the answer is (especially in
its first notes and in points that tend to shift the key) not so much a
transposition of the subject to the key of the dominant as an adaptation
of it from the tonic part to the dominant part of the scale, or vice
versa; in short, the answer is as far as possible _on_ the dominant, not
_in_ the dominant. The modifications this principle produces in the
answer (which have been happily described as resembling
"fore-shortening") are the only distinctive marks of tonal fugue; and
the text-books are half filled with the attempt to reduce them from
matters of ear to rules of thumb, which rules, however, have the merit
(unusual in those of the academic fugue) of being founded on observation
of the practice of great masters. But the same principle as often as not
produces answers that are exact transpositions of the subject; and so
the only kind of real fugue (i.e. fugue with an exact answer) that could
rightly be contrasted with tonal fugue would be that in which the answer
ought to be tonal but is not. It must be admitted that tonal answers are
rare in the modal music of the 16th century, though their melodic
principles are of yet earlier date; still, though tonal fugue does not
become usual until well on in the 17th century, the idea that it is a
separate species is manifestly absurd, unless the term simply means
"fugue in modern tonality or key," whatever the answer may be.
The term "answer" is usually reserved for those entries of the subject
that are placed in what may be called the "complementary" position of
the scale, whether they are "tonally" modified or not. Thus the order of
entries in the exposition of the first fugue of the _Wohltemp_. _Klav_.
is subject, answer, answer, subject; a departure from the usual rule
according to which subject and answer are strictly alternate in the
exposition.
In conclusion we may remind the reader of the most accurate as well as
the most vivid description ever given of the essentials of a fugue, in
the famous lines in _Paradise Lost_, book xi.
"His volant touch,
Instinct through all proportions, low and high,
Fled and pursued transverse the resonant fugue."
It is hard to realize that this description of organ-music was written
in no classical period of instrumental polyphony, but just half-way
between the death of Frescobaldi and the birth of Bach. Every word is a
definition, both retrospective and prophetic; and in "transverse" we see
all that Sir Frederick Gore Ouseley expresses in his popular distinction
between the "perpendicular" or homophonic style in which harmony is
built up in chords, and the "horizontal" or polyphonic style in which it
is woven in threads of independent melody. (D. F. T.)
FOOTNOTE:
[1] An episode occurring during the exposition is sometimes called
_codetta_, a distinction the uselessness of which at once appears on
an analysis of Bach's 2nd fugue in the _Wohltemp_. _Klav_. (the term
codetta is more correctly applied to notes filling in a gap between
subject and its first answer, but such a gap is rare in good
examples).
FUHRICH, JOSEPH VON (1800-1876), Austrian painter, was born at Kratzau
in Bohemia on the 9th of February 1800. Deeply impressed as a boy by
rude pictures adorning the wayside chapels of his native country, his
first attempt at composition was a sketch of the Nativity for the
festival of Christmas in his father's house. He lived to see the day
when, becoming celebrated as a composer of scriptural episodes, his
sacred subjects were transferred in numberless repetitions to the
roadside churches of the Austrian state, where humble peasants thus
learnt to admire modern art reviving the models of earlier ages. Fuhrich
has been fairly described as a "Nazarene," a romantic religious artist
whose pencil did more than any other to restore the old spirit of Durer
and give new shape to countless incidents of the gospel and scriptural
legends. Without the power of Cornelius or the grace of Overbeck, he
composed with great skill, especially in outline. His mastery of
distribution, form, movement and expression was considerable. In its
peculiar way his drapery was perfectly cast. Essentially creative as a
landscape draughtsman, he had still no feeling for colour; and when he
produced monumental pictures he was not nearly so successful as when
designing subjects for woodcuts. Fuhrich's fame extended far beyond the
walls of the Austrian capital, and his illustrations to Tieck's
_Genofeva_, the Lord's Prayer, the Triumph of Christ, the Road to
Bethlehem, the Succession of Christ according to Thomas a Kempis, the
Prodigal Son, and the verses of the Psalter, became well known. His
Prodigal Son, especially, is remarkable for the fancy with which the
spirit of evil is embodied in a figure constantly recurring, and like
that of Mephistopheles exhibiting temptation in a human yet demoniacal
shape. Fuhrich became a pupil at the Academy of Prague in 1816. His
first inspiration was derived from the prints of Durer and the Faust of
Cornelius, and the first fruit of this turn of study was the Genofeva
series. In 1826 he went to Rome, where he added three frescoes to those
executed by Cornelius and Overbeck in the Palazzo Massimi. His subjects
were taken from the life of Tasso, and are almost solitary examples of
his talent in this class of composition. In 1831 he finished the Triumph
of Christ now in the Raczynski palace at Berlin. In 1834 he was made
custos and in 1841 professor of composition in the Academy of Vienna.
After this he completed the monumental pictures of the church of St
Nepomuk, and in 1854-1861 the vast series of wall paintings which cover
the inside of the Lerchenfeld church at Vienna. In 1872 he was pensioned
and made a knight of the order of Franz Joseph; 1875 is the date of his
illustrations to the Psalms. He died on the 13th of March 1876.
His autobiography was published in 1875, and a memoir by his son Lucas
in 1886.
FUJI (Fuji-san, Fujiyama, Fusiyama), a celebrated mountain of Japan,
standing W.S.W. of Tokyo, its base being about 70 m. by rail from that
city. It rises to a height of 12,395 ft. and its southern slopes reach
the shore of Suruga Bay. It is a cone of beautifully simple form, the
more striking to view because it stands isolated; but its summit is not
conical, being broken by a crater some 2000 ft. in diameter, for Fuji is
a quiescent volcano. Small outbursts of steam are still to be observed
at some points. An eruption is recorded so lately as the first decade of
the 18th century. The mountain is the resort of great numbers of
pilgrims (see also JAPAN).
FU-KIEN (formerly MIN), a south-eastern province of China, bounded N. by
the province of Cheh-kiang, S. by that of Kwang-tung, W. by that of
Kiang-si and E. by the sea. It occupies an area of 53,480 sq. m. and its
population is estimated at 20,000,000. The provincial capital is Fuchow
Fu, and it is divided into eleven prefectures, besides that ruled over
by the prefect of the capital city. Fu-kien is generally mountainous,
being overspread by the Nan-shan ranges, which run a general course of
N.E. and S.W. The principal river is the Min, which is formed by the
junction, in the neighbourhood of the city of Yen-p'ing Fu, of three
rivers, namely, the Nui-si, which takes its rise in the mountains on the
western frontier in the prefecture of Kien-ning Fu, the Fuh-tun Ki, the
source of which is found in the district of Kwang-tsih in the north-west
of the province, and the Ta-shi-ki (Shao Ki), which rises in the
mountains in the western district of Ning-hwa. From Yen-p'ing Fu the
river takes a south-easterly course, and after passing along the south
face of the city of Fuchow Fu, empties itself into the sea about 30 m.
below that town. Its upper course is narrow and rocky and abounds in
rapids, but as it approaches Fuchow Fu the channel widens and the
current becomes slow and even. Its depth is very irregular, and it is
navigable only by native boats of a small class. Two other rivers flow
into the sea near Amoy, neither of which, however, is navigable for any
distance from its mouth owing to the shallows and rapids with which they
abound. Thirty-five miles inland from Amoy stands the city of Chang
Chow, famous for the bridge which there spans the Kin-lung river. This
bridge is 800 ft. long, and consists of granite monoliths stretching
from one abutment to another. The soil of the province is, as its name,
"Happy Establishment," indicates, very productive, and the scenery is of
a rich and varied character. Most of the hills are covered with verdure,
and the less rugged are laid out in terraces. The principal products of
the province are tea, of which the best kind is that known as Bohea,
which takes its name, by a mispronunciation, from the Wu-e Mountains, in
the prefecture of Kien-ning Fu, where it is grown; grains of various
kinds, oranges, plantins, lichis, bamboo, ginger, gold, silver, lead,
tin, iron, salt (both marine and rock), deers' horns, beeswax, sugar,
fish, birds' nests, medicine, paper, cloth, timber, &c. Fu-kien has
three open ports, Fuchow Fu opened in 1842, Amoy opened to trade in the
same year and Funing. The latter port was only opened to foreign trade
in 1898, but in 1904 it imported and exported goods to the value of
L7668 and L278,160 respectively.
FUKUI, a town of Japan in the province of Echizen, Nippon, near the west
coast, 20 m. N. by E. of Wakasa Bay. It lies in a volcanic district much
exposed to earthquakes, and suffered severely during the disturbances of
1891-1892, when a chasm over 40 m. long was opened across the Neo valley
from Fukui to Katabira. But Fukui subsequently revived, and is now in a
flourishing condition, with several local industries, especially the
manufacture of paper, and an increasing population exceeding 50,000.
Fukui has railway communication. There are ruins of a castle of the
Daimios of Echizen.
FUKUOKA, a town on the north-west coast of the island of Kiushiu, Japan,
in the province of Chikuzen, 90 m. N.N.E. of Nagasaki by rail. Pop.
about 72,000. With Hakata, on the opposite side of a small coast stream,
it forms a large centre of population, with an increasing export trade
and several local industries. Of these the most important is
silk-weaving, and Hakata especially is noted for its durable silk
fabrics. Fukuoka was formerly the residence of the powerful daimio of
Chikuzen, and played a conspicuous part in the medieval history of
Japan; the renowned temple of Yeiyas in the district was destroyed by
fire during the revolution of 1868. There are several other places of
this name in Japan, the most important being Fukuoka in the province of
Mutsu, North Nippon, a railway station on the main line from Tokyo to
Aimori Ura Bay. Pop. about 5000.
FULA (FULBE, FELLATAH or PEULS), a numerous and powerful African people,
spread over an immense region from Senegal nearly to Darfur. Strictly
they have no country of their own, and nowhere form the whole of the
population, though nearly always the dominant native race. They are most
numerous in Upper Senegal and in the countries under French sway
immediately south of Senegambia, notably Futa Jallon. Farther east they
rule, subject to the control of the French, Segu and Massena, countries
on both banks of the upper Niger, to the south-west of Timbuktu. The
districts within the great bend of the Niger have a large Fula
population. East of that river Sokoto and its tributary emirates are
ruled by Fula princes, subject to the control of the British Nigerian
administration. Fula are settled in Bornu, Bagirmi, Wadai and the upper
Nile Valley,[1] but have no political power in those countries. Their
most southerly emirate is Adamawa, the country on both sides of the
upper Benue. In this vast region of distribution the Fula populations
are most dense towards the west and north, most scattered towards the
east and south. Originally herdsmen in the western and central Sudan,
they extended their sway east of the Niger, under the leadership of
Othman Dan Fodio, during the early years of the 19th century, and having
subdued the Hausa states, founded the empire of Sokoto with the vassal
emirates of Kano, Gando, Nupe, Adamawa, &c.
The question of the ethnic affinities of the Fula has given rise to an
enormous amount of speculation, but the most reasonable theory is that
they are a mixture of Berber and Negro. This is now the most generally
accepted theory. Certainly there is no reason to connect them with the
ancient Egyptians. In the district of Senegal known as Fuladugu or "Fula
Land," where the purest types of the race are found, the people are of a
reddish brown or light chestnut colour, with oval faces, ringlety or
even smooth hair, never woolly, straight and even aquiline noses,
delicately shaped lips and regular features quite differentiating them
from the Negro type. Like most conquering races the Fula are, however,
not of uniform physique, in many districts approximating to the local
type. They nevertheless maintain throughout their widespread territory a
certain national solidarity, thanks to common speech, traditions and
usages. The ruling caste of the Fula differs widely in character from
the herdsmen of the western Sudan. The latter are peaceable, inoffensive
and abstemious. They are mainly monogamous, and by rigidly abstaining
from foreign marriages have preserved racial purity. The ruling caste in
Nigeria, on the other hand, despise their pastoral brethren, and through
generations of polygamy with the conquered tribes have become more
Negroid in type, black, burly and coarse featured. Love of luxury, pomp
and finery is their chief characteristic. Taken as a whole, the Fula
race is distinguished by great intelligence, frankness of disposition
and strength of character. As soldiers they are renowned almost
exclusively as cavalry; and the race has produced several leaders
possessed of much strategical skill. Besides the ordinary Negro weapons,
they use iron spears with leatherbound handles and swords. They are
generally excellent rulers, stern but patient and just. The Nigerian
emirs acquired, however, an evil reputation during the 19th century as
slave raiders. They have long been devout Mahommedans, and mosques and
schools exist in almost all their towns. Tradition says that of old
every Fula boy and girl was a scholar; but during the decadence of their
power towards the close of the 19th century education was not highly
valued. Power seems to have somewhat spoilt this virile race, but such
authorities as Sir Frederick Lugard believe them still capable of a
great future.
The Fula language has as yet found no place in any African linguistic
family. In its rudiments it is akin to the Hamito-Semitic group. It
possesses two grammatical genders, not masculine and feminine, but the
human and the non-human; the adjective agrees in assonance with its
noun, and euphony plays a great part in verbal and nominal inflections.
In some ways resembling the Negro dialects, it betrays non-Negroid
influences in the use of suffixes. The name of the people has many
variations. Fulbe or Fula (sing. Pullo, Peul) is the Mandingan name,
Follani the Hausa, Fellatah the Kanuri, Fullan the Arab, and Fulde on
the Benue. Like the name Abate, "white," given them in Kororofa, all
these seem to refer to their light reddish hue.
See F. Ratzel, _History of Mankind_ (English ed., London, 1896-1898);
Sir F. Lugard, "Northern Nigeria," in _Geographical Journal_ (July
1904); Grimai de Guirodon, _Les Puls_ (1887); E.A. Brackenbury, _A
Short Vocabulary of the Fulani Language_ (Zungeru, 1907); the articles
NIGERIA and SOKOTO and authorities there cited.
FOOTNOTE:
[1] Sir Wm. Wallace in a report on Northern Nigeria ("Colonial
Office" series, No. 551, 1907) calls attention to the exodus "of
thousands of Fulani of all sorts, but mostly Mellawa, from the French
Middle Niger," and states that the majority of the emigrants are
settling in the Nile valley.
FULCHER (or FOUCHER) OF CHARTRES (1058-c. 1130), French chronicler, was
a priest who was present at the council of Clermont in 1095, and
accompanied Robert II., duke of Normandy, on the first crusade in 1096.
Having spent some time in Italy and taken part in the fighting on the
way to the Holy Land, he became chaplain to Baldwin, who was chosen king
of Jerusalem in 1100, and lived with Baldwin at Edessa and then at
Jerusalem. He accompanied this king on several warlike expeditions, but
won more lasting fame by writing his _Historia Hierosolymitana_ or
_Gesta Francorum Jerusalem expugnantium_, one of the most trustworthy
sources for the history of the first crusade. In its final form it is
divided into three books, and covers the period between the council of
Clermont and 1127, and the author only gives details of events which he
himself had witnessed. It was used by William of Tyre. Fulcher died
after 1127, probably at Jerusalem. He has been confused with Foucher of
Mongervillier (d. 1171), abbot of St-Pere-en-Vallee at Chartres, and
also with another person of the same name who distinguished himself at
the siege of Antioch in 1098.
The _Historia_, but in an incomplete form, was first published by J.
Bongars in the _Gesta Dei per Francos_ (Hanover, 1611). The best
edition is in tome iii. of the _Recueil des historiens des croisades,
Historiens occidentaux_ (Paris, 1866); and there is a French
translation in tome xxiv. of Guizot's _Collection des memoires
relatifs a l'histoire de France_ (Paris, 1823-1835).
See H. von Sybel, _Geschichte des ersten Kreuzzuges_ (Leipzig, 1881);
and A. Molinier, _Les Sources de l'histoire de France_, tome ii.
(Paris, 1902).
FULDA, a town and episcopal see of Germany, in the Prussian province of
Hesse-Nassau, between the Rhon and the Vogel-Gebirge, 69 m. N.E. from
Frankfort-on-Main on the railway to Bebra. Although irregularly built
the town is pleasantly situated, and contains two fine squares, on one
of which stands a fine statue of St Boniface. The present cathedral was
built at the beginning of the 18th century on the model of St Peter's at
Rome, but it has an ancient crypt, which contains the bones of St
Boniface and was restored in 1892. Opposite the cathedral is the former
monastery of St Michael, now the episcopal palace. The Michaelskirche,
attached to it, is a small round church built, in imitation of the Holy
Sepulchre, in 822 and restored in 1853. Of other buildings may be
mentioned the Library, with upwards of 80,000 printed books and many
valuable MSS., the stately palace with its gardens and orangery, the
former Benedictine nunnery (founded 1625, and now used as a seminary),
and the Minorite friary (1238) now used as a furniture warehouse. Among
the secular buildings are the fine _Schloss_, the _Bibliothek_, the town
hall and the post office. There are several schools, a hospital founded
in the 13th century, and some new artillery barracks. Many industries
are carried on in Fulda. These include weaving and dyeing, the
manufacture of linen, plush and other textiles and brewing. There are
also railway works in the town. A large trade is done in cattle and
grain, many markets being held here. Fine views are obtained from
several hills in the neighbourhood, among these being the Frauenberg,
the Petersberg and the Kalvarienberg.
Fulda owes its existence to its famous abbey. It became a town in 1208,
and during the middle ages there were many struggles between the abbots
and the townsfolk. During the Peasants' War it was captured by the
rebels and during the Seven Years' War by the Hanoverians. It came
finally into the possession of Prussia in 1866. From 1734 to 1804 Fulda
was the seat of a university, and latterly many assemblies of German
bishops have been held in the town.
The great Benedictine abbey of Fulda occupies the place in the
ecclesiastical history of Germany which Monte Cassino holds in Italy, St
Gall in South Germany, Corvey in Saxony, Tours in France and Iona in
Scotland. Founded in 744 at the instigation of St Boniface by his pupil
Sturm, who was the first abbot, it became the centre of a great
missionary work. It was liberally endowed with land by the princes of
the Carolingian house and others, and soon became one of the most famous
and wealthy establishments of its kind. About 968 the pope declared that
its abbot was primate of all the abbots in Germany and Gaul, and later
he became a prince of the Empire. Fulda was specially famous for its
school, which was the centre of the theological learning of the early
middle ages. Among the teachers here were Alcuin, Hrabanus Maurus, who
was abbot from 822 to 842, and Walafrid Strabo. Early in the 10th
century the monastery was reformed by introducing monks from Scotland,
who were responsible for restoring in its old strictness the Benedictine
rule. Later the abbey lost some of its lands and also its high position,
and some time before the Reformation the days of its glory were over.
Johann von Henneberg, who was abbot from 1529 to 1541, showed some
sympathy with the teaching of the reformers, but the Counter-Reformation
made great progress here under Abbot Balthasar von Dernbach. Gustavus
Adolphus gave the abbey as a principality to William, landgrave of
Hesse, but William's rule only lasted for ten years. In 1752 the abbot
was raised to the rank of a bishop, and Fulda ranked as a
prince-bishopric. This was secularized in 1802, and in quick succession
it belonged to the prince of Orange, the king of France and the
grand-duchy of Frankfort. In 1816 the greater part of the principality
was ceded by Prussia to Hesse-Cassel, a smaller portion being united
with Bavaria. Sharing the fate of Hesse-Cassel, this larger portion was
annexed by Prussia in 1866. In 1829 a new bishopric was founded at
Fulda.
For the town see A. Hartmann, _Zeitgeschichte von Fulda_ (Fulda,
1895); J. Schneider, _Fuhrer durch die Stadt Fulda_ (Fulda, 1899); and
_Chronik von Fulda und dessen Umgebungen_ (1839). For the history of
the abbey see Gegenbaur, _Das Kloster Fulda im Karolinger Zeitalter_
(Fulda, 1871-1874); Arndt, _Geschichte des Hochstifts Fulda_ (Fulda,
1860); and the _Fuldaer Geschichtsblatter_ (1902 fol.).
FULGENTIUS, FABIUS PLANCIADES, Latin grammarian, a native of Africa,
flourished in the first half of the 6th (or the last part of the 5th)
century A.D. He is to be distinguished from Fulgentius, bishop of Ruspe
(468-533), to whom he was probably related, and also from the bishop's
pupil and biographer, Fulgentius Ferrandus. Four extant works are
attributed to him. (1) _Mythologiarum libri iii._, dedicated to a
certain Catus, a presbyter of Carthage, containing 75 myths briefly
told, and then explained in the mystical and allegorical manner of the
Stoics and Neoplatonists. For this purpose the author generally invokes
the aid of etymologies which, borrowed from the philosophers, are highly
absurd. As a Christian, Fulgentius sometimes (but less frequently than
might have been expected) quotes the Bible by the side of the
philosophers, to give a Christian colouring to the moral lesson. (2)
_Expositio Vergilianae continentiae (continentia_ = contents), a sort of
appendix to (1), dedicated to Catus. The poet himself appears to the
author and explains the twelve books of the _Aeneid_ as a picture of
human life. The three words _arma_ (= virtus), _vir_ (= sapientia),
_primus_ (= princeps) in the first line represent respectively
_substantia corporalis, sensualis, ornans_. Book i. symbolizes the birth
and early childhood of man (the shipwreck of Aeneas denotes the peril of
birth), book vi. the plunge into the depths of wisdom. (3) _Expositio
sermonum antiquorum_, explanations of 63 rare and obsolete words,
supported by quotations (sometimes from authors and works that never
existed). It is much inferior to the similar work of Nonius, with which
it is often edited. (4) _Liber absque litteris de aetatibus mundi et
hominis_. In the MS. heading of this work, the name of the author is
given as Fabius Claudius Gordianus Fulgentius (Claudius is the name of
the father, and Gordianus that of the grandfather of the bishop, to whom
some attribute the work). The title _Absque litteris_ indicates that one
letter of the alphabet is wholly omitted in each successive book (A in
bk. i., B in bk. ii.). Only 14 books are preserved. The matter is
chiefly taken from sacred history. In addition to these, Fulgentius
speaks of early poetical attempts after the manner of Anacreon, and of a
work called _Physiologus_, dealing with medical questions, and including
a discussion of the mystical signification of the numbers 7 and 9.
Fulgentius is a representative of the so-called late African style,
taking for his models Apuleius, Tertullian and Martianus Capella. His
language is bombastic, affected and incorrect, while the lengthy and
elaborate periods make it difficult to understand his meaning.
See the edition of the four works by R. Helm (1898, Teubner series);
also M. Zink, _Der Mytholog Fulgentius_ (1867); E. Jungmann, "De
Fulgentii aetate et scriptis," in _Acta Societatis Philologae
Lipsiensis_, i. (1871); A. Ebert, _Allgemeine Geschichte der Litt. des
Mittelalters_, i.; article "Fulgentius" by C.F. Bohr in Ersch and
Gruber's _Allgemeine Encyklopadie_; Teuffel-Schwabe, _History of Roman
Literature_ (Eng. trans.).
FULGINIAE (mod. _Foligno_), an ancient town of Umbria, Italy, on the
later line of the Via Flaminia, 15 m. S. of Nuceria. It appears to have
been of comparatively late origin, inasmuch as it had no city walls,
but, in imperial times especially, owing to its position on the new line
of the Via Flaminia, it must have increased in importance as being the
point of departure of roads to Perusia and to Picenum over the pass of
Plestia. It appears to have had an amphitheatre, and three bridges over
the Topino are attributed to the Roman period. Three miles to the N.
lies the independent community of Forum Flaminii, the site of which is
marked by the church of S. Giovanni Profiamma, at or near which the
newer line of the Via Flaminia rejoined the older. It was no doubt
founded by the builder of the road, C. Flaminius, consul in 220 B.C.
(See FOLIGNO and FLAMINIA, VIA.) (T. As.)
FULGURITE (from Lat. _fulgur_, lightning), in petrology, the name given
to rocks which have been fused on the surface by lightning, and to the
characteristic holes in rocks formed by the same agency. When lightning
strikes the naked surfaces of rocks, the sudden rise of temperature may
produce a certain amount of fusion, especially when the rocks are dry
and the electricity is not readily conducted away. Instances of this
have been observed on Ararat and on several mountains in the Alps,
Pyrenees, &c. A thin glassy crust, resembling a coat of varnish, is
formed; its thickness is usually not more than one-eighth of an inch,
and it may be colourless, white or yellow. When examined under the
microscope, it usually shows no crystallization, and contains minute
bubbles due to the expansion of air or other gases in the fused
pellicle. Occasionally small microliths may appear, but this is uncommon
because so thin a film would cool with extreme rapidity. The minerals of
the rock beneath are in some cases partly fused, but the more refractory
often appear quite unaffected. The glass has arisen from the melting of
the most fusible ingredients alone.
Another type of fulgurite is commonest in dry sands and takes the shape
of vertical tubes which may be nearly half an inch in diameter.
Generally they are elliptical in cross section, or flattened by the
pressure exerted by the surrounding sand on the fulgurite at a time when
it was still very hot and plastic. These tubes are often vertical and
may run downwards for several feet through the sand, branching and
lessening as they descend. Tubular perforations in hard rocks have been
noted also, but these are short and probably follow original cracks. The
glassy material contains grains of sand and many small round or
elliptical cavities, the long axes of which are radial. Minerals like
felspar and mica are fused more readily than quartz, but analysis shows
that some fulgurite glasses are very rich in silica, which perhaps was
dissolved in the glass rather than simply fused. The central cavity of
the tube and the bubbles in its walls point to the expansion of the
gases (air, water, &c.) in the sand by sudden and extreme heating. Very
fine threads of glass project from the surface of the tube as if fused
droplets had been projected outwards with considerable force. Where the
quartz grains have been greatly heated but not melted they become white
and semi-opaque, but where they are in contact with the glass they
usually show partial solution. Occasionally crystallization has begun
before the glass solidified, and small microliths, the nature of which
is undeterminable, occur in streams and wisps in the clear hyaline
matrix. (J. S. F.)
FULHAM, a western metropolitan borough of London, England, bounded N.W.
by Hammersmith, N.E. by Kensington, E. by Chelsea, and S.E., S. and S.W.
by the river Thames. Pop. (1901) 137,289. The principal thoroughfares
are Fulham Palace Road running S. from Hammersmith, Fulham Road and
King's Road, W. from Chelsea, converging and leading to Putney Bridge
over the Thames; North End Road between Hammersmith and Fulham Roads;
Lillie Road between South Kensington and Fulham Palace Road; and
Wandsworth Bridge Road leading S. from New King's Road to Wandsworth
Bridge. In the north Fulham includes the residential district known as
West Kensington, and farther south that of Walham Green. The manor house
or palace of the bishops of London stands in grounds, beautifully
planted and surrounded by a moat, believed to be a Danish work, near the
river west of Putney Bridge. Its oldest portion is the picturesque
western quadrangle, built by Bishop Fitzjames (1506-1522). The parish
church of All Saints, between the bridge and the grounds, was erected in
1881 from designs by Sir Arthur Blomfield. The fine old monuments from
the former building, dating from the 16th to the 18th centuries, are
mostly preserved, and in the churchyard are the memorials of several
bishops of London and of Theodore Hook (1841). The public recreation
grounds include the embankment and gardens between the river and the
palace grounds, and there are also two well-known enclosures used for
sports within the borough. Of these Hurlingham Park is the headquarters
of the Hurlingham Polo Club and a fashionable resort; and Queen's Club,
West Kensington, has tennis and other courts for the use of members, and
is also the scene of important football matches, and of the athletic
meetings between Oxford and Cambridge Universities, and those between
the English and American Universities held in England. In Seagrave Road
is the Western fever hospital. The parliamentary borough of Fulham
returns one member. The borough council consists of a mayor, 6 aldermen
and 36 councillors. Area, 1703.5 acres.
Fulham, or in its earliest form _Fullanham_, is uncertainly stated to
signify "the place" either "of fowls" or "of dirt." The manor is said to
have been given to Bishop Erkenwald about the year 691 for himself and
his successors in the see of London, and Holinshed relates that the
Bishop of London was lodging in his manor place in 1141 when Geoffrey de
Mandeville, riding out from the Tower of London, took him prisoner. At
the Commonwealth the manor was temporarily out of the bishops' hands,
being sold to Colonel Edmund Harvey. There is no record of the first
erection of a parish church, but the first known rector was appointed in
1242, and a church probably existed a century before this. The earliest
part of the church demolished in 1881, however, did not date farther
back than the 15th century. In 879 Danish invaders, sailing up the
Thames, wintered at Fulham and Hammersmith. Near the former wooden
Putney Bridge, built in 1729 and replaced in 1886, the earl of Essex
threw a bridge of boats across the river in 1642 in order to march his
army in pursuit of Charles I., who thereupon fell back on Oxford.
Margravine Road recalls the existence of Bradenburg House, a riverside
mansion built by Sir Nicholas Crispe in the time of Charles I., used as
the headquarters of General Fairfax in 1647 during the civil wars, and
occupied in 1792 by the margrave of Bradenburg-Anspach and Bayreuth and
his wife, and in 1820 by Caroline, consort of George IV.
FULK, king of Jerusalem (b. 1092), was the son of Fulk IV., count of
Anjou, and his wife Bertrada (who ultimately deserted her husband and
became the mistress of Philip I. of France). He became count of Anjou in
1109, and considerably added to the prestige of his house. In particular
he showed himself a doughty opponent to Henry I. of England, against
whom he continually supported Louis VI. of France, until in 1127 Henry
won him over by betrothing his daughter Matilda to Fulk's son Geoffrey
Plantagenet. Already in 1120 Fulk had visited the Holy Land, and become
a close friend of the Templars. On his return he assigned to the order
of the Templars an annual subsidy, while he also maintained two knights
in the Holy Land for a year. In 1128 he was preparing to return to the
East, when he received an embassy from Baldwin II., king of Jerusalem,
who had no male heir to succeed him, offering his daughter Melisinda in
marriage, with the right of eventual succession to the kingdom. Fulk
readily accepted the offer; and in 1129 he came and was married to
Melisinda, receiving the towns of Acre and Tyre as her dower. In 1131,
at the age of thirty-nine, he became king of Jerusalem. His reign is not
marked by any considerable events: the kingdom which had reached its
zenith under Baldwin II., and did not begin to decline till the capture
of Edessa in the reign of Baldwin III., was quietly prosperous under his
rule. In the beginning of his reign he had to act as regent of Antioch,
and to provide a husband, Raymund of Poitou, for the infant heiress
Constance. But the great problem with which he had to deal was the
progress of the atabeg Zengi of Mosul. In 1137 he was beaten near Barin,
and escaping into the fort was surrounded and forced to capitulate. A
little later, however, he greatly improved his position by strengthening
his alliance with the vizier of Damascus, who also had to fear the
progress of Zengi (1140); and in this way he was able to capture the
fort of Banias, to the N. of Lake Tiberias. Fulk also strengthened the
kingdom on the south; while his butler, Paganus, planted the fortress of
Krak to the south of the Dead Sea, and helped to give the kingdom an
access towards the Red Sea, he himself constructed Blanche Garde and
other forts on the S.W. to overawe the garrison of Ascalon, which was
still held by the Mahommedans, and to clear the road towards Egypt.
Twice in Fulk's reign the eastern emperor, John Comnenus, appeared in
northern Syria (1137 and 1142); but his coming did not affect the king,
who was able to decline politely a visit which the emperor proposed to
make to Jerusalem. In 1143 he died, leaving two sons, who both became
kings, as Baldwin III. and Amalric I.
Fulk continued the tradition of good statesmanship and sound
churchmanship which Baldwin I. and Baldwin II. had begun. William of
Tyre speaks of him as a fine soldier, an able politician, and a good son
of the church, and only blames him for partiality to his friends, and a
forgetfulness of names and faces, which placed him at a disadvantage and
made him too dependent on his immediate intimates. Little, perhaps, need
be made of these censures: the real fault of Fulk was his neglect to
envisage the needs of the northern principalities, and to head a
combined resistance to the rising power of Zengi of Mosul.
His reign in Jerusalem is narrated by R. Rohricht (_Geschichte des
Konigreichs Jerusalem_, Innsbruck, 1898), and has been made the
subject of a monograph by G. Dodu (_De Fulconis Hierosolymitani
regno_, Paris, 1894). (E. Br.)
FULK (d. 900), archbishop of Reims, and partisan of Charles the Simple
in his struggle with Odo, count of Paris, was elected to the see as
archbishop in 883 upon the death of Hincmar. In 887 he was engaged in a
struggle with the Normans who invaded his territories. Upon the
deposition of Charles the Fat he sided with Charles the Simple in his
contest for the West Frankish dominions against Count Odo of Paris, and
crowned him king in his own metropolitan church at Reims after most of
the nobles had gone over to Odo (893). Upon the death of Odo he
succeeded in having Charles recognized as king by a majority of the West
Frankish nobility. In 892 he obtained special privileges for his
province from Pope Formosus, who promised that thereafter, when the
archbishopric became vacant, the revenues should not be enjoyed by
anyone while the vacancy existed, but should be reserved for the new
incumbent, provided the election took place within the canonical limit
of three months. From 898 until his death he held the office of
chancellor, which for some time afterwards was regularly filled by the
archbishop of Reims. In his efforts to keep the wealthy abbeys and
benefices of the church out of the hands of the nobles, he incurred the
hatred of Baldwin, count of Flanders, who secured his assassination on
the 17th of June 900, a crime which the weak Carolingian monarch left
unpunished.
Fulk left some letters, which are collected in Migne, _Patrologia
Latina_, vol. cxxxi. 11-14.
FULKE, WILLIAM (1538-1589), Puritan divine, was born in London and
educated at Cambridge. After studying law for six years, he became a
fellow at St John's College, Cambridge, in 1564. He took a leading part
in the "vestiarian" controversy, and persuaded the college to discard
the surplice. In consequence he was expelled from St. John's for a
time, but in 1567 he became Hebrew lecturer and preacher there. After
standing unsuccessfully for the headship of the college in 1569, he
became chaplain to the earl of Leicester, and received from him the
livings of Warley, in Essex, and Dennington in Suffolk. In 1578 he was
elected master of Pembroke Hall, Cambridge. As a Puritan
controversialist he was remarkably active; in 1580 the bishop of Ely
appointed him to defend puritanism against the Roman Catholics, Thomas
Watson, ex-bishop of Lincoln (1513-1584), and John Feckenham, formerly
abbot of Westminster, and in 1581 he was one of the disputants with the
Jesuit, Edmund Campion, while in 1582 he was among the clergy selected
by the privy council to argue against any papist. His numerous polemical
writings include _A Defense of the sincere true Translations of the
holie Scriptures into the English tong_ (London, 1583), and confutations
of Thomas Stapleton (1535-1598), Cardinal Allen and other Roman Catholic
controversialists.
FULK NERRA (c. 970-1040), count of Anjou, eldest son of Count Geoffrey
I., "Grisegonelle" (Grey Tunic) and Adela of Vermandois, was born about
970 and succeeded his father in the countship of Anjou on the 21st of
July 987. He was successful in repelling the attacks of the count of
Rennes and laying the foundations of the conquest of Touraine (see
ANJOU). In this connexion he built a great number of strong castles,
which has led in modern times to his being called "the great builder."
He also founded several religious houses, among them the abbeys of
Beaulieu, near Loches (c. 1007), of Saint-Nicholas at Angers (1020) and
of Ronceray at Angers (1028), and, in order to expiate his crimes of
violence, made three pilgrimages to the Holy Land (in 1002-1003, c. 1008
and in 1039). On his return from the third of these journeys he died at
Metz in Lorraine on the 21st of June 1040. By his first marriage, with
Elizabeth, daughter of Bouchard le Venerable, count of Vendome, he had a
daughter, Adela, who married Boon of Nevers and transmitted to her
children the countship of Vendome. Elizabeth having died in 1000, Fulk
married Hildegarde of Lorraine, by whom he had a son, Geoffrey Martel
(q.v.), and a daughter Ermengarde, who married Geoffrey, count of
Gatinais, and was the mother of Geoffrey "le Barbu" (the Bearded) and of
Fulk "le Rechin" (see ANJOU).
See Louis Halphen, _Le Comte d'Anjou au XI^e siecle_ (Paris, 1906).
The biography of Fulk Nerra by Alexandre de Salies, _Histoire de
Foulques Nerra_ (Angers, 1874) is confused and uncritical. A very
summary biography is given by Celestin Port, _Dictionnaire historique,
geographique et biographique de Maine-et-Loire_ (3 vols.,
Paris-Angers, 1874-1878), vol. ii. pp. 189-192, and there is also a
sketch in Kate Norgate, _England under the Angevin Kings_ (2 vols.,
London, 1887), vol. i. ch. iii. (L. H.*)
FULLEBORN, GEORG GUSTAV (1769-1803), German philosopher, philologist and
miscellaneous writer, was born at Glogau, Silesia, on the 2nd of March
1769, and died at Breslau on the 6th of February 1803. He was educated
at the University of Halle, and was made doctor of philosophy in
recognition of his thesis _De Xenophane, Zenone et Gorgia_. He took
diaconal orders in 1791, but almost immediately became professor of
classics at Breslau. His philosophical works include annotations to
Garve's translation of the _Politics_ of Aristotle (1799-1800), and a
large share in the _Beitrage zur Geschichte der Philosophie_ (published
in twelve parts between 1791 and 1799), in which he collaborated with
Forberg, Reinhold and Niethammer. In philology he wrote _Encyclopaedia
philologica sive primae lineae Isagoges in antiquorum studia_ (1798; 2nd
ed., 1805); _Kurze Theorie des lateinischen Stils_ (1793); _Leitfaden
der Rhetorik_ (1802); and an annotated edition of the _Satires_ of
Persius. Under the pseudonym "Edelwald Justus" he published several
collections of popular tales--_Bunte Blatter_ (1795); _Kleine Schriften
zur Unterhaltung_ (1798); _Nebenstunden_ (1799). After his death were
published _Taschenbuch fur Brunnengaste_ (1806) and _Kanzelreden_
(1807). He was a frequent contributor to the press, where his writings
were very popular.
See Schummel, _Gedachtnisrede_ (1803) and _Garve und Fulleborn_;
Meusel, _Gelehrtes Teutschland_, vol. ii.
FULLER, ANDREW (1754-1815), English Baptist divine, was born on the 6th
of February 1754, at Wicken in Cambridgeshire. In his boyhood and youth
he worked on his father's farm. In his seventeenth year he became a
member of the Baptist church at Soham, and his gifts as an exhorter met
with so much approval that, in the spring of 1775, he was called and
ordained as pastor of that congregation. In 1782 he removed to Kettering
in Northamptonshire, where he became friendly with some of the most
eminent ministers of the denomination. Before leaving Soham he had
written the substance of a treatise in which he had sought to counteract
the prevailing Baptist hyper-Calvinism which, "admitting nothing
spiritually good to be the duty of the unregenerate, and nothing to be
addressed to them in a way of exhortation excepting what related to
external obedience," had long perplexed his own mind. This work he
published, under the title _The Gospel worthy of all Acceptation_, soon
after his settlement in Kettering; and although it immediately involved
him in a somewhat bitter controversy which lasted for nearly twenty
years, it was ultimately successful in considerably modifying the views
prevalent among English dissenters. In 1793 he published a treatise,
_The Calvinistic and Socinian systems examined and compared as to their
moral tendency_, in which he rebutted the accusation of antinomianism
levelled by the Socinians against those who over-emphasized the
doctrines of free grace. This work, along with another against Deism,
entitled _The Gospel its own Witness_, is regarded as the production on
which his reputation as a theologian mainly rests. Fuller also published
an admirable _Memoir of the Rev. Samuel Pearce_, of Birmingham, and a
volume of _Expository Lectures in Genesis_, besides a considerable
number of smaller pieces, chiefly sermons and pamphlets, which were
issued in a collected form after his death. He was a man of forceful
character, more prominent on the practical side of religion than on the
devotional, and accordingly not pre-eminently successful in his local
ministry. His great work was done in connexion with the Baptist
Missionary Society, formed at Kettering in 1792, of which he was
secretary until his death on the 7th of May 1815. Both Princeton and
Yale, U.S.A., conferred on him the degree of D. D., but he never used
it.
Several editions of his collected works have appeared, and a _Memoir_,
principally compiled from his own papers, was published about a year
after his decease by Dr Ryland, his most intimate friend and coadjutor
in the affairs of the Baptist mission. There is also a biography by
the Rev. J.W. Morris (1816); and his son prefixed a memoir to an
edition of his chief works in Bohn's Standard Library (1852).
FULLER, GEORGE (1822-1884), American figure and portrait painter, was
born at Deerfield, Massachusetts, in 1822. At the age of twenty he
entered the studio of the sculptor H.K. Brown, at Albany, New York,
where he drew from the cast and modelled heads. Having attained some
proficiency he went about the country painting portraits, settling at
length in Boston, where he studied the works of the earlier Americans,
Stuart, Copley and Allston. After three years in that city, and twelve
in New York, where in 1857 he was elected a member of the National
Academy of Design, he went to Europe for a brief visit and for study.
During all this time his work had received little recognition and
practically no financial encouragement, and on his return he settled on
the family farm at Deerfield, where he continued to work in his own way
with no thought of the outside world. In 1876, however, he was forced by
pressing needs to dispose of his work, and he sent some pictures to a
dealer in Boston, where he met with immediate success, financial and
artistic, and for the remaining eight years of his life he never lacked
patrons. He died in Boston on the 21st of March 1884. He was a poetic
painter, and a dreamer of delicate fancies and quaint, intangible phases
of nature, his canvases being usually enveloped in a brown mist that
renders the outlines vague. Among his noteworthy canvases are: "The
Turkey Pasture," "Romany Girl," "And she was a Witch," "Nydia,"
"Winifred Dysart" and "The Quadroon."
FULLER, MARGARET, Marchioness Ossoli (1810-1850), American authoress,
eldest child of Timothy Fuller (1778-1835), a lawyer and politician of
some eminence, was born at Cambridgeport, Massachusetts, on the 23rd of
May 1810. Her education was conducted by her father, who, she states,
made the mistake of thinking to "gain time by bringing forward the
intellect as early as possible," the consequence being "a premature
development of brain that made her a youthful prodigy by day, and by
night a victim of spectral illusions, nightmare and somnambulism." At
six years she began to read Latin, and at a very early age she had
selected as her favourite authors Shakespeare, Cervantes and Moliere.
Soon the great amount of study exacted of her ceased to be a burden, and
reading became a habit and a passion. Having made herself familiar with
the masterpieces of French, Italian and Spanish literature, she in 1833
began the study of German, and within the year had read some of the
masterpieces of Goethe, Korner, Novalis and Schiller.
After her father's death in 1835 she went to Boston to teach languages,
and in 1837 she was chosen principal teacher in the Green Street school,
Providence, Rhode Island, where she remained till 1839. From this year
until 1844 she stayed at different places in the immediate neighbourhood
of Boston, forming an intimate acquaintance with the colonists of Brook
Farm, and numbering among her closest friends R.W. Emerson, Nathaniel
Hawthorne and W.H. Channing. In 1839 she published a translation of
Eckermann's _Conversations with Goethe_, which was followed in 1842 by a
translation of the correspondence between Karoline von Gunderode and
Bettina von Arnim, entitled _Gunderode_. Aided by R.W. Emerson and
George Ripley, she in 1840 started _The Dial_, a poetical and
philosophical magazine representing the opinions and aims of the New
England Transcendentalists. This journal she continued to edit for two
years, and while in Boston she also conducted conversation classes for
ladies in which philosophical and social subjects were discussed with a
somewhat over-accentuated earnestness. These meetings may be regarded as
perhaps the beginning of the modern movement in behalf of women's
rights. R.W. Emerson, who had met her as early as 1836, thus describes
her appearance: "She was then twenty-six years old. She had a face and
frame that would indicate fulness and tenacity of life. She was rather
under the middle height; her complexion was fair, with strong fair hair.
She was then, as always, carefully and becomingly dressed, and of
ladylike self-possession. For the rest her appearance had nothing
prepossessing. Her extreme plainness, a trick of incessantly opening and
shutting her eyelids, the nasal tone of her voice, all repelled; and I
said to myself we shall never get far." On better acquaintance this
unprepossessing exterior seemed, however, to melt away, and her
inordinate self-esteem to be lost in the depth and universality of her
sympathy. She possessed an almost irresistible power of winning the
intellectual and moral confidence of those with whom she came in
contact, and "applied herself to her companion as the sponge applies
itself to water." She obtained from each the best they had to give. It
was indeed more as a conversationalist than as a writer that she earned
the title of the Priestess of Transcendentalism. It was her intimate
friends who admired her most. Smart and pungent though she is as a
writer, the apparent originality of her views depends more on
eccentricity than either intellectual depth or imaginative vigour. In
1844 she removed to New York at the desire of Horace Greeley to write
literary criticism for _The Tribune_, and in 1846 she published a
selection from her articles on contemporary authors in Europe and
America, under the title _Papers on Literature and Art_. The same year
she paid a visit to Europe, passing some time in England and France, and
finally taking up her residence in Italy. There she was married in
December 1847 to the marquis Giovanni Angelo Ossoli, a friend of
Mazzini. During 1848-1849 she was present with her husband in Rome, and
when the city was besieged she, at the request of Mazzini, took charge
of one of the two hospitals while her husband fought on the walls. In
May 1850, along with her husband and infant son, she embarked at Leghorn
for America, but when they had all but reached their destination the
vessel was wrecked on Fire Island beach on the 16th of June, and the
Ossolis were among the passengers who perished.
_Life Without and Life Within_ (Boston, 1860) is a collection of
essays, poems, &c., supplementary to her _Collected Works_, printed in
1855. See the _Autobiography of Margaret Fuller Ossoli_, with
additional memoirs by J.F. Clarke, R.W. Emerson and W.H. Channing (2
vols., Boston, 1852); also _Margaret Fuller (Marchesa Ossoli)_, by
Julia Ward Howe (1883), in the "Eminent Women" series; _Margaret
Fuller Ossoli_ (Boston, 1884), by Thomas Wentworth Higginson in the
"American Men of Letters" series, which is based largely on unedited
material; and _The Love Letters of Margaret Fuller, 1845-1846_ (London
and New York, 1903), with an introduction by Julia Ward Howe.
FULLER, MELVILLE WESTON (1833-1910), American jurist, chief justice of
the Supreme Court of the United States, was born at Augusta, Maine, on
the 11th of February 1833. After graduating at Bowdoin College in 1853
he spent a year at the Harvard Law School, and in 1855 began the
practice of law at Augusta, where he was an associate-editor of a
Democratic paper, _The Age_, and served in the city council and as city
attorney. In 1856 he removed to Chicago, Illinois, where he continued to
practise until 1888, rising to a high position at the bar of the
Northwest. For some years he was active in Democratic politics, being a
member of the Illinois Constitutional Convention in 1862 and of the
State House of Representatives from 1863 to 1865. He was a delegate to
various National conventions of his party, and in that of 1876 placed
Thomas A. Hendricks in nomination for the presidency. In 1888, by
President Cleveland's appointment, he succeeded Morrison R. Waite as
chief-justice of the Supreme Court of the United States. In 1899 he was
appointed by President McKinley a member of the arbitration commission
at Paris to settle the Venezuela-British Guiana boundary dispute.
FULLER, THOMAS (1608-1661), English divine and historian, eldest son of
Thomas Fuller, rector of Aldwincle St Peter's, Northamptonshire, was
born at his father's rectory and was baptized on the 19th of June 1608.
Dr John Davenant, bishop of Salisbury, was his uncle and godfather.
According to Aubrey, Fuller was "a boy of pregnant wit." At thirteen he
was admitted to Queens' College, Cambridge, then presided over by Dr
John Davenant. His cousin, Edward Davenant, was a tutor in the same
college. He was apt and quick in study; and in Lent 1624-1625 he became
B.A. and in July 1628 M.A. Being overlooked in an election of fellows of
his college, he was removed by Bishop Davenant to Sidney Sussex College,
November 1628. In 1630 he received from Corpus Christi College the
curacy of St Benet's, Cambridge.
Fuller's quaint and humorous oratory soon attracted attention. He
published in 1631 a poem on the subject of David and Bathsheba, entitled
_David's Hainous Sinne, Heartie Repentance, Heavie Punishment_. In June
of the same year his uncle gave him a prebend in Salisbury, where his
father, who died in the following year, held a canonry. The rectory of
Broadwindsor, Dorsetshire, then in the diocese of Bristol, was his next
preferment (1634); and on the 11th of June 1635 he proceeded B.D. At
Broadwindsor he compiled _The Historie of the Holy Warre_ (1639), a
history of the crusades, and _The Holy State and the Prophane State_
(1642). This work describes the holy state as existing in the family and
in public life, gives rules of conduct, model "characters" for the
various professions and profane biographies. It was perhaps the most
popular of all his writings. He was in 1640 elected proctor for Bristol
in the memorable convocation of Canterbury, which assembled with the
Short Parliament. On the sudden dissolution of the latter he joined
those who urged that convocation should likewise dissolve as usual. That
opinion was overruled; and the assembly continued to sit by virtue of a
royal writ. Fuller has left in his _Church History_ a valuable account
of the proceedings of this synod, for sitting in which he was fined
L200, which, however, was never exacted. His first published volume of
sermons appeared in 1640 under the title of _Joseph's party-coloured
Coat_, which contains many of his quaint utterances and odd conceits.
His grosser mannerisms of style, derived from the divines of the former
generation, disappeared for the most part in his subsequent discourses.
About 1640 he had married Eleanor, daughter of Hugh Grove of Chisenbury,
Wiltshire. She died in 1641. Their eldest child, John, baptized at
Broadwindsor by his father, 6th June 1641, was afterwards rector of
Sidney Sussex College, edited the _Worthies of England_, 1662, and
became rector of Great Wakering, Essex, where he died in 1687.
At Broadwindsor, early in the year 1641, Thomas Fuller, his curate Henry
Sanders, the church wardens, and others, nine persons altogether,
certified that their parish, represented by 242 grown-up male persons,
had taken the Protestation ordered by the speaker of the Long
Parliament. Fuller was not formally dispossessed of his living and
prebend on the triumph of the Presbyterian party, but he relinquished
both preferments about this time. For a short time he preached with
success at the Inns of Court, and thence removed, at the invitation of
the master of the Savoy (Dr Balcanqual) and the brotherhood of that
foundation, to be lecturer at their chapel of St Mary Savoy. Some of the
best discourses of the witty preacher were delivered at the Savoy to
audiences which extended into the chapel-yard. In one he set forth with
searching and truthful minuteness the hindrances to peace, and urged the
signing of petitions to the king at Oxford, and to the parliament, to
continue their care in advancing an accommodation. In his _Appeal of
Injured Innocence_ Fuller says that he was once deputed to carry a
petition to the king at Oxford. This has been identified with a petition
entrusted to Sir Edward Wardour, clerk of the pells, Dr Dukeson, "Dr
Fuller," and four or five others from the city of Westminster and the
parishes contiguous to the Savoy. A pass was granted by the House of
Lords, on the 2nd of January 1643, for an equipage of two coaches, four
or six horses and eight or ten attendants. On the arrival of the
deputation at Uxbridge, on the 4th of January, officers of the
Parliamentary army stopped the coaches and searched the gentlemen; and
they found upon the latter "two scandalous books arraigning the
proceedings of the House," and letters with ciphers to Lord Viscount
Falkland and the Lord Spencer. Ultimately a joint order of both Houses
remanded the party; and Fuller and his friends suffered a brief
imprisonment. The Westminster Petition, notwithstanding, reached the
king's hands; and it was published with the royal reply (see J.E.
Bailey, _Life of Thomas Fuller_, pp. 245 _et seq._). When it was
expected, three months later, that a favourable result would attend the
negotiations at Oxford, Fuller preached a sermon at Westminster Abbey,
on the 27th of March 1643, on the anniversary of Charles I.'s accession,
on the text, "Yea, let him take all, so my Lord the King return in
peace." On Wednesday, the 26th of July, he preached on church
reformation, satirizing the religious reformers, and maintaining that
only the Supreme Power could initiate reforms.
He was now obliged to leave London, and in August 1643 he joined the
king at Oxford. He lived in a hired chamber at Lincoln College for 17
weeks. Thence he put forth a witty and effective reply to John
Saltmarsh, who had attacked his views on ecclesiastical reform. Fuller
subsequently published by royal request a sermon preached on the 10th of
May 1644, at St Mary's, Oxford, before the king and Prince Charles,
called _Jacob's Vow_.
The spirit of Fuller's preaching, always characterized by calmness and
moderation, gave offence to the high royalists, who charged him with
lukewarmness in their cause. To silence unjust censures he became
chaplain to the regiment of Sir Ralph Hopton. For the first five years
of the war, as he said, when excusing the non-appearance of his _Church
History_, "I had little list or leisure to write, fearing to be made a
history, and shifting daily for my safety. All that time I could not
live to study, who did only study to live." After the defeat of Hopton
at Cheriton Down, Fuller retreated to Basing House. He took an active
part in its defence, and his life with the troops caused him to be
afterwards regarded as one of "the great cavalier parsons." In his
marches with his regiment round about Oxford and in the west, he devoted
much time to the collection of details, from churches, old buildings,
and the conversation of ancient gossips, for his _Church-History_ and
_Worthies of England_. He compiled in 1645 a small volume of prayers and
meditations,--the _Good Thoughts in Bad Times_,--which, set up and
printed in the besieged city of Exeter, whither he had retired, was
called by himself "the first fruits of Exeter press." It was inscribed
to Lady Dalkeith, governess to the infant princess, Henrietta Anne (b.
1644), to whose household he was attached as chaplain. The corporation
gave him the Bodleian lectureship on the 21st of March 1645/6, and he
held it until the 17th of June following, soon after the surrender of
the city to the parliament. _The Fear of losing the Old Light_ (1646)
was his farewell discourse to his Exeter friends. Under the Articles of
Surrender Fuller made his composition with the government at London, his
"delinquency" being that he had been present in the king's garrisons. In
_Andronicus, or the Unfortunate Politician_ (1646), partly authentic and
partly fictitious, he satirized the leaders of the Revolution; and for
the comfort of sufferers by the war he issued (1647) a second devotional
manual, entitled _Good Thoughts in Worse Times_, abounding in fervent
aspirations, and drawing moral lessons in beautiful language out of the
events of his life or the circumstances of the time. In grief over his
losses, which included his library and manuscripts (his "upper and
nether millstone"), and over the calamities of the country, he wrote his
work on the _Cause and Cure of a Wounded Conscience_ (1647). It was
prepared at Boughton House in his native county, where he and his son
were entertained by Edward Lord Montagu, who had been one of his
contemporaries at the university and had taken the side of the
parliament.
For the next few years of his life Fuller was mainly dependent upon his
dealings with booksellers, of whom he asserted that none had ever lost
by him. He made considerable progress in an English translation from the
MS. of the _Annales_ of his friend Archbishop Ussher. Amongst his
benefactors it is curious to find Sir John Danvers of Chelsea, the
regicide. Fuller in 1647 began to preach at St Clement's, Eastcheap, and
elsewhere in the capacity of lecturer. While at St Clement's he was
suspended; but speedily recovering his freedom, he preached wherever he
was invited. At Chelsea, where also he occasionally officiated, he
covertly preached a sermon on the death of Charles I., but he did not
break with his Roundhead patrons. James Hay, 2nd earl of Carlisle, made
him his chaplain, and presented him in 1648 or 1649 to the curacy of
Waltham Abbey. His possession of the living was in jeopardy on the
appointment of Cromwell's "Tryers"; but he evaded their inquisitorial
questions by his ready wit. He was not disturbed at Waltham in 1655,
when the Protector's edict prohibited the adherents of the late king
from preaching. Lionel, 3rd earl of Middlesex, who lived at Copt Hall,
near Waltham, gave him what remained of the books of the lord treasurer
his father; and through the good offices of the marchioness of Hertford,
part of his own pillaged library was restored to him. Fuller was thus
able to prosecute his literary labours, producing successively his
descriptive geography of the Holy Land, called _A Pisgah-Sight of
Palestine_ (1650), and his _Church-History of Britain_ (1655), from the
birth of Jesus Christ until the year 1648. With the _Church-History_ was
printed _The History of the University of Cambridge since the Conquest_
and _The History of Waltham Abbey_. These works were furthered in no
slight degree by his connexion with Sion College, London, where he had a
chamber, as well for the convenience of the press as of his city
lectureships. The _Church-History_ was angrily attacked by Dr P. Heylyn,
who, in the spirit of High-Churchmanship, wished, as he said, to
vindicate the truth, the church and the injured clergy. About 1652
Fuller married his second wife, Mary Roper, youngest sister of Thomas,
Viscount Baltinglass, by whom he had several children. At the Oxford Act
of 1657, Robert South, who was _Terrae filius_, lampooned Fuller, whom
he described in this _Oratio_ as living in London, ever scribbling and
each year bringing forth new _folia_ like a tree. At length, continues
South, the _Church-History_ came forth with its 166 dedications to
wealthy and noble friends; and with this huge volume under one arm, and
his wife (said to be little of stature) on the other, he ran up and
down the streets of London, seeking at the houses of his patrons
invitations to dinner, to be repaid by his dull jests at table.
His last and best patron was George Berkeley, 1st Earl Berkeley
(1628-1698), of Cranford House, Middlesex, whose chaplain he was, and
who gave him Cranford rectory (1658). To this nobleman Fuller's reply to
Heylyn's _Examen Historicum_, called _The Appeal of Injured Innocence_
(1659), was inscribed. At the end of the _Appeal_ is an epistle "to my
loving friend Dr Peter Heylyn," conceived in the admirable Christian
spirit which characterized all Fuller's dealings with controversialists.
"Why should _Peter_," he asked, "fall out with _Thomas_, both being
disciples to the same Lord and Master? I assure you, sir, whatever you
conceive to the contrary, I am cordial to the cause of the English
Church, and my hoary hairs will go down to the grave in sorrow for her
sufferings."
In _An Alarum to the Counties of England and Wales_ (1660) Fuller argued
for a free and full parliament--free from force, as he expressed it, as
well as from abjurations or previous engagements. _Mixt Contemplations
in Better Times_ (1660), dedicated to Lady Monk, tendered advice in the
spirit of its motto, "Let your moderation be known to all men: the Lord
is at hand." There is good reason to suppose that Fuller was at the
Hague immediately before the Restoration, in the retinue of Lord
Berkeley, one of the commissioners of the House of Lords, whose last
service to his friend was to interest himself in obtaining him a
bishopric. _A Panegyrick to His Majesty on his Happy Return_ was the
last of Fuller's verse-efforts. On the 2nd of August, by royal letters,
he was admitted D.D. at Cambridge. He resumed his lectures at the Savoy,
where Samuel Pepys heard him preach; but he preferred his conversation
or his books to his sermons. Fuller's last promotion was that of
chaplain in extraordinary to Charles II. In the summer of 1661 he
visited the west in connexion with the business of his prebend, which
had been restored to him. On Sunday, the 12th of August, while preaching
at the Savoy, he was seized with typhus fever, and died at his new
lodgings in Covent Garden on the 16th of August. He was buried in
Cranford church, where a mural tablet was afterwards set up on the north
side of the chancel, with an epitaph which contains a conceit worthy of
his own pen, to the effect that while he was endeavouring (viz. in _The
Worthies_) to give immortality to others, he himself attained it.
Fuller's wit and vivacious good-humour made him a favourite with men of
both sides, and his sense of humour kept him from extremes. Probably
Heylyn and South had some excuse for their attitude towards his very
moderate politics. "By his particular temper and management," said
Echard (_Hist. of England_, iii. 71), "he weathered the late great storm
with more success than many other great men." He was known as "a perfect
walking library." The strength of his memory was proverbial, and some
amusing anecdotes are connected with it.
His writings were the product of a highly original mind. He had a
fertile imagination and a happy faculty of illustration. Antithetic and
axiomatic sentences abound in his pages, embodying literally the wisdom
of the many in the wit of one. He was "quaint," and something more.
"Wit," said Coleridge, in a well-known eulogy, "was the stuff and
substance of Fuller's intellect. It was the element, the earthen base,
the material which he worked in; and this very circumstance has
defrauded him of his due praise for the practical wisdom of the
thoughts, for the beauty and variety of the truths, into which he shaped
the stuff. Fuller was incomparably the most sensible, the least
prejudiced, great man of an age that boasted a galaxy of great men"
(_Literary Remains_, vol. ii. (1836), pp. 389-390). This opinion was
formed after the perusal of the _Church-History_. That work and _The
History of the Worthies of England_ are unquestionably Fuller's greatest
efforts. They embody the collections of an entire life; and since his
day they have been the delight of many readers. The _Holy State_ has
taken rank amongst the best books of "characters." Charles Lamb made
some selections from Fuller, and had a profound admiration for the
"golden works" of the "dear, fine, silly old angel." Since Lamb's time,
mainly through the appreciative criticisms of S.T. Coleridge, Robert
Southey and others, Fuller's works have received much attention.
There is an elaborate account of the life and writings of Fuller by
William Oldys in the _Biographia Britannica_, vol. iii. (1750), based
on Fuller's own works and the anonymous _Life of ... Dr Thomas Fuller_
(1661; reprinted in a volume of selections by A.L.J. Gosset, 1893).
The completest account of him is _The Life of Thomas Fuller, with
Notices of his Books, his Kinsmen and his Friends_ (1874), by J.E.
Bailey, who gives a detailed bibliography (pp. 713-762) of his works.
_The Worthies of England_ was reprinted by John Nichols (1811) and by
P.A. Nuttall (1840). His _Collected Sermons_ were edited by J.E.
Bailey and W.E.A. Axon in 1891. Fuller's quaint wit lends itself to
selection, and there are several modern volumes of extracts from his
works.
FULLER, WILLIAM (1670-c. 1717), English impostor, was born at Milton in
Kent on the 20th of September 1670. His paternity is doubtful, but he
was related to the family of Herbert. After 1688 he served James II.'s
queen, Mary of Modena, and the Jacobites, seeking at the same time to
gain favour with William III.; and after associating with Titus Oates,
being imprisoned for debt and pretending to reveal Jacobite plots, the
House of Commons in 1692 declared he was an "imposter, cheat and false
accuser." Having stood in the pillory he was again imprisoned until
1695, when he was released; and at this time he took the opportunity to
revive the old and familiar story that Mary of Modena was not the mother
of the prince of Wales. In 1701 he published his autobiographical _Life
of William Fuller_ and some _Original Letters of the late King James_.
Unable to prove the assertions made in his writings he was put in the
pillory, whipped and fined. He died, probably in prison, about 1717.
Fuller's other writings are _Mr William Fuller's trip to Bridewell, with
a full account of his barbarous usage in the pillory; The sincere and
hearty confession of Mr William Fuller_ (1704); and _An humble appeal to
the impartial judgment of all parties in Great Britain_ (1716).
He must be distinguished from WILLIAM FULLER (1608-1675), dean of St
Patrick's (1660), bishop of Limerick (1663), and bishop of Lincoln
(1667), the friend of Samuel Pepys; and also from William Fuller (c.
1580-1659), dean of Ely and later dean of Durham.
FULLER'S EARTH (Ger. _Walkererde_, Fr. _terre a foulon_, _argile
smectique_)--so named from its use by fullers as an absorbent of the
grease and oil of cloth,--a clay-like substance, which from its
variability is somewhat difficult to define. In colour it is most often
greenish, olive-green or greenish-grey; on weathering it changes to a
brown tint or it may bleach. As a rule it falls to pieces when placed in
water and is not markedly plastic; when dry it adheres strongly to the
tongue; since, however, these properties are possessed by many clays
that do not exhibit detergent qualities, the only test of value lies in
the capacity to absorb grease or clarify oil. Fuller's earth has a
specific gravity of 1.7-2.4, and a shining streak; it is usually
unctuous to the touch. Microscopically, it consists of minute
irregular-shaped particles of a mineral that appears to be the result of
a chloritic or talcose alteration of a felspar. The small size of most
of the grains, less than .07 mm., makes their determination almost
impossible. Chemical analysis shows that the peculiar properties of this
earth are due to its physical rather than its chemical nature.
The following analyses of the weathered and unweathered condition of
the earth from Nutfield, Surrey, represent the composition of one of
the best known varieties:--
Blue Earth (dried at 100 deg. C.).
Insoluble residue 69.96 | Insoluble residue--
Fe2O3 2.48 | SiO2 62.81
Al2O3 3.46 | Al2O3 3.46
CaO 5.87 | Fe2O3 1.30
MgO 1.41 | CaO 1.53
P2O5 0.27 | MgO 0.86
SO3 0.05 | -----
NaCl 0.05 | 69.96
K2O 0.74 | =====
H2O (combined) 15.57 |
----- |
99.86 |
===== |
Yellow Earth (dried at 100 deg. C.).
Insoluble residue 76.13 | Insoluble residue--
Fe2O3 2.41 | SiO2 59.37
Al2O3 1.77 | Al2O3 10.05
CaO 4.31 | Fe2O3 3.86
MgO 1.05 | CaO 1.86
P2O5 0.14 | MgO 1.04
SO3 0.07 | -----
NaCl 0.14 | 76.18
K2O 0.84 | =====
H2O (combined) 13.19 |
------ |
100.05 |
====== |
(Analysis by P.G. Sanford, _Geol. Mag._, 1889, 6, pp. 456, 526.)
Of other published analyses, not a few show a lower silica content
(44%, 50%), along with a higher proportion of alumina (11%, 23%).
Fuller's earth may occur on any geological horizon; at Nutfield in
Surrey, England, it is in the Cretaceous formations; at Midford near
Bath it is of Jurassic age; at Bala, North Wales, it occurs in
Ordovician strata; in Saxony it appears to be the decomposition product
of a diabasic rock. In America it is found in California in rocks
ranging from Cretaceous to Pleistocene age; in S. Dakota, Custer county
and elsewhere a yellow, gritty earth of Jurassic age is worked; in
Florida and Georgia occurs a brittle, whitish earth of Oligocene age.
Other deposits are worked in Arkansas, Texas, Colorado, Massachusetts
and South Carolina.
Fuller's earth is either mined or dug in the open according to local
circumstances. It is then dried in the sun or by artificial heat and
transported in small lumps in sacks. In other cases it is ground to a
fine powder after being dried; or it is first roughly ground and made
into a slurry with water, which is allowed to carry off the finer from
the coarser particles and deposit them in a creamy state in suitable
tanks. After consolidation this fine material is dried artificially on
drying floors, broken into lumps, and packed for transport. The use of
fuller's earth for cleansing wool and cloth has greatly decreased, but
the demand for the material is as great or greater than it ever was. It
is now used very largely in the filtration of mineral oils, and also for
decolourizing certain vegetable oils. It is employed in the formation of
certain soaps and cleansing preparations.
The term "Fuller's Earth" has a special significance in geology, for it
was applied by W. Smith in 1799 to certain clays in the neighbourhood of
Bath, and the use of the expression is still retained by English
geologists, either in this form or in the generalized "Fullonian." The
Fullonian lies at the base of the Great Oolite or Bathonian series, but
its palaeontological characters place it between that series and the
underlying Inferior Oolite. The zonal fossils are _Perisphinctes
arbustigerus_ and _Macrocephalus subcontractus_ with _Ostrea acuminata_,
_Rhynchonella concinna_ and _Goniomya angulifera_. The formation is in
part the equivalent of the "Vesulien" of J. Marcou (Vesoul in
Haute-Saone). In Dorsetshire and Somersetshire, where it is best
developed, it is represented by an Upper Fuller's Earth Clay, the
Fuller's Earth Rock (an impersistent earthy limestone, usually
fossiliferous), and the Lower Fuller's Earth Clay. Commercial fuller's
earth has been obtained only from the Upper Clay. In eastern
Gloucestershire and northern Oxfordshire the Fuller's Earth passes
downwards without break into the Inferior Oolite; northward it dies out
about Chipping Norton in Oxfordshire and passes laterally into the
Stonesfield Slates series; in the midland counties it may perhaps be
represented by the "Upper Estuarine Series." In parts of Dorsetshire the
clays have been used for brickmaking and the limestone (rock) for local
buildings.
See H.B. Woodward, "Jurassic Rocks of Great Britain," vol. iv. (1894),
_Mem. Geol. Survey_ (London). [J. A. H.]
FULLERTON, LADY GEORGIANA CHARLOTTE (1812-1885), English novelist and
philanthropist, youngest daughter of the 1st Earl Granville, was born at
Tixall Hall in Staffordshire on the 23rd of September 1812. In 1833 she
married Alexander George Fullerton, then an Irish officer in the guards.
After living in Paris for some eight years she and her husband
accompanied Lord Granville to Cannes and thence to Rome. In 1843 her
husband entered the Roman Catholic church, and in the following year
Lady Georgiana Fullerton published her first novel, _Ellen Middleton_,
which attracted W.E. Gladstone's attention in the _English Review_. In
1846 she entered the Roman Catholic church. The death of her only son in
1854 plunged her in grief, and she continued to wear mourning until the
end of her life. In 1856 she became one of the third order of St
Francis, and thenceforward devoted herself to charitable work. In
conjunction with Miss Taylor she founded the religious community known
as "The Poor Servants of the Mother of God Incarnate," and she also took
an active part in bringing to England the sisters of St Vincent of Paul.
Her philanthropic work is described in Mrs Augustus Craven's work _Lady
Georgiana Fullerton, sa vie et ses [oe]uvres_ (Paris, 1888), which was
translated into English by Henry James Coleridge. She died at
Bournemouth on the 19th of January 1885. Among her other novels were
_Grantley Manor_ (1847), _Lady Bird_ (1852), and _Too Strange not to be
True_ (1864).
FULMAR, from the Gaelic _Fulmaire_, the _Fulmarus glacialis_ of modern
ornithologists, one of the largest of the petrels (_Procellariidae_) of
the northern hemisphere, being about the size of the common gull (_Larus
canus_) and not unlike it in general coloration, except that its
primaries are grey instead of black. This bird, which ranges over the
North Atlantic, is seldom seen on the European side below lat. 53 deg.
N., but on the American side comes habitually to lat. 45 deg. or even
lower. In the Pacific it is represented by a scarcely separable form,
_F. glupischa_. It has been commonly believed to have two
breeding-places in the British Islands, namely, St Kilda and South
Barra; but, according to Robert Gray (_Birds of the West of Scotland_,
p. 499), it has abandoned the latter since 1844, though still breeding
in Skye. Northward it established itself about 1838 on Myggenaes Holm,
one of the Faeroes, while it has several stations off the coast of
Iceland and Spitsbergen, as well as at Bear Island. Its range towards
the pole seems to be only bounded by open water, and it is the constant
attendant upon all who are employed in the whale and seal fisheries,
showing the greatest boldness in approaching boats and ships, and
feeding on the offal obtained from them. By British seamen it is
commonly called the "molly mawk"[1] (corrupted from _Mallemuck_), and is
extremely well known to them, its flight, as it skims over the waves,
first with a few beats of the wings and then gliding for a long way,
being very peculiar. It only visits the land to deposit its single white
egg, which is laid on a rocky ledge, where a shallow nest is made in the
turf and lined with a little dried grass. Many of its breeding-places
are a most valuable property to those who live near them and take the
eggs and young, which, from the nature of the locality, are only to be
had at a hazardous risk of life. In St Kilda a large number of the young
are killed in one week of August, the only time when, by the custom of
the community, they are allowed to be taken. These, after the oil is
extracted from them, serve the islanders with food for the winter. The
oil has been chemically analysed and found to be a fish-oil, and to
possess nearly all the qualities of that obtained from the liver of the
cod, with a lighter specific gravity. It, however, has an extremely
strong scent, which is said by those who have visited St Kilda to
pervade every thing and person on the island, and is certainly retained
by an egg or skin of the bird for many years. Whenever a live example is
seized in the hand it ejects a considerable quantity of this oil from
its mouth.
FOOTNOTE:
[1] A name misapplied in the southern hemisphere to _Diomedea
melanophrys_, one of the albatrosses.
FULMINIC ACID, HCNO or H2C2N2O2, an organic acid isomeric with cyanic
and cyanuric acids; its salts, termed fulminates, are very explosive and
are much employed as detonators. The free acid, which is obtained by
treating the salts with acids, is an oily liquid smelling like prussic
acid; it is very explosive, and the vapour is poisonous to about the
same degree as that of prussic acid. The first fulminate prepared was
the "fulminating silver" of L.G. Brugnatelli, who found in 1798 that if
silver be dissolved in nitric acid and the solution added to spirits of
wine, a white, highly explosive powder was obtained. This substance is
to be distinguished from the black "fulminating silver" obtained by
C.L. Berthollet in 1788 by acting with ammonia on precipitated silver
oxide. The next salt to be obtained was the mercuric salt, which was
prepared in 1799 by Edward Charles Howard, who substituted mercury for
silver in Brugnatelli's process. A similar method is that of J. von
Liebig (1823), who heated a mixture of alcohol, nitric acid and mercuric
nitrate; the salt is largely manufactured by processes closely
resembling the last. A laboratory method is to mix solutions of sodium
nitromethane, CH2:NO(ONa), and mercuric chloride, a yellow basic salt
being formed at the same time. Mercuric fulminate is less explosive than
the silver salt, and forms white needles (with 1/2H2O) which are
tolerably soluble in water. The use of mercuric fulminate as a detonator
dates from about 1814, when the explosive cap was invented. It is still
the commonest detonator, but it is now usually mixed with other
substances; the British service uses for percussion caps 6 parts of
fulminate, 6 of potassium chlorate and 4 of antimony sulphide, and for
time fuses 4 parts of fulminate, 6 of potassium chlorate and 4 of
antimony sulphide, the mixture being damped with a shellac varnish; for
use in blasting, a home office order of 1897 prescribes a mixture of 4
parts of fulminate and 1 of potassium chlorate. In 1900 Bielefeldt found
that a fulminate placed on top of an aromatic nitro compound, such as
trinitrotoluene, formed a useful detonator; this discovery has been
especially taken advantage of in Germany, in which country detonators of
this nature are being largely employed. Tetranitromethylaniline (tetryl)
has also been employed (Brit. Pat. 13340 of 1905). It has been proposed
to replace fulminate by silver azoimide (Wohler & Matter, Brit. Pat.
4468 of 1908), and by lead azoimide (Hyronimus, Brit. Pat. 1819 of
1908).
The constitution of fulminic acid has been investigated by many
experimenters, but apparently without definitive results. The
researches of Liebig (1823), Liebig and Gay-Lussac (1824), and of
Liebig again in 1838 showed the acid to be isomeric with cyanic acid,
and probably (HCNO)2, since it gave mixed and acid salts. Kekule, in
1858, concluded that it was nitroacetonitrile, NO2.CH2.CN, a view
opposed by Steiner (1883), E. Divers and M. Kawakita (1884), R. Scholl
(1890), and by J.U. Nef (1894), who proposed the formulae:
C : N.OH / N : CH CH : N.O
.. O / . . C : N.OH.
C : N.OH, \ N : C.OH, CH : N.0,
Steiner, Divers, Scholl, Nef.
The formulae of Kekule, Divers and Armstrong have been discarded, and
it remains to be shown whether Nef's carbonyloxime formula (or the
bimolecular formula of Steiner) or Scholl's glyoxime peroxide formula
is correct. There is some doubt as to the molecular formula of
fulminic acid. The existence of double salts, and the observations of
L. Wohler and K. Theodorovits (_Ber._, 1905, 38, p. 345), that only
compounds containing two carbon atoms yielded fulminates, points to
(HCNO)2; on the other hand, Wohler (_loc. cit._ p. 1351) found that
cryoscopic and electric conductivity measurements showed sodium
fulminate to be NaCNO. Nef based his formula, which involves bivalent
carbon, on many reactions; in particular, that silver fulminate with
hydrochloric acid gave salts of formylchloridoxime, which with water
gave hydroxylamine and formic acid, thus
// NOAg // N.OH
C : NO OAg -> HC // --> HC // --> H.CO2H + H2N.OH,
\ Cl \ OH
and also on the production from sodium nitromethane and mercuric
chloride, thus CH2 : NO.Ohg --> H2O + C : NOhg(hg = 1/2Hg). H. Wieland
and F.C. Palazzo (1907) support this formula, finding that methyl
nitrolic acid, NO2.CH : N.OH, yielded under certain conditions
fulminic acid, and vice versa (Palazzo, 1907). M.Z. Jowitschitsch
(_Ann._, 1906, 347, p. 233) inclines to Scholl's formula; he found
that the synthetic silver salt of glyoxime peroxide resembled silver
fulminate in yielding hydroxylamine with hydrochloric acid, but
differed in being less explosive, and in being soluble in nitric acid.
H. Wieland and his collaborators regard "glyoxime peroxide" as an
oxide of furazane (q.v.), and have shown that a close relationship
exists between the nitrile oxides, furoxane, and fulminic acid (see
Ann. Rep., London Chem. Soc., 1909, p. 84). _Fulminuric acid_,
(HCNO)3, obtained by Liebig by boiling mercuric fulminate with water,
was synthesized in 1905 by C. Ulpiani and L. Bernardini (_Gazetta_,
iii. 35, p. 7), who regard it as NO2.CH(CN).CO.NH2. It deflagrates at
145 deg., and forms a characteristic cuprammonium salt.
The early history of mercuric fulminate and a critical account of its
application as a detonator is given in _The Rise and Progress of the
British Explosives Industry_ (International Congress of Applied
Chemistry, 1909). The manufacture and modern aspects are treated in
Oscar Guttmann, _The Manufacture of Explosives_, and _Manufacture of
Explosives, Twenty Years' Progress_ (1909).
FULTON, ROBERT (1765-1815), American engineer, was born in 1765 in
Little Britain (now Fulton, Lancaster county), Pa. His parents were
Irish, and so poor that they could afford him only a very scanty
education. At an early age he was bound apprentice to a jeweller in
Philadelphia, but subsequently adopted portrait and landscape painting
as his profession. In his twenty-second year, with the object of
studying with his countryman, Benjamin West, he went to England, and
there became acquainted with the duke of Bridgewater, Earl Stanhope and
James Watt. Partly by their influence he was led to devote his attention
to engineering, especially in connexion with canal construction; he
obtained an English patent in 1794 for superseding canal locks by
inclined planes, and in 1796 he published a _Treatise on the Improvement
of Canal Navigation_. He then took up his residence in Paris, where he
projected the first panorama ever exhibited in that city, and
constructed a submarine boat, the "Nautilus," which was tried in Brest
harbour in 1801 before a commission appointed by Napoleon I., and by the
aid of which he was enabled to blow up a small vessel with a torpedo. It
was at Paris also in 1803 that he first succeeded in propelling a boat
by steam-power, thus realizing a design which he had conceived ten years
previously. Returning to America he continued his experiments with
submarine explosives, but failed to convince either the English, French
or United States governments of the adequacy of his methods. With steam
navigation he had more success. In association with Robert R. Livingston
(q.v.), who in 1798 had been granted the exclusive right to navigate the
waters of New York state with steam-vessels, he constructed the
"Clermont," which, engined by Boulton & Watt of Birmingham, began to ply
on the Hudson between New York and Albany in 1807. The privilege
obtained by Livingston in 1798 was granted jointly to Fulton and
Livingston in 1803, and by an act passed in 1808 the monopoly was
secured to them and their associates for a period depending on the
number of steamers constructed, but limited to a maximum of thirty
years. In 1814-1815, on behalf of the United States government, he
constructed the "Fulton," a vessel of 38 tons with central
paddle-wheels, which was the first steam warship. He died at New York on
the 24th of February 1815. Among Fulton's inventions were machines for
spinning flax, for making ropes, and for sawing and polishing marble.
See C.D. Colden, _Life of Robert Fulton_ (New York, 1817); Robert H.
Thurston, _History of the Growth of the Steam-Engine_ (New York,
1878); George H. Preble, _Chronological History of Steam Navigation_
(Philadelphia, 1883); and Mrs A.C. Sutcliffe, _Robert Fulton and the
Clermont_ (New York, 1909).
FULTON, a city and the county-seat of Callaway county, Missouri, U.S.A.,
25 m. N.E. of Jefferson City. Pop. (1890) 4314; (1900) 4883 (1167
negroes); (1910) 5228. It is served by the Chicago & Alton railway. The
city has an important stock market and manufactures fire-brick and
pottery. At Fulton are the Westminster College (Presbyterian, founded in
1853), the Synodical College for Young Women (Pres., founded in 1871),
the William Woods College for Girls (Christian Church, 1890), and the
Missouri school for the deaf (1851). Here, too, is a state hospital for
the insane (1847), the first institution of the kind in Missouri. The
place was laid out as a town in 1825 and named Volney, but in honour of
Robert Fulton the present name was adopted a little later. Fulton was
incorporated in 1859.
FULTON, a city of Oswego county, New York, U.S.A., on the right bank of
the Oswego river, about 10 m. S. by E. of Oswego. Pop. (1900) 5281;
(1905, state census) 8847; (1910) 10,480. Fulton is served by the
Delaware, Lackawanna & Western, the New York Central & Hudson River, and
the New York, Ontario & Western railways, by electric railway to Oswego
and Syracuse and by the Oswego Canal. The city has a Carnegie library.
Ample water-power is furnished by the Oswego river, which here flows in
a series of rapids, and the manufactures are many in kind. On the 3rd of
July 1756, on an island (afterward called Battle Island) 4 m. N. of the
present city of Fulton, a British force of about 300 under Captain John
Bradstreet (1711-1774) defeated an attacking force of French and Indians
(numbering about 700) under De Villiers. Soon after this, Bradstreet
built a fort within the present limits of Fulton. The first civilian
settler came in 1793, and the first survey (which included only a part
of the subsequent village) was made in 1815. Fulton was incorporated as
a village in 1835, and in April 1902 was combined with the village of
Oswego Falls (pop. in 1900, 2925) and was chartered as a city.
FUM, or FUNJ HWANG, one of the four symbolical creatures which in
Chinese mythology are believed to keep watch and ward over the Celestial
Empire. It was begotten by fire, was born in the Hill of the Sun's Halo,
and its body bears inscribed on it the five cardinal virtues. It has the
breast of a goose, the hindquarters of a stag, a snake's neck, a fish's
tail, a fowl's forehead, a duck's down, the marks of a dragon, the back
of a tortoise, the face of a swallow, the beak of a cock, is about six
cubits high, and perches only on the woo-tung tree. The appearance of
Fum heralds an age of universal virtue. Its figure is that which is
embroidered on the dresses of some mandarins.
FUMARIC AND MALEIC ACIDS, two isomeric unsaturated acids of composition
C4H4O4. _Fumaric acid_ is found in fumitory (_Fumaria officinalis_), in
various fungi (_Agaricus piperatus_, &c.), and in Iceland moss. It is
obtained by heating malic acid alone to 150 deg. C., or by heating it
with hydrochloric acid (V. Dessaignes, _Jahresb_., 1856, p. 463) or with
a large quantity of hydrobromic acids (A. Kekule, _Ann._, 1864, 130, p.
21). It may also be obtained by boiling monobromsuccinic acid with
water; by the action of dichloracetic acid and water on silver malonate
(T. Komnenos, _Ann._, 1883, 218, p. 169); by the cyanide synthesis from
acetylene di-iodide; and by heating maleic acid to 210 deg. C. (Z.
Skraup, _Monats. f. Chemie_, 1891, 12, p. 112). It crystallizes in small
prisms or needles, and is practically insoluble in cold water. It
sublimes to some extent at about 200 deg. C., being partially converted
into maleic anhydride and water, the reaction becoming practically
quantitative if dehydrating agents be used. Reducing agents (zinc and
caustic alkali, hydriodic acid, sodium amalgam, &c.) convert it into
succinic acid. Bromine converts it into dibromsuccinic acid. Potassium
permanganate oxidizes it to racemic acid (A. Kekule and R. Anschutz,
_Ber._, 1881, 14, p. 713). By long-continued heating with caustic soda
at 100 deg. C. it is converted into inactive malic acid.
_Maleic acid_ is obtained by distilling malic or fumaric acids; by
heating fumaric acid with acetyl chloride to 100 deg. C; or by the
hydrolysis of trichlorphenomalic acid ([beta]-trichloraceto-acrylic
acid) [A. Kekule, _Ann._, 1884, 223, p. 185]. It crystallizes in
monoclinic prisms, which are easily soluble in water, melt at 130 deg.
C., and boil at 160 deg. C., decomposing into water and maleic
anhydride. When heated with concentrated hydrobromic or hydriodic acids,
it is converted into fumaric acid. It yields an anilide; oxidation
converts it into mesotartaric acid. Maleic anhydride is obtained by
distilling fumaric acid with phosphorus pentoxide. It forms triclinic
crystals which melt at 60 deg. C. and boil at 196 deg. C.
Both acids are readily esterified by the action of alkyl halides on
their silver salts, and the maleic ester is readily transformed into
the fumaric ester by warming with iodine, the same result being
obtained by esterification of maleic acid in alcoholic solution by
means of hydrochloric acid. Both acids yield acetylene by the
electrolysis of aqueous solutions of their alkali salts, and on
reduction both yield succinic acid, whilst by the addition of
hydrobromic acid they both yield monobromsuccinic acid (R. Fittig,
_Ann._, 1877, 188, p. 98). From these results it follows that the two
acids are structurally identical, and the isomerism has consequently
to be explained on other grounds. This was accomplished by W.
Wislicenus ["Uber die raumliche Anordnung der Atome," &c., _Trans, of
the Saxon Acad. of Sciences_ (Math. Phys. Section), 1887, p. 14] by an
extension of the van't Hoff hypothesis (see STEREO-ISOMERISM). The
formulae of the acids are written thus:
HC.CO2H HC.CO2H
.. Maleic acid. .. Fumaric acid.
HC.CO2H HO2C.C.H
These account for maleic acid readily yielding an anhydride, whereas
fumaric acid does not, and for the behaviour of the acids towards
bromine, fumaric acid yielding ordinary dibromsuccinic acid, and
maleic acid the isomeric isodibromsuccinic acid.
FUMAROLE, a vent from which volcanic vapours issue, named indirectly
from the Lat. _fumariolum_, a smoke-hole. The vapours from fumaroles
were studied first by R.W. Bunsen, on his visit to Iceland, and
afterwards by H. Sainte-Claire Deville and other chemists and geologists
in France, who examined the vapours from Santorin, Etna, &c. The hottest
vapours issue from dry fumaroles, at temperatures of at least 500 deg.
C., and consist chiefly of anhydrous chlorides, notably sodium chloride.
The acid fumaroles yield vapours of lower temperature (300 deg. to 400
deg.) containing much water vapour, with hydrogen chloride and sulphur
dioxide. The alkaline fumaroles are still cooler, though above 100 deg.,
and evolve ammonium chloride with other vapours. Cold fumaroles, below
100 deg., discharge principally aqueous vapour, with carbon dioxide, and
perhaps hydrogen sulphide. The fumaroles of Mont Pele in Martinique
during the eruption of 1902 were examined by A. Lacroix, and the vapours
analysed by H. Moissan, who found that they consisted chiefly of water
vapour, with hydrogen chloride, sulphur, carbon dioxide, carbon
monoxide, methane, hydrogen, nitrogen, oxygen and argon. These vapours
issued at a temperature of about 400 deg. Armand Gautier has pointed out
that these gases are practically of the same composition as those which
he obtained on heating granite and certain other rocks. (See VOLCANO).
FUMIGATION (from Lat. _fumigare_, to smoke), the process of producing
smoke or fumes, as by burning sulphur, frankincense, tobacco, &c.,
whether as a ceremony of incantation, or for perfuming a room, or for
purposes of disinfection or destruction of vermin. In medicine the term
has been used of the exposure of the body, or a portion of it, to fumes
such as those of nitre, sal-ammoniac, mercury, &c.; fumigation, by the
injection of tobacco smoke into the great bowel, was a recognized
procedure in the 18th century for the resuscitation of the apparently
drowned. "Fumigated" or "fumed" oak is oak which has been darkened by
exposure to ammonia vapour.
FUMITORY, in botany, the popular name for the British species of
_Fumaria_, a genus of small, branched, often climbing annual herbs with
much-divided leaves and racemes of small flowers. The flowers are
tubular with a spurred base, and in the British species are pink to
purplish in colour. They are weeds of cultivation growing in fields and
waste places. _F. capreolata_ climbs by means of twisting petioles. In
past times fumitory was in esteem for its reputed cholagogue and other
medicinal properties; and in England, boiled in water, milk or whey, it
was used as a cosmetic. The root of the allied species (_Corydalis cava_
or _tuberosa_) is known as _radix aristolochia_, and has been used
medicinally for various cutaneous and other disorders, in doses of 10 to
30 grains. Some eleven alkaloids have been isolated from it. The herbage
of _Fumaria officinalis_ and _F. racemosa_ is used in China under the
name of _Tsze-hwa-ti-ting_ as an application for glandular swellings,
carbuncles and abscesses, and was formerly valued in jaundice, and in
cases of accidental swallowing of the beard of grain (see F. Porter
Smith, _Contrib. towards the Mat. Medica ... of China_, p. 99, 1871).
The name fumitory, Latin _fumus terrae_, has been supposed to be derived
from the fact that its juice irritates the eyes like smoke (see Fuchs,
_De historia stirpium_, p. 338, 1542); but _The Grete Herball_, cap.
clxix., 1529, fol., following the _De simplici medicina_ of Platearius,
fo. xciii. (see in _Nicolai Praepositi dispensatorium ad aromatarios_,
1536), says: "It is called Fumus terre fume or smoke of the erthe
bycause it is engendred of a cours fumosyte rysynge frome the erthe in
grete quantyte lyke smoke: this grosse or cours fumosyte of the erthe
wyndeth and wryeth out: and by workynge of the ayre and sonne it turneth
into this herbe."
FUNCHAL, the capital of the Portuguese archipelago of the Madeiras; on
the south coast of Madeira, in 32 deg. 37' N. and 16 deg. 54' W. Pop.
(1900) 20,850. Funchal is the see of a bishop, in the archiepiscopal
province of Lisbon; it is also the administrative centre of the
archipelago, and the residence of the governor and foreign consuls. The
city has an attractive appearance from the sea. Its whitewashed houses,
in their gardens full of tropical plants, are built along the curving
shore of Funchal Bay, and on the lower slopes of an amphitheatre of
mountains, which form a background 4000 ft. high. Numerous country
houses (_quintas_), with terraced gardens, vineyards and sugar-cane
plantations occupy the surrounding heights. Three mountain streams
traverse the city through deep channels, which in summer are dry, owing
to the diversion of the water for irrigation. A small fort, on an
isolated rock off shore, guards the entrance to the bay, and a larger
and more powerfully armed fort crowns an eminence inland. The chief
buildings include the cathedral, Anglican and Presbyterian churches,
hospitals, opera-house, museum and casino. There are small public
gardens and a meteorological observatory. In the steep and narrow
streets, which are lighted by electricity, wheeled traffic is
impossible; sledges drawn by oxen, and other primitive conveyances are
used instead (see MADEIRA). In winter the fine climate and scenery
attract numerous invalids and other visitors, for whose accommodation
there are good hotels; many foreigners engaged in the coal and wine
trades also reside here permanently. The majority of these belong to the
British community, which was first established here in the 18th century.
Funchal is the headquarters of Madeiran industry and commerce (see
MADEIRA). It has no docks and no facilities for landing passengers or
goods; vessels are obliged to anchor in the roadstead, which, however,
is sheltered from every wind except the south. Funchal is connected by
cable with Carcavellos (for Lisbon), Porthcurnow (for Falmouth, England)
and St Vincent in the Cape Verde Islands (for Pernambuco, Brazil).
FUNCTION,[1] in mathematics, a variable number the value of which
depends upon the values of one or more other variable numbers. The
theory of functions is conveniently divided into (I.) Functions of Real
Variables, wherein real, and only real, numbers are involved, and (II.)
Functions of Complex Variables, wherein complex or imaginary numbers are
involved.
I. FUNCTIONS OF REAL VARIABLES
1. _Historical._--The word function, defined in the above sense, was
introduced by Leibnitz in a short note of date 1694 concerning the
construction of what we now call an "envelope" (_Leibnizens
mathematische Schriften_, edited by C.I. Gerhardt, Bd. v. p. 306), and
was there used to denote a variable length related in a defined way to a
variable point of a curve. In 1698 James Bernoulli used the word in a
special sense in connexion with some isoperimetric problems (Joh.
Bernoulli, _Opera_, t. i. p. 255). He said that when it is a question of
selecting from an infinite set of like curves that one which best
fulfils some function, then of two curves whose intersection determines
the thing sought one is always the "line of the function" (_Linea
functionis_). In 1718 John Bernoulli (_Opera_, t. ii. p. 241) defined a
"function of a variable magnitude" as a quantity made up in any way of
this variable magnitude and constants; and in 1730 (Opera, t. iii. p.
174) he noted a distinction between "algebraic" and "transcendental"
functions. By the latter he meant integrals of algebraic functions. The
notation [f](x) for a function of a variable x was introduced by
Leonhard Euler in 1734 (_Comm. Acad. Petropol._ t. vii. p. 186), in
connexion with the theorem of the interchange of the order of
differentiations. The notion of functionality or functional relation of
two magnitudes was thus of geometrical origin; but a function soon came
to be regarded as an analytical expression, not necessarily an algebraic
expression, containing the variable or variables. Thus we may have
rational integral algebraic functions such as _ax_^2 + _bx_ + c, or
rational algebraic functions which are not integral, such as
a1x^n + a2x^(n - 1) + ... + a_n
-------------------------------,
b1x^m + b2x^(m - 1) + ... + b_m
or irrational algebraic functions, such as [root]x, or, more generally
the algebraic functions that are determined implicitly by an algebraic
equation, as, for instance,
[f]_n(x, y) + [f]_(n - 1) (x, y) + ... + [f]0 = 0
where [f]_n(x, y), ... mean homogeneous expressions in x and y having
constant coefficients, and having the degrees indicated by the suffixes,
and [f]0 is a constant. Or again we may have trigonometrical functions,
such as sin x and tan x, or inverse trigonometrical functions, such as
sin^(-1)x, or exponential functions, such as e^x and a^x, or logarithmic
functions, such as log x and log (1 + x). We may have these functional
symbols combined in various ways, and thus there arises a great number
of functions. Further we may have functions of more than one variable,
as, for instance, the expression xy/(x^2 + y^2), in which both x and y
are regarded as variable. Such functions were introduced into analysis
somewhat unsystematically as the need for them arose, and the later
developments of analysis led to the introduction of other classes of
functions.
2. _Graphic Representation._--In the case of a function of one variable
x, any value of x and the corresponding value y of the function can be
the co-ordinates of a point in a plane. To any value of x there
corresponds a point N on the axis of x, in accordance with the rule that
x is the abscissa of N. The corresponding value of y determines a point
P in accordance with the rule that x is the abscissa and y the ordinate
of P. The ordinate y gives the value of the function which corresponds
to that value of the variable x which is specified by N; and it may be
described as "the value of the function at N." Since there is a
one-to-one correspondence of the points N and the numbers x, we may also
describe the ordinate as "the value of the function at x." In simple
cases the aggregate of the points P which are determined by any
particular function (of one variable) is a curve, called the "graph of
the function" (see S 14). In like manner a function of two variables
defines a surface.
3. _The Variable._--Graphic methods of representation, such as those
just described, enabled mathematicians to deal with irrational values of
functions and variables at the time when there was no theory of
irrational numbers other than Euclid's theory of incommensurables. In
that theory an irrational number was the ratio of two incommensurable
geometric magnitudes. In the modern theory of number irrational numbers
are defined in a purely arithmetical manner, independent of the
measurement of any quantities or magnitudes, whether geometric or of any
other kind. The definition is effected by means of the system of
_ordinal_ numbers (see NUMBER). When this formal system is established,
the theory of measurement may be founded upon it; and, in particular,
the co-ordinates of a point are defined as numbers (not lengths), which
are assigned in accordance with a rule. This rule involves the
measurement of lengths. The theory of functions can be developed without
any reference to graphs, or co-ordinates or lengths. The process by
which analysis has been freed from any consideration of measurable
quantities has been called the "arithmetization of analysis." In the
theory so developed, the variable upon which a function depends is
always to be regarded as a number, and the corresponding value of the
function is also a number. Any reference to points or co-ordinates is to
be regarded as a picturesque mode of expression, pointing to a possible
application of the theory to geometry. The development of "arithmetized
analysis" in the 19th century is associated with the name of Karl
Weierstrass.
All possible values of a variable are numbers. In what follows we shall
confine our attention to the case where the numbers are real. When
complex numbers are introduced, instead of real ones, the theory of
functions receives a wide extension, which is accompanied by appropriate
limitations (see below, II. Functions of Complex Variables). The set of
all real numbers forms a _continuum_. In fact the notion of a
one-dimensional continuum first becomes precise in virtue of the
establishment of the system of real numbers.
4. _Domain of a Variable.--Theory of Aggregates._--The notion of a
"variable" is that of a number to which we may assign at pleasure any
one of the values that belong to some chosen set, or _aggregate_, of
numbers; and this set, or aggregate, is called the "domain of the
variable." This domain may be an "interval," that is to say it may
consist of two terminal numbers, all the numbers between them and no
others. When this is the case the number is said to be "continuously
variable." When the domain consists of all real numbers, the variable is
said to be "unrestricted." A domain which consists of all the real
numbers which exceed some fixed number may be described as an "interval
unlimited towards the right"; similarly we may have an interval
"unlimited towards the left."
In more complicated cases we must have some rule or process for
assigning the aggregate of numbers which constitute the domain of a
variable. The methods of definition of particular types of aggregates,
and the theorems relating to them, form a branch of analysis called
the "theory of aggregates" (_Mengenlehre, Theorie des ensembles,
Theory of sets of points_). The notion of an "aggregate" in general
underlies the system of ordinal numbers. An aggregate is said to be
"infinite" when it is possible to effect a one-to-one correspondence
of all its elements to some of its elements. For example, we may make
all the integers correspond to the even integers, by making 1
correspond to 2, 2 to 4, and generally n to 2n. The aggregate of
positive integers is an infinite aggregate. The aggregates of all
rational numbers and of all real numbers and of points on a line are
other examples of infinite aggregates. An aggregate whose elements are
real numbers is said to "extend to infinite values" if, after any
number N, however great, is specified, it is possible to find in the
aggregate numbers which exceed N in absolute value. Such an aggregate
is always infinite. The "neighbourhood of a number (or point) a for a
positive number h" is the aggregate of all numbers (or points) x for
which the absolute value of x - a denoted by |x - a|, does not exceed
h.
5. _General Notion of Functionality._--A function of one variable was
for a long time commonly regarded as the ordinate of a curve; and the
two notions (1) that which is determined by a curve supposed drawn, and
(2) that which is determined by an analytical expression supposed
written down, were not for a long time clearly distinguished. It was for
this reason that Fourier's discovery that a single analytical expression
is capable of representing (in different parts of an interval) what
would in his time have been called different functions so profoundly
struck mathematicians (S 23). The analysts who, in the middle of the
19th century, occupied themselves with the theory of the convergence of
Fourier's series were led to impose a restriction on the character of a
function in order that it should admit of such representation, and thus
the door was opened for the introduction of the general notion of
functional dependence. This notion may be expressed as follows: We have
a variable number, y, and another variable number, x, a domain of the
variable x, and a rule for assigning one or more definite values to y
when x is any point in the domain; then y is said to be a "function" of
the variable x, and x is called the "argument" of the function.
According to this notion a function is, as it were, an indefinitely
extended table, like a table of logarithms; to each point in the domain
of the argument there correspond values for the function, but it remains
arbitrary what values the function is to have at any such point.
For the specification of any particular function two things are
requisite: (1) a statement of the values of the variable, or of the
aggregate of points, to which values of the function are to be made to
correspond, i.e. of the "domain of the argument"; (2) a rule for
assigning the value or values of the function that correspond to any
point in this domain. We may refer to the second of these two
essentials as "the rule of calculation." The relation of functions to
analytical expressions may then be stated in the form that the rule of
calculation is: "Give the function the value of the expression at any
point at which the expression has a determinate value," or again more
generally, "Give the function the value of the expression at all
points of a definite aggregate included in the domain of the
argument." The former of these is the rule of those among the earlier
analysts who regarded an analytical expression and a function as the
same thing, and their usage may be retained without causing confusion
and with the advantage of brevity, the analytical expression serving
to specify the domain of the argument as well as the rule of
calculation, e.g. we may speak of "the function 1/x." This function is
defined by the analytical expression 1/x at all points except the
point x = 0. But in complicated cases separate statements of the
domain of the argument and the rule of calculation cannot be dispensed
with. In general, when the rule of calculation is determined as above
by an analytical expression at any aggregate of points, the function
is said to be "represented" by the expression at those points.
When the rule of calculation assigns a single definite value for a
function at each point in the domain of the argument the function is
"uniform" or "one-valued." In what follows it is to be understood that
all the functions considered are one-valued, and the values assigned
by the rule of calculation real. In the most important cases the
domain of the argument of a function of one variable is an interval,
with the possible exception of isolated points.
6. _Limits._--Let [f](x) be a function of a variable number x; and let a
be a point such that there are points of the domain of the argument x in
the neighbourhood of a for any number h, however small. If there is a
number L which has the property that, after any positive number
[epsilon], however small, has been specified, it is possible to find a
positive number h, so that |L - [f](x)| < [epsilon] for all points x of
the domain (other than a) for which |x-a| < h, then L is the "limit of
[f](x) at the point a." The condition for the existence of L is that,
after the positive number [epsilon] has been specified, it must be
possible to find a positive number h, so that |[f](x') - [f](x)| <
[epsilon] for all points x and x' of the domain (other than a) for which
|x - a| < h and |x' - a| < h.
It is a fundamental theorem that, when this condition is satisfied,
there exists a perfectly definite number L which is the limit of [f](x)
at the point a as defined above. The limit of [f](x) at the point a is
denoted by Lt_(x = a)[f](x), or by lim_(x = a)[f](x).
If [f](x) is a function of one variable x in a domain which extends to
infinite values, and if, after [epsilon] has been specified, it is
possible to find a number N, so that |[f](x') - [f](x)| <[epsilon] for
all values of x and x' which are in the domain and exceed N, then
there is a number L which has the property that |[f](x) - L| <
[epsilon] for all such values of x. In this case [f](x) has a limit L
at x = [oo]. In like manner [f](x) may have a limit at x = -[oo]. This
statement includes the case where the domain of the argument consists
exclusively of positive integers. The values of the function then form
a "sequence," u1, u2, ... u_n, ..., and this sequence can have a limit
at n = [oo].
The principle common to the above definitions and theorems is called,
after P. du Bois Reymond, "the general principle of convergence to a
limit."
It must be understood that the phrase "x = [oo]" does not mean that x
takes some particular value which is infinite. There is no such value.
The phrase always refers to a limiting process in which, as the
process is carried out, the variable number x increases without limit:
it may, as in the above example of a sequence, increase by taking
successively the values of all the integral numbers; in other cases it
may increase by taking the values that belong to any domain which
"extends to infinite values."
A very important type of limits is furnished by _infinite series_.
When a sequence of numbers u1, u2, ... u_n, ... is given, we may form
a new sequence s1, s2, ... s_n, ... from it by the rules s1 = u1, s2 =
u1 + u2, ... s_n = u1 + u2 + ... + u_n or by the equivalent rules s1 =
u, s_n - s_(n - 1) = u_n(n = 2, 3, ...). If the new sequence has a
limit at n = [oo], this limit is called the "sum of the infinite
series" u1 + u2 + ..., and the series is said to be "convergent" (see
SERIES).
A function which has not a limit at a point a may be such that, if a
certain aggregate of points is chosen out of the domain of the
argument, and the points x in the neighbourhood of a are restricted to
belong to this aggregate, then the function has a limit at a. For
example, sin(1/x) has limit zero at 0 if x is restricted to the
aggregate 1/[pi], 1/2[pi], ... 1/n[pi], ... or to the aggregate
1/2[pi], 2/5[pi], ... n/(n^2 + 1)[pi], ..., but if x takes all values
in the neighbourhood of 0, sin (1/x) has not a limit at 0. Again,
there may be a limit at a if the points x in the neighbourhood of a
are restricted by the condition that x - a is positive; then we have a
"limit on the right" at a; similarly we may have a "limit on the left"
at a point. Any such limit is described as a "limit for a restricted
domain." The limits on the left and on the right are denoted by [f](a
- 0) and [f](a + 0).
The limit L of [f](x) at a stands in no necessary relation to the
value of [f](x) at a. If the point a is in the domain of the argument,
the value of [f](x) at a is assigned by the rule of calculation, and
may be different from L. In case [f](a) = L the limit is said to be
"attained." If the point a is not in the domain of the argument, there
is no value for [f](x) at a. In the case where [f](x) is defined for
all points in an interval containing a, except the point a, and has a
limit L at a, we may arbitrarily annex the point a to the domain of
the argument and assign to [f](a) the value L; the function may then
be said to be "extrinsically defined." The so-called "indeterminate
forms" (see INFINITESIMAL CALCULUS) are examples.
7. _Superior and Inferior Limits; Infinities._--The value of a function
at every point in the domain of its argument is finite, since, by
definition, the value can be assigned, but this does not necessarily
imply that there is a number N which exceeds all the values (or is less
than all the values). It may happen that, however great a number N we
take, there are among the values of the function numbers which exceed N
(or are less than -N).
If a number can be found which is greater than every value of the
function, then either ([alpha]) there is one value of the function
which exceeds all the others, or ([beta]) there is a number S which
exceeds every value of the function but is such that, however small a
positive number [epsilon] we take, there are values of the function
which exceed S -[epsilon]. In the case ([alpha]) the function has a
greatest value; in case ([beta]) the function has a "superior limit" S,
and then there must be a point a which has the property that there are
points of the domain of the argument, in the neighbourhood of a for any
h, at which the values of the function differ from S by less than
[epsilon]. Thus S is the limit of the function at a, either for the
domain of the argument or for some more restricted domain. If a is in
the domain of the argument, and if, after omission of a, there is a
superior limit S which is in this way the limit of the function at a, if
further [f](a) = S, then S is the greatest value of the function: in
this case the greatest value is a limit (at any rate for a restricted
domain) which is attained; it may be called a "superior limit which is
attained." In like manner we may have a "smallest value" or an "inferior
limit," and a smallest value may be an "inferior limit which is
attained."
All that has been said here may be adapted to the description of
greatest values, superior limits, &c., of a function in a restricted
domain contained in the domain of the argument. In particular, the
domain of the argument may contain an interval; and therein the
function may have a superior limit, or an inferior limit, which is
attained. Such a limit is a _maximum_ value or a _minimum_ value of
the function.
Again, if, after any number N, however great, has been specified, it
is possible to find points of the domain of the argument at which the
value of the function exceeds N, the values of the function are said
to have an "infinite superior limit," and then there must be a point a
which has the property that there are points of the domain, in the
neighbourhood of a for any h, at which the value of the function
exceeds N. If the point a is in the domain of the argument the
function is said to "tend to become infinite" at a; it has of course a
finite value at a. If the point a is not in the domain of the argument
the function is said to "become infinite" at a; it has of course no
value at a. In like manner we may have a (negatively) infinite
inferior limit. Again, after any number N, however great, has been
specified and a number h found, so that all the values of the
function, at points in the neighbourhood of a for h, exceed N in
absolute value, all these values may have the same sign; the function
is then said to become, or to tend to become, "determinately
(positively or negatively) infinite"; otherwise it is said to become
or to tend to become, "indeterminately infinite."
All the infinities that occur in the theory of functions are of the
nature of variable finite numbers, with the single exception of the
infinity of an infinite aggregate. The latter is described as an
"actual infinity," the former as "improper infinities." There is no
"actual infinitely small" corresponding to the actual infinity. The
only "infinitely small" is zero. All "infinite values" are of the
nature of superior and inferior limits which are not attained.
8. _Increasing and Decreasing Functions._--A function [f](x) of one
variable x, defined in the interval between a and b, is "increasing
throughout the interval" if, whenever x and x' are two numbers in the
interval and x' > x, then [f](x') > [f](x); the function "never
decreases throughout the interval" if, x' and x being as before, [f](x')
> [f](x). Similarly for decreasing functions, and for functions which
never increase throughout an interval. A function which either never
increases or never diminishes throughout an interval is said to be
"monotonous throughout" the interval. If we take in the above definition
b > a, the definition may apply to a function under the restriction that
x' is not b and x is not a; such a function is "monotonous within" the
interval. In this case we have the theorem that the function (if it
never decreases) has a limit on the left at b and a limit on the right
at a, and these are the superior and inferior limits of its values at
all points within the interval (the ends excluded); the like holds
_mutatis mutandis_ if the function never increases. If the function is
monotonous throughout the interval, [f](b) is the greatest (or least)
value of [f](x) in the interval; and if [f](b) is the limit of [f](x) on
the left at b, such a greatest (or least) value is an example of a
superior (or inferior) limit which is attained. In these cases the
function tends continually to its limit.
These theorems and definitions can be extended, with obvious
modifications, to the cases of a domain which is not an interval, or
extends to infinite values. By means of them we arrive at sufficient,
but not necessary, criteria for the existence of a limit; and these
are frequently easier to apply than the general principle of
convergence to a limit (S 6), of which principle they are particular
cases. For example, the function represented by x log (1/x)
continually diminishes when 1/e > x > 0 and x diminishes towards
zero, and it never becomes negative. It therefore has a limit on the
right at x = 0. This limit is zero. The function represented by x sin
(1/x) does not continually diminish towards zero as x diminishes
towards zero, but is sometimes greater than zero and sometimes less
than zero in any neighbourhood of x = 0, however small. Nevertheless,
the function has the limit zero at x = 0.
9. _Continuity of Functions._--A function [f](x) of one variable x is
said to be continuous at a point a if (1) [f](x) is defined in an
interval containing a; (2) [f](x) has a limit at a; (3) [f](a) is equal
to this limit. The limit in question must be a limit for continuous
variation, not for a restricted domain. If [f](x) has a limit on the
left at a and [f](a) is equal to this limit, the function may be said to
be "continuous to the left" at a; similarly the function may be
"continuous to the right" at a.
A function is said to be "continuous throughout an interval" when it is
continuous at every point of the interval. This implies continuity to
the right at the smaller end-value and continuity to the left at the
greater end-value. When these conditions at the ends are not satisfied
the function is said to be continuous "within" the interval. By a
"continuous function" of one variable we always mean a function which is
continuous throughout an interval.
The principal properties of a continuous function are:
1. The function is practically constant throughout sufficiently small
intervals. This means that, after any point a of the interval has been
chosen, and any positive number [epsilon], however small, has been
specified, it is possible to find a number h, so that the difference
between any two values of the function in the interval between a-h and
a + h is less than [epsilon]. There is an obvious modification if a is
an end-point of the interval.
2. The continuity of the function is "uniform." This means that the
number h which corresponds to any [epsilon] as in (1) may be the same
at all points of the interval, or, in other words, that the numbers h
which correspond to [epsilon] for different values of a have a
positive inferior limit.
3. The function has a greatest value and a least value in the
interval, and these are superior and inferior limits which are
attained.
4. There is at least one point of the interval at which the function
takes any value between its greatest and least values in the interval.
5. If the interval is unlimited towards the right (or towards the
left), the function has a limit at [oo] (or at -[oo]).
10. _Discontinuity of Functions._--The discontinuities of a function of
one variable, defined in an interval with the possible exception of
isolated points, may be classified as follows:
(1) The function may become infinite, or tend to become infinite, at a
point.
(2) The function may be undefined at a point.
(3) The function may have a limit on the left and a limit on the right
at the same point; these may be different from each other, and at least
one of them must be different from the value of the function at the
point.
(4) The function may have no limit at a point, or no limit on the left,
or no limit on the right, at a point.
In case a function [f](x), defined as above, has no limit at a point
a, there are four limiting values which come into consideration.
Whatever positive number h we take, the values of the function at
points between a and a + h (a excluded) have a superior limit (or a
greatest value), and an inferior limit (or a least value); further, as
h decreases, the former never increases and the latter never
decreases; accordingly each of them tends to a limit. We have in this
way two limits on the right--the inferior limit of the superior limits
in diminishing neighbourhoods, and the superior limit of the inferior
limits in diminishing neighbourhoods. These are denoted by /{[f](a +
o)} and {[f](a + 0)}/, and they are called the "limits of
indefiniteness" on the right. Similar limits on the left are denoted
by /{[f](a - 0)} and {[f](a - 0)}/. Unless [f](x) becomes, or tends to
become, infinite at a, all these must exist, any two of them may be
equal, and at least one of them must be different from [f](a), if
[f](a) exists. If the first two are equal there is a limit on the
right denoted by [f](a + 0); if the second two are equal, there is a
limit on the left denoted by [f](a - 0). In case the function becomes,
or tends to become, infinite at a, one or more of these limits is
infinite in the sense explained in S 7; and now it is to be noted
that, e.g. the superior limit of the inferior limits in diminishing
neighbourhoods on the right of a may be negatively infinite; this
happens if, after any number N, however great, has been specified, it
is possible to find a positive number h, so that all the values of the
function in the interval between a and a + h (a excluded) are less
than -N; in such a case [f](x) tends to become negatively infinite
when x decreases towards a; other modes of tending to infinite limits
may be described in similar terms.
11. _Oscillation of Functions._--The difference between the greatest and
least of the numbers [f](a), /{[f](a + 0)}, {[f](a + 0)}/, /{[f](a -
0)}, {[f](a - 0)}/, when they are all finite, is called the
"oscillation" or "fluctuation" of the function [f](x) at the point a.
This difference is the limit for h = 0 of the difference between the
superior and inferior limits of the values of the function at points in
the interval between a - h and a + h. The corresponding difference for
points in a finite interval is called the "oscillation of the function
in the interval." When any of the four limits of indefiniteness is
infinite the oscillation is infinite in the sense explained in S 7.
For the further classification of functions we divide the domain of
the argument into partial intervals by means of points between the
end-points. Suppose that the domain is the interval between a and b.
Let intermediate points x1, x2 ... x{n - 1}_, be taken so that b >
X_(n - 1) > x_(n - 2) ... > X1 > a_. We may devise a rule by which, as
n increases indefinitely, all the differences b - x_(n - 1), x_(n - 1)
- x_(n - 2), ... x1 - a tend to zero as a limit. The interval is then
said to be divided into "indefinitely small partial intervals."
A function defined in an interval with the possible exception of
isolated points may be such that the interval can be divided into a
set of finite partial intervals within each of which the function is
monotonous (S 8). When this is the case the sum of the oscillations of
the function in those partial intervals is finite, provided the
function does not tend to become infinite. Further, in such a case the
sum of the oscillations will remain below a fixed number for any mode
of dividing the interval into indefinitely small partial intervals. A
class of functions may be defined by the condition that the sum of the
oscillations has this property, and such functions are said to have
"restricted oscillation." Sometimes the phrase "limited fluctuation"
is used. It can be proved that any function with restricted
oscillation is capable of being expressed as the sum of two monotonous
functions, of which one never increases and the other never diminishes
throughout the interval. Such a function has a limit on the right and
a limit on the left at every point of the interval. This class of
functions includes all those which have a finite number of maxima and
minima in a finite-interval, and some which have an infinite number.
It is to be noted that the class does not include all continuous
functions.
12. _Differentiable Function._--The idea of the differentiation of a
continuous function is that of a process for measuring the rate of
growth; the increment of the function is compared with the increment of
the variable. If _[f](x)_ is defined in an interval containing the point
a, and _a - k_ and _a + k_ are points of the interval, the expression
[f](a + h) - [f](a)
------------------- (1)
h
represents a function of h, which we may call [phi](h), defined at all
points of an interval for h between -k and k except the point 0. Thus
the four limits /[phi](+0), [phi](+0)/, /[phi](-0), [phi](-0)/ exist,
and two or more of them may be equal. When the first two are equal
either of them is the "progressive differential coefficient" of [f](x)
at the point a; when the last two are equal either of them is the
"regressive differential coefficient" of [f](x) at a; when all four are
equal the function is said to be "differentiable" at a, and either of
them is the "differential coefficient" of [f](x) at a, or the "first
derived function" of [f](x) at a. It is denoted by d[f](x) / dx or by
[f]'(x). In this case [phi](h) has a definite limit at h = 0, or is
determinately infinite at h = 0 (S 7). The four limits here in question
are called, after Dini, the "four derivates" of [f](x) at a. In
accordance with the notation for derived functions they may be denoted
by
---------- ----------
[f]' + (a), [f]' + (a), [f]' - (a), f' - (a).
--------- --------
A function which has a finite differential coefficient at all points
of an interval is continuous throughout the interval, but if the
differential coefficient becomes infinite at a point of the interval
the function may or may not be continuous throughout the interval; on
the other hand a function may be continuous without being
differentiable. This result, comparable in importance, from the point
of view of the general theory of functions, with the discovery of
Fourier's theorem, is due to G.F.B. Riemann; but the failure of an
attempt made by Ampere to prove that every continuous function must be
differentiable may be regarded as the first step in the theory.
Examples of analytical expressions which represent continuous
functions that are not differentiable have been given by Riemann,
Weierstrass, Darboux and Dini (see S 24). The most important theorem
in regard to differentiable functions is the "theorem of intermediate
value." (See INFINITESIMAL CALCULUS.)
13. _Analytic Function._--If [f](x) and its first n differential
coefficients, denoted by[f]'(x), [f]''(x), ... [f](^n)(x), are
continuous in the interval between a and a + h, then
h^2
[f](a + h) = [f](a) + h[f]'(a) + --- f''(a) + ...
2!
h^(n - 1)
+ --------- [f]^(n - 1)(a) + R_n,
(n - 1)!
where R_n may have various forms, some of which are given in the article
INFINITESIMAL CALCULUS. This result is known as "Taylor's theorem."
When Taylor's theorem leads to a representation of the function by means
of an infinite series, the function is said to be "analytic" (cf. S 21).
14. _Ordinary Function._--The idea of a curve representing a continuous
function in an interval is that of a line which has the following
properties: (1) the co-ordinates of a point of the curve are a value x
of the argument and the corresponding value y of the function; (2) at
every point the curve has a definite tangent; (3) the interval can be
divided into a finite number of partial intervals within each of which
the function is monotonous; (4) the property of monotony within partial
intervals is retained after interchange of the axes of co-ordinates x
and y. According to condition (2) y is a continuous and differentiable
function of x, but this condition does not include conditions (3) and
(4): there are continuous partially monotonous functions which are not
differentiable, there are continuous differentiable functions which are
not monotonous in any interval however small; and there are continuous,
differentiable and monotonous functions which do not satisfy condition
(4) (cf. S 24). A function which can be represented by a curve, in the
sense explained above, is said to be "ordinary," and the curve is the
graph of the function (S2). All analytic functions are ordinary, but not
all ordinary functions are analytic.
15. _Integrable Function._--The idea of integration is twofold. We may
seek the function which has a given function as its differential
coefficient, or we may generalize the question of finding the area of a
curve. The first inquiry leads directly to the indefinite integral, the
second directly to the definite integral. Following the second method we
define "the definite integral of the function [f](x) through the
interval between a and b" to be the limit of the sum
_n
\ [f](x'_r)(x_r - x_(r - 1))
/_
1
when the interval is divided into ultimately indefinitely small partial
intervals by points x1, x2, ... x_(n - 1). Here x'_r denotes any point
in the rth partial interval, x0 is put for a, and x_n for b. It can be
shown that the limit in question is finite and independent of the mode
of division into partial intervals, and of the choice of the points such
as x'_r, provided (1) the function is defined for all points of the
interval, and does not tend to become infinite at any of them; (2) for
any one mode of division of the interval into ultimately indefinitely
small partial intervals, the sum of the products of the oscillation of
the function in each partial interval and the difference of the
end-values of that partial interval has limit zero when n is increased
indefinitely. When these conditions are satisfied the function is said
to be "integrable" in the interval. The numbers a and b which limit the
interval are usually called the "lower and upper limits." We shall call
them the "nearer and further end-values." The above definition of
integration was introduced by Riemann in his memoir on trigonometric
series (1854). A still more general definition has been given by
Lebesgue. As the more general definition cannot be made intelligible
without the introduction of some rather recondite notions belonging to
the theory of aggregates, we shall, in what follows, adhere to Riemann's
definition.
We have the following theorems:--
1. Any continuous function is integrable.
2. Any function with restricted oscillation is integrable.
3. A discontinuous function is integrable if it does not tend to
become infinite, and if the points at which the oscillation of the
function exceeds a given number [sigma], however small, can be
enclosed in partial intervals the sum of whose breadths can be
diminished indefinitely.
These partial intervals must be a set chosen out of some complete set
obtained by the process used in the definition of integration.
4. The sum or product of two integrable functions is integrable.
As regards integrable functions we have the following theorems:
1. If S and I are the superior and inferior limits (or greatest and
least values) of [f](x) in the interval between a and b, [int] [a to
b] [f](x)dx is intermediate between S(b - a) and I(b - a).
2. The integral is a continuous function of each of the end-values.
3. If the further end-value b is variable, and if [int] [a to x]
[f](x)_dx_ = F(x), then if [f](x) is continuous at b, F(x) is
differentiable at b, and F'(b) = [f](b).
4. In case [f](x) is continuous throughout the interval F(x) is
continuous and differentiable throughout the interval, and F'(x) =
[f](x) throughout the interval.
5. In case [f]'(x) is continuous throughout the interval between a and
b,
_
/ b
| [f]'(x)dx = [f](b) - [f](a).
_/a
6. In case [f](x) is discontinuous at one or more points of the
interval between a and b, in which it is integrable,
_
/ x
| [f](x)dx
_/a
is a function of x, of which the four derivates at any point of the
interval are equal to the limits of indefiniteness of [f](x) at the
point.
7. It may be that there exist functions which are differentiable
throughout an interval in which their differential coefficients are
not integrable; if, however, F(x) is a function whose differential
coefficient, F'(x), is integrable in an interval, then
_
/ x
F(x) = | F'(x)dx + const.,
_/a
where a is a fixed point, and x a variable point, of the interval.
Similarly, if any one of the four derivates of a function is
integrable in an interval, all are integrable, and the integral of
either differs from the original function by a constant only.
The theorems (4), (6), (7) show that there is some discrepancy between
the indefinite integral considered as the function which has a given
function as its differential coefficient, and as a definite integral
with a variable end-value.
We have also two theorems concerning the integral of the product of
two integrable functions [f](x) and [phi](x); these are known as "the
first and second theorems of the mean." The first theorem of the mean
is that, if [phi](x) is one-signed throughout the interval between a
and b, there is a number M intermediate between the superior and
inferior limits, or greatest and least values, of [f](x) in the
interval, which has the property expressed by the equation
_ _
/ b / b
M | [phi](x)dx = | [f](x)[phi](x)dx.
_/a _/a
The second theorem of the mean is that, if [f](x) is monotonous
throughout the interval, there is a number [xi] between a and b which
has the property expressed by the equation
_ _ _
/ b /[xi] / b
| [f](x)[phi](x)dx = [f](a) | [phi](x)dx + [f](b) | [phi](x)dx.
_/a _/a _/[xi]
(_See_ FOURIER'S SERIES.)
16. _Improper Definite Integrals._--We may extend the idea of
integration to cases of functions which are not defined at some point,
or which tend to become infinite in the neighbourhood of some point, and
to cases where the domain of the argument extends to infinite values. If
c is a point in the interval between a and b at which [f](x) is not
defined, we impose a restriction on the points x'_r of the definition:
none of them is to be the point c. This comes to the same thing as
defining [int] [a to b] [f](x)dx to be
_ _
/ c-[epsilon] / b
Lt | [f](x)dx + Lt | [f](x)dx, (1)
_/a _/c+[epsilon]'
[epsilon]=0 [epsilon]'=0
where, to fix ideas, b is taken > a, and [epsilon] and [epsilon]' are
positive. The same definition applies to the case where [f](x) becomes
infinite, or tends to become infinite, at c, provided both the limits
exist. This definition may be otherwise expressed by saying that a
partial interval containing the point c is omitted from the interval of
integration, and a limit taken by diminishing the breadth of this
partial interval indefinitely; in this form it applies to the cases
where c is a or b.
Again, when the interval of integration is unlimited to the right, or
extends to positively infinite values, we have as a definition
_ _
/ [oo] / h
| [f](x)dx = Lt | [f](x)dx,
_/a _/a
h=[oo]
provided this limit exists. Similar definitions apply to
_ _
/-[oo] / [oo]
| [f](x)dx and to | [f](x)dx.
_/a _/-[oo]
All such definite integrals as the above are said to be "improper." For
example, [int] {0 to [oo]} (sin x / x)dx is improper in two ways. It
means
_
/ h sin x
Lt Lt | ----- dx,
h=[oo] [epsilon]=0 _/[epsilon] x
in which the positive number [epsilon] is first diminished indefinitely,
and the positive number h is afterwards increased indefinitely.
The "theorems of the mean" (S 15) require modification when the
integrals are improper (see FOURIER'S SERIES).
When the improper definite integral of a function which becomes, or
tends to become, infinite, exists, the integral is said to be
"convergent." If [f](x) tends to become infinite at a point c in the
interval between a and b, and the expression (1) does not exist, then
the expression [int] [a to b][f](x)_dx_, which has no value, is called a
"divergent integral, "and it may happen that there is a definite value
for
_ _ _ _
| / c-[epsilon] / b |
Lt | | [f](x) dx + | [f](x) dx |
|_ _/a _/c+[epsilon]' _|
provided that [epsilon] and [epsilon]' are connected by some definite
relation, and both, remaining positive, tend to limit zero. The value of
the above limit is then called a "principal value" of the divergent
integral. Cauchy's principal value is obtained by making [epsilon]' =
[epsilon], i.e. by taking the omitted interval so that the infinity is
at its middle point. A divergent integral which has one or more
principal values is sometimes described as "semi-convergent."
17. _Domain of a Set of Variables._--The numerical continuum of n
dimensions (C_n) is the aggregate that is arrived at by attributing
simultaneous values to each of n variables x1, x2, ... x_n, these values
being any real numbers. The elements of such an aggregate are called
"points," and the numbers x1, x2 ... x_n the "co-ordinates" of a point.
Denoting in general the points (x1, x2, ... x_n) and (x'1, x'2 ... x'_n)
by x and x', the sum of the differences |x1 - x'1| + |x2 - x'2| + ... +
|x_n - x'_n| may be denoted by |x - x'| and called the "difference of
the two points." We can in various ways choose out of the continuum an
aggregate of points, which may be an infinite aggregate, and any such
aggregate can be the "domain" of a "variable point." The domain is said
to "extend to an infinite distance" if, after any number N, however
great, has been specified, it is possible to find in the domain points
of which one or more co-ordinates exceed N in absolute value. The
"neighbourhood" of a point a for a (positive) number h is the aggregate
constituted of all the points x, which are such that the "difference"
denoted by |x - a| < h. If an infinite aggregate of points does not
extend to an infinite distance, there must be at least one point a,
which has the property that the points of the aggregate which are in the
neighbourhood of a for any number h, however small, themselves
constitute an infinite aggregate, and then the point a is called a
"limiting point" of the aggregate; it may or may not be a point of the
aggregate. An aggregate of points is "perfect" when all its points are
limiting points of it, and all its limiting points are points of it; it
is "connected" when, after taking any two points a, b of it, and
choosing any positive number [epsilon], however small, a number m and
points x', x", ... x^(m) of the aggregate can be found so that all the
differences denoted by |x' - a|, |x" - x'|, ... |b - x^(m)| are less
than [epsilon]. A perfect connected aggregate is a _continuum_. This is
G. Cantor's definition.
The definition of a continuum in C_n leaves open the question of the
number of dimensions of the continuum, and a further explanation is
necessary in order to define arithmetically what is meant by a
"homogeneous part" H_n of C_n. Such a part would correspond to an
interval in C1, or to an area bounded by a simple closed contour in
C2; and, besides being perfect and connected, it would have the
following properties: (1) There are points of C_n, which are not
points of H_n; these form a complementary aggregate H'_n. (2) There
are points "within" H_n; this means that for any such point there is a
neighbourhood consisting exclusively of points of H_n. (3) The points
of H_n which do not lie "within" H_n are limiting points of H'_n; they
are not points of H'_n, but the neighbourhood of any such point for
any number h, however small, contains points within H_n and points of
H'_n: the aggregate of these points is called the "boundary" of H_n.
(4) When any two points a, b within H_n are taken, it is possible to
find a number [epsilon] and a corresponding number m, and to choose
points x', x", ... x^m, so that the neighbourhood of a for [epsilon]
contains x', and consists exclusively of points within H_n, and
similarly for x' and x", x" and x"', ... x^m and b. Condition (3)
would exclude such an aggregate as that of the points within and upon
two circles external to each other and a line joining a point on one
to a point on the other, and condition (4) would exclude such an
aggregate as that of the points within and upon two circles which
touch externally.
18. Functions of Several Variables.--A function of several variables
differs from a function of one variable in that the argument of the
function consists of a set of variables, or is a variable point in a C_n
when there are n variables. The function is definable by means of the
domain of the argument and the rule of calculation. In the most
important cases the domain of the argument is a homogeneous part H_n of
C_n with the possible exception of isolated points, and the rule of
calculation is that the value of the function in any assigned part of
the domain of the argument is that value which is assumed at the point
by an assigned analytical expression. The limit of a function at a point
a is defined in the same way as in the case of a function of one
variable.
We take a positive fraction [epsilon] and consider the neighbourhood
of a for h, and from this neighbourhood we exclude the point a, and we
also exclude any point which is not in the domain of the argument.
Then we take x and x' to be any two of the retained points in the
neighbourhood. The function [f] has a limit at a if for any positive
[epsilon], however small, there is a corresponding h which has the
property that |[f](x') - [f](x)| < [epsilon], whatever points x, x' in
the neighbourhood of a for h we take (a excluded). For example, when
there are two variables x1, x2, and both are unrestricted, the domain
of the argument is represented by a plane, and the values of the
function are correlated with the points of the plane. The function has
a limit at a point a, if we can mark out on the plane a region
containing the point a within it, and such that the difference of the
values of the function which correspond to any two points of the
region (neither of the points being a) can be made as small as we
please in absolute value by contracting all the linear dimensions of
the region sufficiently. When the domain of the argument of a function
of n variables extends to an infinite distance, there is a "limit at
an infinite distance" if, after any number [epsilon], however small,
has been specified, a number N can be found which is such that
|[f](x') - [f](x)| < [epsilon], for all points x and x' (of the
domain) of which one or more co-ordinates exceed N in absolute value.
In the case of functions of several variables great importance
attaches to limits for a restricted domain. The definition of such a
limit is verbally the same as the corresponding definition in the case
of functions of one variable (S 6). For example, a function of x1 and
x2 may have a limit at (x1 = 0, x2 = 0) if we first diminish x1
without limit, keeping x2 constant, and afterwards diminish x2 without
limit. Expressed in geometrical language, this process amounts to
approaching the origin along the axis of x2. The definitions of
superior and inferior limits, and of maxima and minima, and the
explanations of what is meant by saying that a function of several
variables becomes infinite, or tends to become infinite, at a point,
are almost identical verbally with the corresponding definitions and
explanations in the case of a function of one variable (S 7). The
definition of a continuous function (S 9) admits of immediate
extension; but it is very important to observe that a function of two
or more variables may be a continuous function of each of the
variables, when the rest are kept constant, without being a continuous
function of its argument. For example, a function of x and y may be
defined by the conditions that when x = 0 it is zero whatever value y
may have, and when x [/=] 0 it has the value of sin {4tan^(-1)(y/x)}.
When y has any particular value this function is a continuous function
of x, and, when x has any particular value this function is a
continuous function of y; but the function of x and y is discontinuous
at (x = 0, y = 0).
19. _Differentiation and Integration._--The definition of partial
differentiation of a function of several variables presents no
difficulty. The most important theorems concerning differentiable
functions are the "theorem of the total differential," the theorem of
the interchangeability of the order of partial differentiations, and the
extension of Taylor's theorem (see INFINITESIMAL CALCULUS).
With a view to the establishment of the notion of integration through a
domain, we must define the "extent" of the domain. Take first a domain
consisting of the point a and all the points x for which |x - a| < 1/2h,
where h is a chosen positive number; the extent of this domain is h^n, n
being the number of variables; such a domain may be described as
"square," and the number h may be called its "breadth"; it is a
homogeneous part of the numerical continuum of n dimensions, and its
boundary consists of all the points for which |x - a| = 1/2h. Now the
points of any domain, which does not extend to an infinite distance, may
be assigned to a finite number m of square domains of finite breadths,
so that every point of the domain is either within one of these square
domains or on its boundary, and so that no point is within two of the
square domains; also we may devise a rule by which, as the number m
increases indefinitely, the breadths of all the square domains are
diminished indefinitely. When this process is applied to a homogeneous
part, H, of the numerical continuum C_n, then, at any stage of the
process, there will be some square domains of which all the points
belong to H, and there will generally be others of which some, but not
all, of the points belong to H. As the number m is increased
indefinitely the sums of the extents of both these categories of square
domains will tend to definite limits, which cannot be negative; when the
second of these limits is zero the domain H is said to be "measurable,"
and the first of these limits is its "extent"; it is independent of the
rule adopted for constructing the square domains and contracting their
breadths. The notion thus introduced may be adapted by suitable
modifications to continua of lower dimensions in C_n.
The integral of a function f(x) through a measurable domain H, which
is a homogeneous part of the numerical continuum of n dimensions, is
defined in just the same way as the integral through an interval, the
extent of a square domain taking the place of the difference of the
end-values of a partial interval; and the condition of integrability
takes the same form as in the simple case. In particular, the
condition is satisfied when the function is continuous throughout the
domain. The definition of an integral through a domain may be adapted
to any domain of measurable extent. The extensions to "improper"
definite integrals may be made in the same way as for a function of
one variable; in the particular case of a function which tends to
become infinite at a point in the domain of integration, the point is
enclosed in a partial domain which is omitted from the integration,
and a limit is taken when the extent of the omitted partial domain is
diminished indefinitely; a divergent integral may have different
(principal) values for different modes of contracting the extent of
the omitted partial domain. In applications to mathematical physics
great importance attaches to convergent integrals and to principal
values of divergent integrals. For example, any component of magnetic
force at a point within a magnet, and the corresponding component of
magnetic induction at the same point are expressed by different
principal values of the same divergent integral. Delicate questions
arise as to the possibility of representing the integral of a function
of n variables through a domain H_n, as a repeated integral, of
evaluating it by successive integrations with respect to the variables
one at a time and of interchanging the order of such integrations.
These questions have been discussed very completely by C. Jordan, and
we may quote the result that all the transformations in question are
valid when the function is continuous throughout the domain.
20. _Representation of Functions in General._--We have seen that the
notion of a function is wider than the notion of an analytical
expression, and that the same function may be "represented" by one
expression in one part of the domain of the argument and by some other
expression in another part of the domain (S 5). Thus there arises the
general problem of the representation of functions. The function may be
given by specifying the domain of the argument and the rule of
calculation, or else the function may have to be determined in
accordance with certain conditions; for example, it may have to satisfy
in a prescribed domain an assigned differential equation. In either case
the problem is to determine, when possible, a single analytical
expression which shall have the same value as the function at all points
in the domain of the argument. For the representation of most functions
for which the problem can be solved recourse must be had to limiting
processes. Thus we may utilize infinite series, or infinite products, or
definite integrals; or again we may represent a function of one variable
as the limit of an expression containing two variables in a domain in
which one variable remains constant and another varies. An example of
this process is afforded by the expression Lt_y = [oo]xy/(x^2y + 1),
which represents a function of x vanishing at x = 0 and at all other
values of x having the value of 1/x. The method of series falls under
this more general process (cf. S 6). When the terms u1, u2, ... of a
series are functions of a variable x, the sum s_n of the first n terms
of the series is a function of x and n; and, when the series is
convergent, its sum, which is Lt_n = [oo]s_n, can represent a function
of x. In most cases the series converges for some values of x and not
for others, and the values for which it converges form the "domain of
convergence." The sum of the series represents a function in this
domain.
The apparently more general method of representation of a function of
one variable as the limit of a function of two variables has been
shown by R. Baire to be identical in scope with the method of series,
and it has been developed by him so as to give a very complete account
of the possibility of representing functions by analytical
expressions. For example, he has shown that Riemann's totally
discontinuous function, which is equal to 1 when x is rational and to
0 when x is irrational, can be represented by an analytical
expression. An infinite process of a different kind has been adapted
to the problem of the representation of a continuous function by T.
Broden. He begins with a function having a graph in the form of a
regular polygon, and interpolates additional angular points in an
ordered sequence without limit. The representation of a function by
means of an infinite product falls clearly under Baire's method, while
the representation by means of a definite integral is analogous to
Broden's method. As an example of these two latter processes we may
cite the Gamma function [[Gamma](x)] defined for positive values of x
by the definite integral
_
/ [oo]
| e^(-t)t^(x - 1)dt,
_/0
or by the infinite product
/ x \
L t_(n = [oo]) n^x/x (1 + x)(1 + 1/2x) ... ( 1 + ----- ).
\ n - 1 /
The second of these expressions avails for the representation of the
function at all points at which x is not a negative integer.
21. _Power Series._--Taylor's theorem leads in certain cases to a
representation of a function by an infinite series. We have under
certain conditions (S 13)
_n-1
\ (x - a)^r
[f](x) = [f](a) + /_ --------- [f]^(r) (a) + R_n;
r=1 r!
and this becomes
_[oo]
\ (x - a)^r
[f](x) = [f](a) + /_ --------- [f]^(r) (a),
r=1 r!
provided that ([alpha]) a positive number k can be found so that at all
points in the interval between a and a + k (except these points) [f](x)
has continuous differential coefficients of all finite orders, and at a
has progressive differential coefficients of all finite orders; ([beta])
Cauchy's form of the remainder R_n, viz. [(x - a)^n / (n - 1)!] (1 -
[theta])^(n - 1)[f]^n {a + [theta](x - a)}, has the limit zero when n
increases indefinitely, for all values of [theta] between 0 and 1, and
for all values of x in the interval between a and a + k, except possibly
a + k. When these conditions are satisfied, the series (1) represents
the function at all points of the interval between a and a + k, except
possibly a + k, and the function is "analytic" (S 13) in this domain.
Obvious modifications admit of extension to an interval between a and a
- k, or between a - k and a + k. When a series of the form (1)
represents a function it is called "the Taylor's series for the
function."
Taylor's series is a power series, i.e. a series of the form
_[oo]
\ a_n (x - a)^n.
/_
n=0
As regards power series we have the following theorems:
1. If the power series converges at any point except a there is a
number k which has the property that the series converges absolutely
in the interval between a - k and a + k, with the possible exception
of one or both end-points.
2. The power series represents a continuous function in its domain of
convergence (the end-points may have to be excluded).
3. This function is analytic in the domain, and the power series
representing it is the Taylor's series for the function.
The theory of power series has been developed chiefly from the point
of view of the theory of functions of complex variables.
22. _Uniform Convergence._--We shall suppose that the domain of
convergence of an infinite series of functions is an interval with the
possible exception of isolated points. Let [f](x) be the sum of the
series at any point x of the domain, and [f]_n(x) the sum of the first n
+ 1 terms. The condition of convergence at a point a is that, after any
positive number [epsilon], however small, has been specified, it must be
possible to find a number n so that |[f]_m(a) - [f]_p(a)| < [epsilon]
for all values of m and p which exceed n. The sum, [f](a), is the limit
of the sequence of numbers [f]_n(a) at n = [oo]. The convergence is
said to be "uniform" in an interval if, after specification of
[epsilon], the same number n suffices at all points of the interval to
make |[f](x) - [f]_m(x)| < [epsilon] for all values of m which exceed n.
The numbers n corresponding to any [epsilon], however small, are all
finite, but, when [epsilon] is less than some fixed finite number, they
may have an infinite superior limit (S 7); when this is the case there
must be at least one point, a, of the interval which has the property
that, whatever number N we take, [epsilon] can be taken so small that,
at some point in the neighbourhood of a, n must be taken > N to make
|[f](x) - f_m(x)| < [epsilon] when m > n; then the series does not
converge uniformly in the neighbourhood of a. The distinction may be
otherwise expressed thus: Choose a first and [epsilon] afterwards, then
the number n is finite; choose [epsilon] first and allow a to vary, then
the number n becomes a function of a, which may tend to become infinite,
or may remain below a fixed number; if such a fixed number exists, ho
wever small [epsilon] may be, the convergence is uniform.
For example, the series sin x - 1/2 sin 2x + {1/3} sin 3x - ... is
convergent for all real values of x, and, when [pi] > x > -[pi] its
sum is 1/2x; but, when x is but a little less than [pi], the number of
terms which must be taken in order to bring the sum at all near to the
value of 1/2x is very large, and this number tends to increase
indefinitely as x approaches [pi]. This series does not converge
uniformly in the neighbourhood of x = [pi]. Another example is
afforded by the series
_[oo] nx (n + 1)x
\ ---------- - ---------------- ,
/_ n^2x^2 + 1 (n + 1)^2x^2 + 1
n=0
of which the remainder after n terms is nx/(n^2x^2 + 1). If we put x =
1/n, for any value of n, however great, the remainder is 1/2; and the
number of terms required to be taken to make the remainder tend to
zero depends upon the value of x when x is near to zero--it must, in
fact, be large compared with 1/x. The series does not converge
uniformly in the neighbourhood of x = 0.
As regards series whose terms represent continuous functions we have the
following theorems:
(1) If the series converges uniformly in an interval it represents a
function which is continuous throughout the interval.
(2) If the series represents a function which is discontinuous in an
interval it cannot converge uniformly in the interval.
(3) A series which does not converge uniformly in an interval may
nevertheless represent a function which is continuous throughout the
interval.
(4) A power series converges uniformly in any interval contained within
its domain of convergence, the end-points being excluded.
(5) If [Sigma] (r=0 to [oo]) [f]_r(x) = [f](x) converges uniformly in
the interval between a and b
_ _[oo] _
/ b \ / b
| [f](x)dx = /_ | [f]_r(x)dx,
_/ a r=0 _/a
or a series which converges uniformly may be integrated term by term.
(6) If [Signa] (r=0 to [oo]) [f]'_r(x) converges uniformly in an
interval, then [Signa] (r=o to [oo]) [f]_r(x) converges in the interval,
and represents a continuous differentiable function, [phi](x); in fact
we have
_[oo]
[phi]'(x) = \ [f]'_r(x),
/_
r=0
or a series can be differentiated term by term if the series of derived
functions converges uniformly.
A series whose terms represent functions which are not continuous
throughout an interval may converge uniformly in the interval. If
[Signa] (r=0 to [oo]) [f]_r(x) = [f](x), is such a series, and if all
the functions [f]_r(x) have limits at a, then [f](x) has a limit at a,
which is [Signa] (r=0 a=0 to [oo]) Lt [f]_r(x). A similar theorem holds
for limits on the left or on the right.
23. _Fourier's Series._--An extensive class of functions admit of being
represented by series of the form
_[oo] / n[pi]x n[pi]x \
a0 + \ ( a_n cos ------ + b_n sin ------ ), (i.)
/_ \ c c /
n=1
and the rule for determining the coefficients a_n, b_n of such a series,
in order that it may represent a given function [f](x) in the interval
between -c and c, was given by Fourier, viz. we have
_ _
1 / c 1 / c n[pi]x
a0 = --- | [f](x)dx, a_n = -- | [f](x)cos ------ dx,
2c _/-c c _/-c c
_
/ c 1 n[pi]x
b_n = | -- [f](x)sin ------ dx.
_/-c c c
The interval between -c and c may be called the "periodic interval," and
we may replace it by any other interval, e.g. that between 0 and 1,
without any restriction of generality. When this is done the sum of the
series takes the form
_ _r=n
/ 1 \ [f](z)cos {2r[pi](z - x)} dz,
Lt | /_
n=[oo] _/0 r=-n
and this is
_
/ 1 sin {(2n + 1)(z - x)[pi]}
Lt | [f](z) ------------------------ dz. (ii.)
n=[oo] _/0 sin {(z - x)[pi]}
Fourier's theorem is that, if the periodic interval can be divided into
a finite number of partial intervals within each of which the function
is ordinary (S 14), the series represents the function within each of
those partial intervals. In Fourier's time a function of this character
was regarded as completely arbitrary.
By a discussion of the integral (ii.) based on the Second Theorem of
the Mean (S 15) it can be shown that, if [f](x) has restricted
oscillation in the interval (S 11), the sum of the series is equal to
1/2{[f](x + 0) + [f](x - 0)} at any point x within the interval, and
that it is equal to 1/2{[f]( + 0) + [f](1 - 0} at each end of the
interval. (See the article FOURIER'S SERIES.) It therefore represents
the function at any point of the periodic interval at which the
function is continuous (except possibly the end-points), and has a
definite value at each point of discontinuity. The condition of
restricted oscillation includes all the functions contemplated in the
statement of the theorem and some others. Further, it can be shown
that, in any partial interval throughout which [f](x) is continuous,
the series converges uniformly, and that no series of the form (i),
with coefficients other than those determined by Fourier's rule, can
represent the function at all points, except points of discontinuity,
in the same periodic interval. The result can be extended to a
function [f](x) which tends to become infinite at a finite number of
points a of the interval, provided (1) [f](x) tends to become
determinately infinite at each of the points a, (2) the improper
definite integral of [f](x) through the interval is convergent, (3)
[f](x) has not an infinite number of discontinuities or of maxima or
minima in the interval.
24. _Representation of Continuous Functions by Series._--If the series
for [f](x) formed by Fourier's rule converges at the point a of the
periodic interval, and if [f](x) is continuous at a, the sum of the
series is [f](a); but it has been proved by P. du Bois Reymond that the
function may be continuous at a, and yet the series formed by Fourier's
rule may be divergent at a. Thus some continuous functions do not admit
of representation by Fourier's series. All continuous functions,
however, admit of being represented with arbitrarily close approximation
in either of two forms, which may be described as "terminated Fourier's
series" and "terminated power series," according to the two following
theorems:
(1) If [f](x) is continuous throughout the interval between 0 and 2[pi],
and if any positive number [epsilon] however small is specified, it is
possible to find an integer n, so that the difference between the value
of [f](x) and the sum of the first n terms of the series for [f](x),
formed by Fourier's rule with periodic interval from 0 to 2[pi], shall
be less than [epsilon] at all points of the interval. This result can be
extended to a function which is continuous in any given interval.
(2) If [f](x) is continuous throughout an interval, and any positive
number [epsilon] however small is specified, it is possible to find an
integer n and a polynomial in x of the nth degree, so that the
difference between the value of [f](x) and the value of the polynomial
shall be less than [epsilon] at all points of the interval.
Again it can be proved that, if [f](x) is continuous throughout a given
interval, polynomials in x of finite degrees can be found, so as to form
an infinite series of polynomials whose sum is equal to [f](x) at all
points of the interval. Methods of representation of continuous
functions by infinite series of rational fractional functions have also
been devised.
Particular interest attaches to continuous functions which are not
differentiable. Weierstrass gave as an example the function
represented by the series [Sigma] (n=0 to [oo]) a^n cos(b^[n] x[pi]),
where a is positive and less than unity, and b is an odd integer
exceeding (1 + (3/2)[pi]) / a. It can be shown that this series is
uniformly convergent in every interval, and that the continuous
function [f](x) represented by it has the property that there is, in
the neighbourhood of any point x0, an infinite aggregate of points x',
having x0 as a limiting point, for which {[f](x') - [f](x0)} / (x' -
x0) tends to become infinite with one sign when x' - x0 approaches
zero through positive values, and infinite with the opposite sign when
x' - x0 approaches zero through negative values. Accordingly the
function is not differentiable at any point. The definite integral of
such a function [f](x) through the interval between a fixed point and
a variable point x, is a continuous differentiable function F(x), for
which F'(x) = [f](x); and, if [f](x) is one-signed throughout any
interval F(x) is monotonous throughout that interval, but yet F(x)
cannot be represented by a curve. In any interval, however small, the
tangent would have to take the same direction for infinitely many
points, and yet there is no interval in which the tangent has
everywhere the same direction. Further, it can be shown that all
functions which are everywhere continuous and nowhere differentiable
are capable of representation by series of the form [Sigma][a]_n
[phi]_n (x), where [Sigma][a]_n is an absolutely convergent series of
numbers, and [phi]_n(x) is an analytic function whose absolute value
never exceeds unity.
25. _Calculations with Divergent Series._--When the series described in
(1) and (2) of S 24 diverge, they may, nevertheless, be used for the
approximate numerical calculation of the values of the function,
provided the calculation is not carried beyond a certain number of
terms. Expansions in series which have the property of representing a
function approximately when the expansion is not carried too far are
called "asymptotic expansions." Sometimes they are called
"semi-convergent series"; but this term is avoided in the best modern
usage, because it is often used to describe series whose convergence
depends upon the order of the terms, such as the series 1 - 1/2 + 1/3 -
...
In general, let [f]0(x) + [f]1(x) + ... be a series of functions which
does not converge in a certain domain. It may happen that, if any
number [epsilon], however small, is first specified, a number n can
afterwards be found so that, at a point a of the domain, the value
[f](a) of a certain function [f](x) is connected with the sum of the
first n + 1 terms of the series by the relation |[f](a) - [Sigma] (r=0
to n) [f]_r(a) | < [epsilon]. It must also happen that, if any number
N, however great, is specified, a number n'(>n) can be found so that,
for all values of m which exceed n', | [Sigma](r=0 to m) [f]_r(a) | >
N. The divergent series [f]0(x) + [f]1(x) + ... is then an asymptotic
expansion for the function f(x) in the domain.
The best known example of an asymptotic expansion is Stirling's
formula for n! when n is large, viz.
_____
/
n! = \/(2[pi]) 1/2n^(n + 1/2) e^(-n + [theta] / 12n),
where [theta] is some number lying between 0 and 1. This formula is
included in the asymptotic expansion for the Gamma function. We have
in fact
log {[Gamma](x)} = (x - 1/2) log x - x + 1/2 log 2[pi] + [~omega](x),
where [~omega](x) is the function defined by the definite integral
_
/ [oo]
~[omega](x) = | {[1 - e^(-t)]^(-1) - t^(-1) - 1/2} t^(-1) e^(-tx)dt.
_/0
The multiplier of e^(-tx) under the sign of integration can be
expanded in the power series
B1 B2 B3
---- - ---- t^2 + ---- t^4 - ...,
2! 4! 6!
where B1, B2, ... are "Bernoulli's numbers" given by the formula
_[oo]
\
B_m = 2.2m! (2[pi])^(-2m) /_ [r^(-2m)].
r=1
When the series is integrated term by term, the right-hand member of
the equation for [~omega](x) takes the form
B1 1 B2 1 B3 1
---- --- - ---- --- + ---- --- - ...,
1.2 x 3.4 x^3 5.6 x^5
This series is divergent; but, if it is stopped at any term, the
difference between the sum of the series so terminated and the value
of [~omega](x) is less than the last of the retained terms. Stirling's
formula is obtained by retaining the first term only. Other well-known
examples of asymptotic expansions are afforded by the descending
series for Bessel's functions. Methods of obtaining such expansions
for the solutions of linear differential equations of the second order
were investigated by G.G. Stokes (_Math. and Phys. Papers_, vol. ii.
p. 329), and a general theory of asymptotic expansions has been
developed by H. Poincare. A still more general theory of divergent
series, and of the conditions in which they can be used, as above, for
the purposes of approximate calculation has been worked out by E.
Borel. The great merit of asymptotic expansions is that they admit of
addition, subtraction, multiplication and division, term by term, in
the same way as absolutely convergent series, and they admit also of
integration term by term; that is to say, the results of such
operations are asymptotic expansions for the sum, difference,
product, quotient, or integral, as the case may be.
26. _Interchange of the Order of Limiting Operations._--When we require
to perform any limiting operation upon a function which is itself
represented by the result of a limiting process, the question of the
possibility of interchanging the order of the two processes always
arises. In the more elementary problems of analysis it generally happens
that such an interchange is possible; but in general it is not possible.
In other words, the performance of the two processes in different orders
may lead to two different results; or the performance of them in one of
the two orders may lead to no result. The fact that the interchange is
possible under suitable restrictions for a particular class of
operations is a theorem to be proved.
Among examples of such interchanges we have the differentiation and
integration of an infinite series term by term (S 22), and the
differentiation and integration of a definite integral with respect to
a parameter by performing the like processes upon the subject of
integration (S 19). As a last example we may take the limit of the sum
of an infinite series of functions at a point in the domain of
convergence. Suppose that the series [Sigma] (r=0 to [oo]) [f]_r(x)
represents a function ([f]x) in an interval containing a point a, and
that each of the functions [f]_r(x) has a limit at a. If we first put
x = a, and then sum the series, we have the value [f](a); if we first
sum the series for any x, and afterwards take the limit of the sum at
x = a, we have the limit of [f](x) at a; if we first replace each
function [f]_r(x) by its limit at a, and then sum the series, we may
arrive at a value different from either of the foregoing. If the
function [f](x) is continuous at a, the first and second results are
equal; if the functions [f]_r(x) are all continuous at a, the first
and third results are equal; if the series is uniformly convergent,
the second and third results are equal. This last case is an example
of the interchange of the order of two limiting operations, and a
sufficient, though not always a necessary, condition, for the validity
of such an interchange will usually be found in some suitable
extension of the notion of uniform convergence.
AUTHORITIES.--Among the more important treatises and memoirs connected
with the subject are: R. Baire, _Fonctions discontinues_ (Paris,
1905); O. Biermann, _Analytische Functionen_ (Leipzig, 1887); E.
Borel, _Theorie des fonctions_ (Paris, 1898) (containing an
introductory account of the Theory of Aggregates), and _Series
divergentes_ (Paris, 1901), also _Fonctions de variables reelles_
(Paris, 1905); T.J. I'A. Bromwich, _Introduction to the Theory of
Infinite Series_ (London, 1908); H.S. Carslaw, _Introduction to the
Theory of Fourier's Series and Integrals_ (London, 1906); U. Dini,
_Functionen e. reellen Grosse_ (Leipzig, 1892), and _Serie di Fourier_
(Pisa, 1880); A. Genocchi u. G. Peano, _Diff.- u. Int.-Rechnung_
(Leipzig, 1899); J. Harkness and F. Morley, _Introduction to the
Theory of Analytic Functions_ (London, 1898); A. Harnack, _Diff. and
Int. Calculus_ (London, 1891); E.W. Hobson, _The Theory of Functions
of a real Variable and the Theory of Fourier's Series_ (Cambridge,
1907); C. Jordan, _Cours d'analyse_ (Paris, 1893-1896); L. Kronecker,
_Theorie d. einfachen u. vielfachen Integrale_ (Leipzig, 1894); H.
Lebesgue, _Lecons sur l'integration_ (Paris, 1904); M. Pasch, _Diff.-
u. Int.-Rechnung_ (Leipzig, 1882); E. Picard, _Traite d'analyse_
(Paris, 1891); O. Stolz, _Allgemeine Arithmetik_ (Leipzig, 1885), and
_Diff.- u. Int.-Rechnung_ (Leipzig, 1893-1899); J. Tannery, _Theorie
des fonctions_ (Paris, 1886); W.H. and G.C. Young, _The Theory of Sets
of Points_ (Cambridge, 1906); Broden, "Stetige Functionen e. reellen
Veranderlichen," _Crelle_, Bd. cxviii.; G. Cantor, A series of memoirs
on the "Theory of Aggregates" and on "Trigonometric series" in _Acta
Math_. tt. ii., vii., and _Math. Ann_. Bde. iv.-xxiii.; Darboux,
"Fonctions discontinues," _Ann. Sci. Ecole normale sup_. (2), t. iv.;
Dedekind, _Was sind u. was sollen d. Zahlen_? (Brunswick, 1887), and
_Stetigkeit u. irrationale Zahlen_ (Brunswick, 1872); Dirichlet,
"Convergence des series trigonometriques," _Crelle_, Bd. iv.; P. Du
Bois Reymond, _Allgemeine Functionentheorie_ (Tubingen, 1882), and
many memoirs in _Crelle_ and in _Math. Ann_.; Heine,
"Functionenlehre," _Crelle_, Bd. lxxiv.; J. Pierpont, _The Theory of
Functions of a real Variable_ (Boston, 1905); F. Klein, "Allgemeine
Functionsbegriff," _Math. Ann_. Bd. xxii.; W.F. Osgood, "On Uniform
Convergence," _Amer. J. of Math_. vol. xix.; Pincherle, "Funzioni
analitiche secondo Weierstrass," _Giorn. di mat_. t. xviii.;
Pringsheim, "Bedingungen d. Taylorschen Lehrsatzes," _Math. Ann_. Bd.
xliv.; Riemann, "Trigonometrische Reihe," _Ges. Werke_ (Leipzig,
1876); Schoenflies, "Entwickelung d. Lehre v. d.
Punktmannigfaltigkeiten," _Jahresber. d. deutschen Math.-Vereinigung_,
Bd. viii.; Study, Memoir on "Functions with Restricted Oscillation,"
_Math. Ann_. Bd. xlvii.; Weierstrass, Memoir on "Continuous Functions
that are not Differentiable," _Ges. math. Werke_, Bd. ii. p. 71
(Berlin, 1895), and on the "Representation of Arbitrary Functions,"
ibid. Bd. iii. p. 1; W.H. Young, "On Uniform and Non-uniform
Convergence," _Proc. London Math. Soc._ (Ser. 2) t. 6. Further
information and very full references will be found in the articles by
Pringsheim, Schoenflies and Voss in the _Encyclopadie der math.
Wissenschaften_, Bde. i., ii. (Leipzig, 1898, 1899). (A. E. H. L.)
II.--FUNCTIONS OF COMPLEX VARIABLES
In the preceding section the doctrine of functionality is discussed with
respect to real quantities; in this section the theory when complex or
imaginary quantities are involved receives treatment. The following
abstract explains the arrangement of the subject matter: (S 1), _Complex
numbers_, states what a complex number is; (S 2), _Plotting of simple
expressions involving complex numbers_, illustrates the meaning in some
simple cases, introducing the notion of conformal representation and
proving that an algebraic equation has complex, if not real, roots; (S
3), _Limiting operations_, defines certain simple functions of a complex
variable which are obtained by passing to a limit, in particular the
exponential function, and the generalized logarithm, here denoted by
[lambda](z); (S 4), _Functions of a complex variable in general_, after
explaining briefly what is to be understood by a region of the complex
plane and by a path, and expounding a logical principle of some
importance, gives the accepted definition of a function of a complex
variable, establishes the existence of a complex integral, and proves
Cauchy's theorem relating thereto; (S 5), _Applications_, considers the
differentiation and integration of series of functions of a complex
variable, proves Laurent's theorem, and establishes the expansion of a
function of a complex variable as a power series, leading, in (S 6),
_Singular points_, to a definition of the region of existence and
singular points of a function of a complex variable, and thence, in (S
7), _Monogenic Functions_, to what the writer believes to be the
simplest definition of a function of a complex variable, that of
Weierstrass; (S 8), _Some elementary properties of single valued
functions_, first discusses the meaning of a pole, proves that a single
valued function with only poles is rational, gives Mittag-Leffler's
theorem, and Weierstrass's theorem for the primary factors of an
integral function, stating generalized forms for these, leading to the
theorem of (S 9), _The construction of a monogenic function with a given
region of existence_, with which is connected (S10), _Expression of a
monogenic function by rational functions in a given region_, of which
the method is applied in (S 11), _Expression of_ (1 - z)^(-1) _by
polynomials_, to a definite example, used here to obtain (S 12), _An
expansion of an arbitrary function by means of a series of polynomials,
over a star region_, also obtained in the original manner of
Mittag-Leffler; (S 13), _Application of Cauchy's theorem to the
determination of definite integrals_, gives two examples of this method;
(S 14), _Doubly Periodic Functions_, is introduced at this stage as
furnishing an excellent example of the preceding principles. The reader
who wishes to approach the matter from the point of view of Integral
Calculus should first consult the section (S 20) below, dealing with
_Elliptic Integrals_; (S 15), _Potential Functions, Conformal
representation in general_, gives a sketch of the connexion of the
theory of potential functions with the theory of conformal
representation, enunciating the Schwarz-Christoffel theorem for the
representation of a polygon, with the application to the case of an
equilateral triangle; (S 16), _Multiple-valued Functions, Algebraic
Functions_, deals for the most part with algebraic functions, proving
the residue theorem, and establishing that an algebraic function has a
definite Order; (S 17), _Integrals of Algebraic Functions_, enunciating
Abel's theorem; (S 18), _Indeterminateness of Algebraic Integrals_,
deals with the periods associated with an algebraic integral,
establishing that for an elliptic integral the number of these is two;
(S 19), _Reversion of an algebraic integral_, mentions a problem
considered below in detail for an elliptic integral; (S 20), _Elliptic
Integrals_, considers the algebraic reduction of any elliptic integral
to one of three standard forms, and proves that the function obtained by
reversion is single-valued; (S 21), _Modular Functions_, gives a
statement of some of the more elementary properties of some functions of
great importance, with a definition of Automorphic Functions, and a hint
of the connexion with the theory of linear differential equations; (S
22), _A property of integral functions, deduced from the theory of
modular functions_, proves that there cannot be more than one value not
assumed by an integral function, and gives the basis of the well-known
expression of the modulus of the elliptic functions in terms of the
ratio of the periods; (S 23), _Geometrical applications of Elliptic
Functions_, shows that any plane curve of deficiency unity can be
expressed by elliptic functions, and gives a geometrical proof of the
addition theorem for the function RN(u); (S 24), _Integrals of Algebraic
Functions in connexion with the theory of plane curves_, discusses the
generalization to curves of any deficiency; (S 25), _Monogenic Functions
of several independent variables_, describes briefly the beginnings of
this theory, with a mention of some fundamental theorems: (S 26),
_Multiply-Periodic Functions and the Theory of Surfaces_, attempts to
show the nature of some problems now being actively pursued.
Beside the brevity necessarily attaching to the account here given of
advanced parts of the subject, some of the more elementary results are
stated only, without proof, as, for instance: the monogeneity of an
algebraic function, no reference being made, moreover, to the cases of
differential equations whose integrals are monogenic; that a function
possessing an algebraic addition theorem is necessarily an elliptic
function (or a particular case of such); that any area can be
conformally represented on a half plane, a theorem requiring further
much more detailed consideration of the meaning of _area_ than we have
given; while the character and properties, including the connectivity,
of a Riemann surface have not been referred to. The theta functions are
referred to only once, and the principles of the theory of Abelian
Functions have been illustrated only by the developments given for
elliptic functions.
S 1. _Complex Numbers._--Complex numbers are numbers of the form x + iy,
where x, y are ordinary real numbers, and i is a symbol imagined capable
of combination with itself and the ordinary real numbers, by way of
addition, subtraction, multiplication and division, according to the
ordinary commutative, associative and distributive laws; the symbol i is
further such that i^2 = -1.
Taking in a plane two rectangular axes Ox, Oy, we assume that every
point of the plane is definitely associated with two real numbers x, y
(its co-ordinates) and conversely; thus any point of the plane is
associated with a single complex number; in particular, for every
point of the axis Ox, for which y = O, the associated number is an
ordinary real number; the complex numbers thus include the real
numbers. The axis Ox is often called the real axis, and the axis Oy
the imaginary axis. If P be the point associated with the complex
variable z = x + iy, the distance OP be called r, and the positive
angle less than 2[pi] between Ox and OP be called [theta], we may
write z = r(cos[theta] + i sin[theta]); then r is called the modulus
or absolute value of z and often denoted by |z| and [theta] is called
the phase or amplitude of z, and often denoted by ph (z); strictly the
phase is ambiguous by additive multiples of 2[pi]. If z' = x' + iy' be
represented by P', the complex argument z' + z is represented by a
point P" obtained by drawing from P' a line equal to and parallel to
OP; the geometrical representation involves for its validity certain
properties of the plane; as, for instance, the equation z' + z = z +
z' involves the possibility of constructing a parallelogram (with OP"
as diagonal). It is important constantly to bear in mind, what is
capable of easy algebraic proof (and geometrically is Euclid's
proposition III. 7), that the modulus of a sum or difference of two
complex numbers is generally less than (and is never greater than) the
sum of their moduli, and is greater than (or equal to) the difference
of their moduli; the former statement thus holds for the sum of any
number of complex numbers. We shall write E(i[theta]) for cos[theta] +
i sin [theta]; it is at once verified that E(i[alpha]). E(i[beta]) =
E[i([alpha] + [beta])], so that the phase of a product of complex
quantities is obtained by addition of their respective phases.
S 2. _Plotting and Properties of Simple Expressions involving a Complex
Number._--If we put [zeta] = (z-i)/(z + i), and, putting [zeta] = [xi] +
i[eta], take a new plane upon which [xi], [eta] are rectangular
co-ordinates, the equations [xi] = (x^2 + y^2-1)/[x^2 + (y + 1)^2],
[eta] = -2xy/[x^2 + (y + i)^2] will determine, corresponding to any
point of the first plane, a point of the second plane. There is the one
exception of z = -i, that is, x = 0, y = -1, of which the corresponding
point is at infinity. It can now be easily proved that as z describes
the real axis in its plane the point [zeta] describes once a circle of
radius unity, with centre at [zeta] = 0, and that there is a definite
correspondence of point to point between points in the z-plane which are
above the real axis and points of the [zeta]-plane which are interior to
this circle; in particular z = i corresponds to [zeta] = 0.
Moreover, [zeta] being a rational function of z, both [xi] and [eta]
are continuous differentiable functions of x and y, save when [zeta]
is infinite; writing [zeta] = [f](x, y) = [f](z - iy, y), the fact
that this is really independent of y leads at once to (Pd)f/(Pd)x +
i(Pd)[f]/(Pd)y = 0, and hence to
(Pd)[xi] (Pd)[eta] (Pd)[xi] (Pd)[eta] (Pd)^2[xi] (Pd)^2[xi]
-------- = ---------, -------- = - ---------, ---------- + ---------- = 0;
(Pd)x (Pd)y' (Pd)y (Pd)x' (Pd)x^2 (Pd)y^2
so that [xi] is not any arbitrary function of x, y, and when [xi] is
known [eta] is determinate save for an additive constant. Also, in
virtue of these equations, if [zeta], [zeta]' be the values of [zeta]
corresponding to two near values of z, say z and z', the ratio
([zeta]'-[zeta])/(z'- z) has a definite limit when z' = z, independent
of the ultimate phase of z'- z, this limit being therefore equal to
(Pd)[zeta]/(Pd)x, that is, (Pd)[xi]/(Pd)x + i(Pd)[eta])/(Pd)x.
Geometrically this fact is interpreted by saying that if two curves in
the z-plane intersect at a point P, at which both the differential
coefficients (Pd)[xi]/(Pd)x, (Pd)[eta]/(Pd)x are not zero, and P', P"
be two points near to P on these curves respectively, and the
corresponding points of the [zeta]-plane be Q, Q', Q", then (1) the
ratios PP"/PP', QQ"/QQ' are ultimately equal, (2) the angle P'PP" is
equal to Q'QQ", (3) the rotation from PP' to PP" is in the same sense
as from QQ' to QQ", it being understood that the axes of [xi], [eta]
in the one plane are related as are the axes of x, y. Thus any diagram
of the z-plane becomes a diagram of the [zeta]-plane with the same
angles; the magnification, however, which is equal to
_ _
| /(Pd)[xi]\^2 /(Pd)[xi]\^2 | 1/2
| ( -------- ) + ( -------- ) |
|_ \ (Pd)x / \ (Pd)y / _|
varies from point to point. Conversely, it appears subsequently that
the expression of any copy of a diagram (say, a map) which preserves
angles requires the intervention of the complex variable.
As another illustration consider the case when [zeta] is a polynomial
in z,
[zeta] = p0 z^n + p1 z^(n - 1) + ... + p_n;
H being an arbitrary real positive number, it can be shown that a
radius R can be found such for every |z| > R we have |[zeta]| > H;
consider the lower limit of |[zeta]| for |z| < R; as [xi]^2 + [eta]^2
is a real continuous function of x, y for |z| < R, there is a point
(x, y), say (x0, y0), at which |[zeta]| is least, say equal to [rho],
and therefore within a circle in the [zeta]-plane whose centre is the
origin, of radius [rho], there are no points [zeta] representing
values corresponding to |z| < R. But if [zeta]0 be the value of [zeta]
corresponding to (x0, y0), and the expression of [zeta] - [zeta]0 near
z0 = x0 + iy0, in terms of z - z0, be A(z - z0)^m + B(z - z0)^(m + 1)
+ ..., where A is not zero, to two points near to (x0, y0), say (x1,
y1) or z1 and z2 = z0 + (z1 - z0)(cos [pi]/m + i sin [pi]/m), will
correspond two points near to [zeta]0, say [zeta]1, and 2[zeta]0
-[zeta]'1, situated so that [zeta]0 is between them. One of these must
be within the circle ([rho]). We infer then that [rho] = 0, and have
proved that every polynomial in z vanishes for some value of z, and
can therefore be written as a product of factors of the form z -
[alpha], where [alpha] denotes a complex number. This proposition
alone suffices to suggest the importance of complex numbers.
S 3. _Limiting Operations._--In order that a complex number [zeta] =
[xi] + i[eta] may have a limit it is necessary and sufficient that each
of [xi] and [eta] has a limit. Thus an infinite series w0 + w1 + w2 +
..., whose terms are complex numbers, is convergent if the real series
formed by taking the real parts of its terms and that formed by the
imaginary terms are both convergent. The series is also convergent if
the real series formed by the moduli of its terms is convergent; in that
case the series is said to be absolutely convergent, and it can be shown
that its sum is unaltered by taking the terms in any other order.
Generally the necessary and sufficient condition of convergence is that,
for a given real positive [epsilon], a number m exists such that for
every n > m, and every positive p, the batch of terms w_n + w_(n + 1) +
... + w_(n + p) is less than [epsilon] in absolute value. If the terms
depend upon a complex variable z, the convergence is called _uniform_
for a range of values of z, when the inequality holds, for the same
[epsilon] and m, for all the points z of this range.
The infinite series of most importance are those of which the general
term is a_nz^n, wherein a_n is a constant, and z is regarded as
variable, n = 0, 1, 2, 3, ... Such a series is called a power series,
if a real and positive number M exists such that for z = z0 and every
n, |a_n z0^n| < M, a condition which is satisfied, for instance, if
the series converges for z = z0, then it is at once proved that the
series converges absolutely for every z for which |z| < |z0|, and
converges uniformly over every range |z| < r' for which r' < |z0|. To
every power series there belongs then a circle of convergence within
which it converges absolutely and uniformly; the function of z
represented by it is thus continuous within the circle (this being the
result of a general property of uniformly convergent series of
continuous functions); the sum for an interior point z is, however,
continuous with the sum for a point z0 on the circumference, as z
approaches to z0 provided the series converges for z = z0, as can be
shown without much difficulty. Within a common circle of convergence
two power series [Sigma] a_n z^n, [Sigma] b_n z^n can be multiplied
together according to the ordinary rule, this being a consequence of a
theorem for absolutely convergent series. If r1 be less than the
radius of convergence of a series [Sigma] a_nz^n and for |z| = r1, the
sum of the series be in absolute value less than a real positive
quantity M, it can be shown that for |z| = r1 every term is also less
than M in absolute value, namely, |a_n| < Mr1^(-n). If in every
arbitrarily small neighbourhood of z = 0 there be a point for which
two converging power series [Sigma]a_nz^n, [Sigma]b_n z^n agree in
value, then the series are identical, or a_n = b_n; thus also if
[Sigma]a_nz^n vanish at z = 0 there is a circle of finite radius about
z = 0 as centre within which no other points are found for which the
sum of the series is zero. Considering a power series [f](z) =
[Sigma]a_nz^n of radius of convergence R, if |z0| < R and we put z =
z0 + t with |t| < R-|z0|, the resulting series [Sigma]a_n (z0 + t)^n
may be regarded as a double series in z0 and t, which, since |z0| + t
< R, is absolutely convergent; it may then be arranged according to
powers of t. Thus we may write [f](z) = [Sigma]A_n t^n; hence A0 =
[f](z0), and we have [[f](z0 + t) - [f](z0)]/t = [Sigma](n=1) A_n
t^(n-l), wherein the continuous series on the right reduces to A1 for
t = 0; thus the ratio on the left has a definite limit when t = 0,
equal namely to A1 or [Sigma]na_nz0^(n - 1). In other words, the
original series may legitimately be differentiated at any interior
point z0 of its circle of convergence. Repeating this process we find
[f](z0 + t) = [Sigma]t^n [f]^(n) (z0)/n!, where [f]^(n) (z0) is the
nth differential coefficient. Repeating for this power series, in t,
the argument applied about z = 0 for [Sigma]a_n z^n, we infer that for
the series [f](z) every point which reduces it to zero is an isolated
point, and of such points only a finite number lie within a circle
which is within the circle of convergence of [f](z).
Perhaps the simplest possible power series is e^z = exp(z) = 1 +
z^2/2! + z^3/3! + ... of which the radius of convergence is infinite.
By multiplication we have exp(z).exp(z^1) = exp(z + z^1). In
particular when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy).
Now the functions
U0 = sin y, V0 = 1 - cos y, U1 = y - sin y,
V1 = 1/2y^2 - 1 + cos y, U2 = (1/6)y^3 - y + sin y,
V2 = (1/24)y^4 - 1/2y^2 + 1 - cos y, ...
all vanish for y = 0, and the differential coefficient of any one
after the first is the preceding one; as a function (of a real
variable) is increasing when its differential coefficient is positive,
we infer, for y positive, that each of these functions is positive;
proceeding to a limit we hence infer that
cos y = 1 - 1/2y^2 + (1/24)y^4 - ..., sin y = y - (1/6)y^3 +
(1/120)y^5 - ...,
for positive, and hence, for all values of y. We thus have exp(iy) =
cos y + i sin y, and exp (z) = exp (x).(cos y + i sin y). In other
words, the modulus of exp (z) is exp (x) and the phase is y. Hence
also
exp(z + 2[pi]i) = exp(x) [cos (y + 2[pi]) + i sin(y + 2[pi])],
which we express by saying that exp (z) has the period 2[pi]i, and
hence also the period 2k[pi]i, where k is an arbitrary integer. From
the fact that the constantly increasing function exp (x) can vanish
only for x = 0, we at once prove that exp (z) has no other periods.
Taking in the plane of z an infinite strip lying between the lines y =
0, y = 2[pi] and plotting the function [zeta] = exp (z) upon a new
plane, it follows at once from what has been said that every complex
value of [zeta] arises when z takes in turn all positions in this
strip, and that no value arises twice over. The equation [zeta] =
exp(z) thus defines z, regarded as depending upon [zeta], with only an
additive ambiguity 2k[pi]i, where k is an integer. We write z =
[lambda]([zeta]); when [zeta] is real this becomes the logarithm of
[zeta]; in general [lambda]([zeta]) = log |[zeta]| + i ph ([zeta]) +
2k[pi]i, where k is an integer; and when [zeta] describes a closed
circuit surrounding the origin the phase of [zeta] increases by 2[pi],
or k increases by unity. Differentiating the series for [zeta] we have
d[zeta]/dz = [zeta], so that z, regarded as depending upon [zeta], is
also differentiable, with dz/d[zeta] = [zeta]^(-1). On the other hand,
consider the series [zeta] - 1 - 1/2([zeta] - 1)^2 + 1/3([zeta] - 1)^3
- ...; it converges when [zeta] = 2 and hence converges for |[zeta] -
1| < 1; its differential coefficient is, however, 1 - ([zeta] - 1) +
([zeta] - 1)^2 - ..., that is, (1 + [zeta] - 1)^(-1). Wherefore if
[phi]([zeta]) denote this series, for |[zeta] - 1| < 1, the difference
[lambda]([zeta]) - [phi]([zeta]), regarded as a function of [xi] and
[eta], has vanishing differential coefficients; if we take the value
of [lambda]([zeta]) which vanishes when [zeta] = 1 we infer thence
that for |[zeta] - 1| < 1, [lambda]([zeta]) = [Sigma][n = 1] [(-1)^(n
- 1)]/n ([zeta] - 1)^n. It is to be remarked that it is impossible for
[zeta] while subject to |[zeta] - 1| < 1 to make a circuit about the
origin. For values of [zeta] for which |[zeta] - 1| [not less than] 1,
we can also calculate [lambda]([zeta]) with the help of infinite
series, utilizing the fact that [lambda]([zeta][zeta]') =
[lambda]([zeta]) + [lambda]([zeta]').
The function [lambda]([zeta]) is required to define [zeta]^a when
[zeta] and a are complex numbers; this is defined as exp
[a[lambda]([zeta])], that is as [Sigma] (n=0) a^n[[lambda]
([zeta])]^n/n!. When a is a real integer the ambiguity of
[lambda]([zeta]) is immaterial here, since exp [a[lambda]([zeta]) +
2ka[pi]i] = exp[a[lambda]([zeta])]; when a is of the form 1/q, where q
is a positive integer, there are q values possible for [zeta]^(1/q),
of the form exp [1/q [lambda]([zeta])] exp(2k[pi]i/q), with k = 0, 1,
... q - 1, all other values of k leading to one of these; the qth
power of any one of these values is [zeta]; when a = p/q, where p, q
are integers without common factor, q being positive, we have
[zeta]^(p/q) = ([zeta]^(1/q))^p. The definition of the symbol [zeta]^a
is thus a generalization of the ordinary definition of a power, when
the numbers are real. As an example, let it be required to find the
meaning of i^i; the number i is of modulus unity and phase 1/2[pi];
thus [lambda](i) = i(1/2[pi] + 2k[pi]); thus
i^i = exp(-1/2[pi] - 2k[pi]) = exp(-1/2[pi]) exp(-2k[pi]),
is always real, but has an infinite number of values.
The function exp (z) is used also to define a generalized form of the
cosine and sine functions when z is complex; we write, namely, cos z =
1/2[exp(iz) + exp(-iz)] and sin z = -1/2i[exp(iz) - exp(-iz)]. It will
be found that these obey the ordinary relations holding when z is
real, except that their moduli are not inferior to unity. For example,
cos i = 1 + 1/2! + 1/4! + ... is obviously greater than unity.
S4. _Of Functions of a Complex Variable in General._--We have in what
precedes shown how to generalize the ordinary rational, algebraic and
logarithmic functions, and considered more general cases, of functions
expressible by power series in z. With the suggestions furnished by
these cases we can frame a general definition. So far our use of the
plane upon which z is represented has been only illustrative, the
results being capable of analytical statement. In what follows this
representation is vital to the mode of expression we adopt; as then the
properties of numbers cannot be ultimately based upon spatial
intuitions, it is necessary to indicate what are the geometrical ideas
requiring elucidation.
Consider a square of side a, to whose perimeter is attached a definite
direction of description, which we take to be counter-clockwise;
another square, also of side a, may be added to this, so that there is
a side common; this common side being erased we have a composite
region with a definite direction of perimeter; to this a third square
of the same size may be attached, so that there is a side common to it
and one of the former squares, and this common side may be erased. If
this process be continued any number of times we obtain a region of
the plane bounded by one or more polygonal closed lines, no two of
which intersect; and at each portion of the perimeter there is a
definite direction of description, which is such that the region is on
the left of the describing point. Similarly we may construct a region
by piecing together triangles, so that every consecutive two have a
side in common, it being understood that there is assigned an upper
limit for the greatest side of a triangle, and a lower limit for the
smallest angle. In the former method, each square may be divided into
four others by lines through its centre parallel to its sides; in the
latter method each triangle may be divided into four others by lines
joining the middle points of its sides; this halves the sides and
preserves the angles. When we speak of a _region_ of the plane in
general, unless the contrary is stated, we shall suppose it capable of
being generated in this latter way by means of a finite number of
triangles, there being an upper limit to the length of a side of the
triangle and a lower limit to the size of an angle of the triangle. We
shall also require to speak of a _path_ in the plane; this is to be
understood as capable of arising as a limit of a polygonal path of
finite length, there being a definite direction or sense of
description at every point of the path, which therefore never meets
itself. From this the meaning of a closed path is clear. The boundary
points of a region form one or more closed paths, but, in general, it
is only in a limiting sense that the interior points of a closed path
are a region.
There is a logical principle also which must be referred to. We
frequently have cases where, about every interior or boundary, point
z0 of a certain region a circle can be put, say of radius r0, such
that for all points z of the region which are interior to this circle,
for which, that is, |z - z0| < r0, a certain property holds. Assuming
that to r0 is given the value which is the upper limit for z0, of the
possible values, we may call the points |z - z0| < r0, the
neighbourhood belonging to or _proper_ to z0, and may speak of the
property as the property (z, z0). The value of r0 will in general vary
with z0; what is in most cases of importance is the question whether
the lower limit of r0 for all positions is zero or greater than zero.
(A) This lower limit is certainly greater than zero provided the
property (z, z0) is of a kind which we may call extensive; such,
namely, that if it holds, for some position of z0 and all positions of
z, within a certain region, then the property (z, z1) holds within a
circle of radius R about any interior point z1 of this region for all
points z for which the circle |z - z1| = R is within the region. Also
in this case r0 varies continuously with z0. (B) Whether the property
is of this extensive character or not we can prove that the region can
be divided into a finite number of sub-regions such that, for every
one of these, the property holds, (1) for _some_ point z0 within or
upon the boundary of the sub-region, (2) for _every_ point z within or
upon the boundary of the sub-region.
We prove these statements (A), (B) in reverse order. To prove (B) let
a region for which the property (z, z0) holds for all points z and
some point z0 of the region, be called _suitable_: if each of the
triangles of which the region is built up be suitable, what is desired
is proved; if not let an unsuitable triangle be subdivided into four,
as before explained; if one of these subdivisions is unsuitable let it
be again subdivided; and so on. Either the process terminates and then
what is required is proved; or else we obtain an indefinitely
continued sequence of unsuitable triangles, each contained in the
preceding, which converge to a point, say [zeta]; after a certain
stage all these will be interior to the proper region of [zeta]; this,
however, is contrary to the supposition that they are all unsuitable.
We now make some applications of this result (B). Suppose a definite
finite real value attached to every interior or boundary point of the
region, say [f](x, y). It may have a finite upper limit H for the
region, so that no point (x, y) exists for which [f](x, y) > H, but
points (x, y) exist for which [f](x, y) > H - [epsilon], however small
[epsilon] may be; if not we say that its upper limit is infinite.
There is then at least one point of the region such that, for points
of the region within a circle about this point, the upper limit of
[f](x, y) is H, however small the radius of the circle be taken; for
if not we can put about every point of the region a circle within
which the upper limit of [f](x, y) is less than H; then by the result
(B) above the region consists of a finite number of sub-regions within
each of which the upper limit is less than H; this is inconsistent
with the hypothesis that the upper limit for the whole region is H. A
similar statement holds for the lower limit. A case of such a function
[f](x, y) is the radius r0 of the neighbourhood proper to any point
z0, spoken of above. We can hence prove the statement (A) above.
Suppose the property (z, z0) extensive, and, if possible, that the
lower limit of r0 is zero. Let then [zeta] be a point such that the
lower limit of r0 is zero for points z0 within a circle about [zeta]
however small; let r be the radius of the neighbourhood proper to
[zeta]; take z0 so that |z0 - [zeta]| < 1/2r; the property (z, z0),
being extensive, holds within a circle, centre z0, of radius r - |z0 -
[zeta]|, which is greater than |z0 - [zeta]|, and increases to r as
|z0 - [zeta]| diminishes; this being true for all points z0 near
[zeta], the lower limit of r0 is not zero for the neighbourhood of
[zeta], contrary to what was supposed. This proves (A). Also, as is
here shown that r0 [ = >] r - |z0-[zeta]|, may similarly be shown that
r [=>] r0 - |z0 - [zeta]|. Thus r0 differs arbitrarily little from r
when |z0-[zeta]| is sufficiently small; that is, r0 varies
continuously with z0. Next suppose the function [f](x, y), which has a
definite finite value at every point of the region considered, to be
continuous but not necessarily real, so that about every point z0,
within or upon the boundary of the region, [eta] being an arbitrary
real positive quantity assigned beforehand, a circle is possible, so
that for all points z of the region interior to this circle, we have
|[f](x, y) - [f](x0, y0)| < 1/2[eta], and therefore (x', y') being any
other point interior to this circle, |[f](x', y') - [f](x, y)| <
[eta]. We can then apply the result (A) obtained above, taking for the
neighbourhood proper to any point z0 the circular area within which,
for any two points (x, y), (x', y'), we have |[f](x', y') - [f](x, y)|
< [eta]. This is clearly an extensive property. Thus, a number r is
assignable, greater than zero, such that, for any two points (x, y),
(x', y') within a circle |z - z0| = r about any point z0, we have
|[f](x', y') - [f](x, y)| < [eta], and, in particular, |[f](x, y) -
[f](x0, y0)| < [eta], where [eta] is an arbitrary real positive
quantity agreed upon beforehand.
Take now any path in the region, whose extreme points are z0, z, and
let z1, ... z_(n - 1) be intermediate points of the path, in order;
denote the continuous function [f](x, y) by [f](z), and let [f]_r
denote any quantity such that |[f]_r - [f](z_r)| [=<] |[f](z_(r + 1))
- [f](z_r)|; consider the sum
(z1 - z0)[f]0 + (z2 - z1)[f]1 + ... + (z - z_(n - 1))[f](n - 1).
By the definition of a path we can suppose, n being large enough, that
the intermediate points z1, ... z_(n - 1) are so taken that if z_i,
z_(i + 1) be any two points intermediate, in order, to z_r and z_(r +
1), we have |z_(i + i) - z_i| < |z_(r + 1) - z_r|; we can thus suppose
|z1 - z0|, |z2 - z1|, ... |z - z_(n - 1)|all to converge constantly
to zero. This being so, we can show that the sum above has a definite
limit. For this it is sufficient, as in the case of an integral of a
function of one real variable, to prove this to be so when the
convergence is obtained by taking new points of division intermediate
to the former ones. If, however, z_(r, 1), z_(r, 2), ... z_(r, m - 1)
be intermediate in order to z_r and z_(r + 1), and |[f]_(r, i) -
[f](z_(r, i))| < |[f](z_(r, i + 1)) - [f](z_(r, i))|, the difference
between [Sigma](z_(r + 1) - z_r)[f]_r and
[Sigma]{(z_(r, 1) - z_r)[f]_(r, 0) + (z_(r, 2) - z{r, 1})[f]_(r, 1)
+ ... + (z_(r + 1) - z_(r, m - 1))[f]_(r, m - 1)},
which is equal to
[Sigma]_r [Sigma]_i (z_(r, i + 1) - z_(r, i))([f]_(r, i) - [f]_r),
is, when |z_(r + 1) - z_r| is small enough, to ensure |[f](z_(r +
1)) - [f](z_r)| < [eta], less in absolute value than
[Sigma]2[eta] [Sigma] |z_(r, i + 1) - z{r, i}|,
which, if S be the upper limit of the perimeter of the polygon from
which the path is generated, is < 2[eta]S, and is therefore
arbitrarily small.
The limit in question is called [int](z_0 to z) [f](z)dz. In
particular when [f](z) = 1, it is obvious from the definition that its
value is z - z0; when [f](z) = z, by taking [f]_r = 1/2(z_(r + 1) -
z_r), it is equally clear that its value is 1/2(z^2 - z0^2); these
results will be applied immediately.
Suppose now that to every interior and boundary point z0 of a certain
region there belong two definite finite numbers [f](z0), F(z0), such
that, whatever real positive quantity [eta] may be, a real positive
number [epsilon] exists for which the condition
| [f](z) - [f](z0) |
| ---------------- - F(z0) | < [eta],
| z-z0 |
which we describe as the condition (z, z0), is satisfied for every
point z, within or upon the boundary of the region, satisfying the
limitation |z - z0| < [epsilon]. Then [f](z0) is called a
differentiable function of the complex variable z0 over this region,
its differential coefficient being F(z0). The function [f](z0) is thus
a continuous function of the real variables x0, y0, where z0 = x0 +
iy0, over the region; it will appear that F(z0) is also continuous and
in fact also a differentiable function of z0.
Supposing [eta] to be retained the same for all points z0 of the
region, and [sigma]0 to be the upper limit of the possible values of
[epsilon] for the point z0, it is to be presumed that [sigma]0 will
vary with z0, and it is not obvious as yet that the lower limit of the
values of [sigma]0 as z0 varies over the region may not be zero. We
can, however, show that the region can be divided into a finite number
of sub-regions for each of which the condition (z, z0), above, is
satisfied for all points z, within or upon the boundary of this
sub-region, for an appropriate position of z0, within or upon the
boundary of this sub-region. This is proved above as result (B).
Hence it can be proved that, for a differentiable function [f](z), the
integral [int](z_1 to z) [f](z)dz has the same value by whatever path
within the region we pass from z1 to z. This we prove by showing that
when taken round a closed path in the region the integral
[int][f](z)dz vanishes. Consider first a triangle over which the
condition (z, z0) holds, for some position of z0 and every position of
z, within or upon the boundary of the triangle. Then as
[f](z) = [f](z0) + (z - z0)F(z0) + [eta][theta](z - z0), where |[theta]| < 1,
we have
_ _ _ _
/ / / /
|[f](z)dz = [[f](z0) - z0F(z0)] |dz + F(z0) |zdz + [eta] |[theta](z - z0)dz,
_/ _/ _/ _/
which, as the path is closed, is [eta] [int][theta](z-z0)dz. Now, from
the theorem that the absolute value of a sum is less than the sum of
the absolute values of the terms, this last is less, in absolute
value, than [eta]ap, where a is the greatest side of the triangle and
p is its perimeter; if [Delta] be the area of the triangle, we have
[Delta] = 1/2ab sin C > ([alpha]/[pi])ba, where [alpha] is the least
angle of the triangle, and hence a(a + b + c) < 2a(b + c) <
4[pi][Delta]/[alpha]; the integral [int][f](z)dz round the perimeter
of the triangle is thus < 4[pi][eta][Delta]/[alpha]. Now consider any
region made up of triangles, as before explained, in each of which the
condition (z, z0) holds, as in the triangle just taken. The integral
[int][f](z)dz round the boundary of the region is equal to the sum of
the values of the integral round the component triangles, and thus
less in absolute value than 4[pi][eta]K/[alpha], where K is the whole
area of the region, and [alpha] is the smallest angle of the component
triangles. However small [eta] be taken, such a division of the region
into a finite number of component triangles has been shown possible;
the integral round the perimeter of the region is thus arbitrarily
small. Thus it is actually zero, which it was desired to prove. Two
remarks should be added: (1) The theorem is proved only on condition
that the closed path of integration belongs to the region at every
point of which the conditions are satisfied. (2) The theorem, though
proved only when the region consists of triangles, holds also when the
boundary points of the region consist of one or more closed paths, no
two of which meet.
Hence we can deduce the remarkable result that the value of [f](z) at
any interior point of a region is expressible in terms of the value of
[f](z) at the boundary points. For consider in the original region the
function [f](z)/(z - z0), where z0 is an interior point: this
satisfies the same conditions as [f](z) except in the immediate
neighbourhood of z0. Taking out then from the original region a small
regular polygonal region with z0 as centre, the theorem holds for the
remaining portion. Proceeding to the limit when the polygon becomes a
circle, it appears that the integral [int] dz[f](z)/(z - z0) round the
boundary of the original region is equal to the same integral taken
counter-clockwise round a small circle having z0 as centre; on this
circle, however, if z - z0 = rE(i[theta]), dz/(z - z0) = id[theta],
and [f](z) differs arbitrarily little from f(z0) if r is sufficiently
small; the value of the integral round this circle is therefore,
ultimately, when r vanishes, equal to 2[pi]i[f](z0). Hence [f](z0) = 1
/ 2[pi]i [int] [dt[f](t)/(t - z0)], where this integral is round the
boundary of the original region. From this it appears that
_
[f](z) - [f](z0) 1 / dt[f](t)
F(z0) = lim. ---------------- = ------ | ---------
z - z0 2[pi]i _/ (t-z0)^2
also round the boundary of the original region. This form shows,
however, that F(z0) is a continuous, finite, differentiable function
of z0 over the whole interior of the original region.
S 5. _Applications._--The previous results have manifold applications.
(1) If an infinite series of differentiable functions of z be
uniformly convergent along a certain path lying with the region of
definition of the functions, so that S(2) = u0(z) + u1(z) + ... + u_(n
- 1)(z) + R_n(z), where |R_n(z)| < [epsilon] for all points of the
path, we have
_ _ _ _ _
/z /z /z /z /z
| S(z)dz = | u0(z)dz + | u1(z)dz + ... + | u_(n - 1)(z)dz + | R_n(z)dz,
_/z0 _/z0 _/z0 _/z0 _/z0
wherein, in absolute value, [int](z_0 to z) R_n(z)dz < [epsilon]L, if
L be the length of the path. Thus the series may be integrated, and
the resulting series is also uniformly convergent.
(2) If [f](x, y) be definite, finite and continuous at every point of
a region, and over any closed path in the region [int][f](x, y)dz = 0,
then [psi](z) = [int](z_0 to z) [f](x, y)dz, for interior points z0,
z, is a differentiable function of z, having for its differential
coefficient the function [f](x, y), which is therefore also a
differentiable function of z at interior points.
(3) Hence if the series u0(z) + u1(z) + ... to [oo] be uniformly
convergent over a region, its terms being differentiable functions of
z, then its sum S(z) is a differentiable function of z, whose
differential coefficient, given by (1 / 2[pi]i) [int] (2[pi]i /
(t - z)^2), is obtainable by differentiating the series. This theorem,
unlike (1), does not hold for functions of a real variable.
(4) If the region of definition of a differentiable function [f](z)
include the region bounded by two concentric circles of radii r, R,
with centre at the origin, and z0 be an interior point of this region,
_ _
1 / [f](t)dt 1 / [f](t)dt
[f](z0) = ------ | -------- - ------ | --------,
2[pi]i _/R t - z0 2[pi]i _/r t - z0
where the integrals are both counter-clockwise round the two
circumferences respectively; putting in the first (t - z0)^(-1) =
[Sigma]_(n=0) z0^n/t^(n + 1), and in the second (t - z0)^(-1) =
[Sigma]_(n=0) t^n/z0^(n + 1), we find [f](z0) = [Sigma] (-[oo] to
[oo]) A_nz0^n, wherein A_n = (1 / 2[pi]i) [int] [f(t) / t^(n + 1)] dt,
taken round any circle, centre the origin, of radius intermediate
between r and R. Particular cases are: ([alpha]) when the region of
definition of the function includes the whole interior of the outer
circle; then we may take r = 0, the coefficients A_n for which n < 0
all vanish, and the function [f](z0) is expressed for the whole
interior |z0| < R by a power series [Sigma] (0 to [oo]) A_n z0^n. In
other words, _about every interior point c of the region of definition
a differentiable function of z is expressible by a power series in z -
c; a very important result.
([beta]) If the region of definition, though not including the origin,
extends to within arbitrary nearness of this on all sides, and at the
same time the product z^m [f](z) has a finite limit when |z|
diminishes to zero, all the coefficients A_n for which n < -m vanish,
and we have
f(z0) = A_(-m) z0^(-m) + A_(-m + 1) z0^(-m + 1)
+ ... + A_(-1) z0^(-1) + A0 + A1z0 ... to [oo].
Such a case occurs, for instance, when [f](z) = cosec z, the number m
being unity.
S 6. _Singular Points._--The _region of existence_ of a differentiable
function of z is an unclosed aggregate of points, each of which is an
interior point of a neighbourhood consisting wholly of points of the
aggregate, at every point of which the function is definite and finite
and possesses a unique finite differential coefficient. Every point of
the plane, not belonging to the aggregate, which is a limiting point of
points of the aggregate, such, that is, that points of the aggregate lie
in every neighbourhood of this, is called a _singular point_ of the
function.
About every interior point z0 of the region of existence the function
may be represented by a power series in z-z0, and the series converges
and represents the function over any circle centre at z0 which
contains no singular point in its interior. This has been proved
above. And it can be similarly proved, putting z = 1/[zeta], that if
the region of existence of the function contains all points of the
plane for which |z| > R, then the function is representable for all
such points by a power series in z^(-1) or [zeta]; in such case we say
that the region of existence of the function contains the point z =
[oo]. A series in z^(-1) has a finite limit when |z| = [oo]; a series
in z cannot remain finite for all points z for which |z| > R; for if,
for |z| = R, the sum of a power series [Sigma]a_n z^n in z is in
absolute value less than M, we have |a_n| < Mr(-n), and therefore, if
M remains finite for all values of r however great, a_n = 0. Thus the
region of existence of a function if it contains all finite points of
the plane cannot contain the point z = [oo]; such is, for instance,
the case of the function exp (z) = [Sigma]z^n/n!. This may be regarded
as a particular case of a well-known result (S 7), that the
circumference of convergence of any power series representing the
function contains at least one singular point. As an extreme case
functions exist whose region of existence is circular, there being a
singular point in every arc of the circumference, however small; for
instance, this is the case for the functions represented for |z| < 1
by the series [Sigma]_(n=0) z^m, where m = n^2, the series
[Sigma]_(n=0) z^m where m = n!, and the series [Sigma](n=1 to 0)
z^m/(m + 1)(m + 2) where m = a^n, a being a positive integer, although
in the last case the series actually converges for every point of the
circle of convergence |z| = 1. If z be a point interior to the circle
of convergence of a series representing the function, the series may
be rearranged in powers of z - z0; as z0 approaches to a singular
point of the function, lying on the circle of convergence, the radii
of convergence of these derived series in z - z0 diminish to zero;
when, however, a circle can be put about z0, not containing any
singular point of the function, but containing points outside the
circle of convergence of the original series, then the series in z -
z0 gives the value of the function for these external points. If the
function be supposed to be given only for the interior of the original
circle, by the original power series, the series in z - z0 converging
beyond the original circle gives what is known as an _analytical
continuation_ of the function. It appears from what has been proved
that the value of the function at all points of its region of
existence can be obtained from its value, supposed given by a series
in one original circle, by a succession of such processes of
analytical continuation.
S 7. _Monogenic Functions._--This suggests an entirely different way of
formulating the fundamental parts of the theory of functions of a
complex variable, which appears to be preferable to that so far followed
here.
Starting with a convergent power series, say in powers of z, this
series can be arranged in powers of z - z0, about any point z0
interior to its circle of convergence, and the new series converges
certainly for |z - z0| < r - |z0|, if r be the original radius of
convergence. If for every position of z0 this is the greatest radius
of convergence of the derived series, then the original series
represents a function existing only within its circle of convergence.
If for some position of z0 the derived series converges for |z - z0| <
r - |z0| + D, then it can be shown that for points z, interior to the
original circle, lying in the annulus r - |z0| < |z - z0| < r - |z0| +
D, the value represented by the derived series agrees with that
represented by the original series. If for another point z1 interior
to the original circle the derived series converges for |z - z1| < r -
|z1| + E, and the two circles |z - z0| = r - |z0| + D, |z - z1| = r -
|z1| + E have interior points common, lying beyond |z| = r, then it
can be shown that the values represented by these series at these
common points agree. Either series then can be used to furnish an
analytical continuation of the function as originally defined.
Continuing this process of continuation as far as possible, we arrive
at the conception of the function as defined by an aggregate of power
series of which every one has points of convergence common with some
one or more others; the whole aggregate of points of the plane which
can be so reached constitutes the region of existence of the function;
the limiting points of this region are the points in whose
neighbourhood the derived series have radii of convergence diminishing
indefinitely to zero; these are the singular points. The circle of
convergence of any of the series has at least one such singular point
upon its circumference. So regarded the function is called a
_monogenic_ function, the epithet having reference to the single
origin, by one power series, of the expressions representing the
function; it is also sometimes called a _monogenic analytical_
function, or simply an _analytical_ function; all that is necessary to
define it is the value of the function and of all its differential
coefficients, at some one point of the plane; in the method previously
followed here it was necessary to suppose the function differentiable
at every point of its region of existence. The theory of the
integration of a monogenic function, and Cauchy's theorem, that
[int][f](z)dz = 0 over a closed path, are at once deducible from the
corresponding results applied to a single power series for the
interior of its circle of convergence. There is another advantage
belonging to the theory of monogenic functions: the theory as
originally given here applies in the first instance only to single
valued functions; a monogenic function is by no means necessarily
single valued--it may quite well happen that starting from a
particular power series, converging over a certain circle, and
applying the process of analytical continuation over a closed path
back to an interior point of this circle, the value obtained does not
agree with the initial value. The notion of basing the theory of
functions on the theory of power series is, after Newton, largely due
to Lagrange, who has some interesting remarks in this regard at the
beginning of his _Theorie des fonctions analytiques_. He applies the
idea, however, primarily to functions of a real variable for which the
expression by power series is only of very limited validity; for
functions of a complex variable probably the systematization of the
theory owes most to Weierstrass, whose use of the word monogenic is
that adopted above. In what follows we generally suppose this point of
view to be regarded as fundamental.
S 8. _Some Elementary Properties of Single Valued Functions._--A _pole_
is a singular point of the function [f](z) which is not a singularity of
the function 1/[f](z); this latter function is therefore, by the
definition, capable of representation about this point, z0, by a series
[[f](z)]^(-1 ) = [Sigma]a_n (z - z0)^n. If herein a0 is not zero we can
hence derive a representation for [f](z) as a power series about z0,
contrary to the hypothesis that z0 is a singular point for this
function. Hence a0 = 0; suppose also a1 = 0, a2 = 0, ... a_(m - 1) = 0,
but a_m [+-] 0. Then [[f](z)]^(-1) = (z - z0)^m [a_m + a_(m + 1)(z - z0)
+ ...], and hence (z - z0)^m [f](z) = a_m^(-1) + [Sigma]b_n (z - z0)^n,
namely, the expression of [f](z) about z = z0 contains a finite number
of negative powers of z - z0 and a (finite or) infinite number of
positive powers. Thus a pole is always an isolated singularity.
The integral [int][f](z)dz taken by a closed circuit about the pole
not containing any other singularity is at once seen to be 2[pi]iA1,
where A1 is the coefficient of (z - z0)^(-1) in the expansion of
[f](z) at the pole; this coefficient has therefore a certain
uniqueness, and it is called the _residue of [f](z) at the pole_.
Considering a region in which there are no other singularities than
poles, all these being interior points, _the integral (1 / 2[pi]i)
[int][f](z)dz round the boundary of this region is equal to the sum
of the residues at the included poles_, a very important result. Any
singular point of a function which is not a pole is called an
_essential singularity_; if it be isolated the function is capable, in
the neighbourhood of this point, of approaching arbitrarily near to
any assigned value. For, the point being isolated, the function can be
represented, in its neighbourhood, as we have proved, by a series
[Sigma] (-[oo] to [oo]) a_n(z - z0)^n; it thus cannot remain finite in
the immediate neighbourhood of the point. The point is necessarily an
isolated essential singularity also of the function {[f](z) - A}^(-1)
for if this were expressible by a power series about the point, so
would also the function [f](z) be; as {[f](z) - A}^(-1) approaches
infinity, so does [f](z) approach the arbitrary value A. Similar
remarks apply to the point z = [oo], the function being regarded as a
function of [zeta] = z^(-1). In the neighbourhood of an essential
singularity, which is a limiting point also of poles, the function
clearly becomes infinite. For an essential singularity which is not
isolated the same result does not necessarily hold.
A single valued function is said to be an _integral_ function when it
has no singular points except z = [oo]. Such is, for instance, an
integral polynomial, which has z = [oo] for a pole, and the functions
exp (z) which has z = [oo] as an essential singularity. A function which
has no singular points for finite values of z other than poles is called
a _meromorphic_ function. If it also have a pole at z = [oo] it is a
_rational_ function; for then, if a1, ... a_s be its finite poles, of
orders m1; m2, ... m_s, the product (z - a1)^m1 ... (z - a_s)^m_s[f](z)
is an integral function with a pole at infinity, capable therefore, for
large values of z, of an expression (z^ - 1)^(-m) [Sigma]_(r=0) a_r(z^ -
1)^r; thus (z - a1)^m1 ... (z - a_s)^m_s[f](z) is capable of a form
[Sigma]_(r=0) b_r z^r, but z^(-m) [Sigma]_(r=0) b_r z^r remains finite
for z = [oo]. Therefore b_(r + 1) = b_(r + 2) = ... = 0, and[f](z) is a
rational function.
If for a single valued function F(z) every singular point in the
finite part of the plane is isolated there can only be a finite number
of these in any finite part of the plane, and they can be taken to be
a1, a2, a3, ... with |a1| [=<] |a2| [=<] |a3| ... and limit |a_n| =
[oo]. About a_s the function is expressible as [Sigma] (-[oo] to [oo])
A_n(z - a_s)^n; let [f]_s(z) = [Sigma] (-[oo] to 1) A^n(z - a_s)^n be
the sum of the negative powers in this expansion. Assuming z = 0 not
to be a singular point, let [f]_s(z) be expanded in powers of z, in
the form [Sigma]_(n=0) C_n z^n, and [mu]_s be chosen so that F_s(z) =
[f]_s(z) - [Sigma] (1 to [mu]_s-1) C_nz^n = [Sigma] ([mu]_s to [oo])
C_n z^n is, for |z| < r_s < |a_s|, less in absolute value than the
general term [epsilon]_s of a fore-agreed convergent series of real
positive terms. Then the series [phi](z) = [Sigma] (s=1 to [oo])
F_s(z) converges uniformly in any finite region of the plane, other
than at the points a_s, and is expressible about any point by a power
series, and near a_s, [phi](z) - f_s(z) is expressible by a power
series in z-a_s. Thus F(z) - [phi](z) is an integral function. In
particular when all the finite singularities of F(z) are poles, F(z)
is hereby expressed as the sum of an integral function and a series of
rational functions. The condition |F_s(z)| < [epsilon]_s is imposed
only to render the series [Sigma]F_s(z) uniformly convergent; this
condition may in particular cases be satisfied by a series [Sigma]
G_s(z) where G_s(z) = [f]_s(z) - [Sigma] (1 to [nu]_s-1) C_nz^n and
[nu]_s < [mu]_s. An example of the theorem is the function [pi] cot
[pi]z - z^(-1) for which, taking at first only half the poles,
[f]_s(z) = 1/(z-s); in this case the series [Sigma]F_s(z) where F_s(z)
= (z - s)^-1 + s^-1 is uniformly convergent; thus [pi]cot[pi]z - z^-1
- [Sigma] (-[oo] to [oo]) [(z - s)^-1 + s^-1], where s = 0 is excluded
from the summation, is an integral function. It can be proved that
this integral function vanishes.
Considering an integral function [f](z), if there be no finite
positions of z for which this function vanishes, the function
[lambda][[f](z)] is at once seen to be an integral function, [phi](z),
or [f](z) = exp[[phi](z)]; if however great R may be there be only a
finite number of values of z for which [f](z) vanishes, say z = a1,
... a_m, then it is at once seen that [f](z) = exp [[phi](z)].(z -
a1)^h1...(z - a_m)^h_m, where [phi](z) is an integral function, and
h1, ... h_m are positive integers. If, however, [f](z) vanish for z =
a1, a2 ... where |a1| [=<] |a2| [=<] ... and limit |a_n| = [oo], and
if for simplicity we assume that z - 0 is not a zero and all the zeros
a1, a2, ... are of the first order, we find, by applying the preceding
theorem to the function (1 / [f](z)) (d[f](z) / dz), that [f](z) =
exp[[phi](z)] [Pi] (n=1 to [oo]) {(1 - z/a_n) exp[phi]_n(z)}, where
[phi](z) is an integral function, and [phi]_n(z) is an integral
polynomial of the form
z z^2 z^s
[phi]_n(z) = --- + ------ + ... + ------.
a_n 2a^2_n sa_n^s
The number s may be the same for all values of n, or it may increase
indefinitely with n; it is sufficient in any case to take s = n. In
particular for the function sin[pi]x / [pi]x, we have
_ _
sin[pi]x [oo] | / x \ / x \ |
-------- = [Pi] | ( 1 - -- ) exp ( -- ) |,
[pi]x -[oo] |_ \ n / \ n / _|
where n = 0 is excluded from the product. Or again we have
_ _
1 [oo] | / x \ / x \ |
---------- = xe^C_x [Pi] | ( 1 - -- ) exp ( - -- ) |,
[Gamma](x) n=1 |_ \ n / \ n / _|
where C is a constant, and [Gamma](x) is a function expressible when x
is real and positive by the integral [int] (0 to [oo])
e^(-t) t^(x - 1)dt.
There exist interesting investigations as to the connexion of the
value of s above, the law of increase of the modulus of the integral
function [f](z), and the law of increase of the coefficients in the
series [f](z) = [Sigma] a_n z^n as n increases (see the bibliography
below under _Integral Functions_). It can be shown, moreover, that an
integral function actually assumes every finite complex value, save,
in exceptional cases, one value at most. For instance, the function
exp (z) assumes every finite value except zero (see below under S 21,
_Modular Functions_).
The two theorems given above, the one, known as Mittag-Leffler's
theorem, relating to the expression as a sum of simpler functions of a
function whose singular points have the point z = [oo] as their only
limiting point, the other, Weierstrass's factor theorem, giving the
expression of an integral function as a product of factors each with
only one zero in the finite part of the plane, may be respectively
generalized as follows:--
I. If a1, a2, a3, ... be an infinite series of isolated points having
the points of the aggregate (c) as their limiting points, so that in
any neighbourhood of a point of (c) there exists an infinite number of
the points a1, a2, ..., and with every point a_i there be associated a
polynomial in (z - a_i)^-1, say g_i; then there exists a single valued
function whose region of existence excludes only the points (a) and
the points (c), having in a point a_i a pole whereat the expansion
consists of the terms g_i, together with a power series in z - a_i;
the function is expressible as an infinite series of terms g_i -
[gamma]_i, where [gamma]_i is also a rational function.
II. With a similar aggregate (a), with limiting points (c), suppose
with every point a_i there is associated a positive integer r_i. Then
there exists a single valued function whose region of existence
excludes only the points (c), vanishing to order r_i at the point a_i,
but not elsewhere, expressible in the form
[oo] / a_n - c_n \^r_n
[Pi] ( 1 - --------- ) exp(g_n),
n=1 \ z - c_n /
where with every point a_n is associated a proper point c_n of (c),
and
_[mu]_n
\ 1 / a_n - c_n \^s
g_n = r_n /_ -- ( --------- ),
s=1 s \ z - c_n /
[mu]_n being a properly chosen positive integer.
If it should happen that the points (c) determine a path dividing the
plane into separated regions, as, for instance, if a_n = R(1 - n^-1)
exp(i[pi] [root]2.n), when (c) consists of the points of the circle
|z| = R, the product expression above denotes different monogenic
functions in the different regions, not continuable into one another.
S 9. _Construction of a Monogenic Function with a given Region of
Existence._--A series of isolated points interior to a given region can
be constructed in infinitely many ways whose limiting points are the
boundary points of the region, or are boundary points of the region of
such denseness that one of them is found in the neighbourhood of every
point of the boundary, however small. Then the application of the last
enunciated theorem gives rise to a function having no singularities in
the interior of the region, but having a singularity in a boundary point
in every small neighbourhood of every boundary point; this function has
the given region as region of existence.
S 10. _Expression of a Monogenic Function by means of Rational Functions
in a given Region._--Suppose that we have a region R0 of the plane, as
previously explained, for all the interior or boundary points of which z
is finite, and let its boundary points, consisting of one or more closed
polygonal paths, no two of which have a point in common, be called C0.
Further suppose that all the points of this region, including the
boundary points, are interior points of another region R, whose boundary
is denoted by C. Let z be restricted to be within or upon the boundary
of C0; let a, b, ... be finite points upon C or outside R. Then when b
is near enough to a, the fraction (a - b)/(z - b) is arbitrarily small
for all positions of z; say
| a - b |
| ----- | < [epsilon], for |a - b| < [eta];
| z - b |
the rational function of the complex variable t,
_ _
1 | / a - b \^n |
----- |1 - ( ----- ) |,
t - a |_ \ t - b / _|
in which n is a positive integer, is not infinite at t = a, but has a
pole at t = b. By taking n large enough, the value of this function, for
all positions z of t belonging to R0, differs as little as may be
desired from (t - a)^-1. By taking a sum of terms such as
_ _ _
\ { 1 | / a - b \^n | }^p
F = /_ A_p { ----- |1 - ( ----- ) | },
{ t - a |_ \ t - b / _| }
we can thus build a rational function differing, in value, in R0, as
little as may be desired from a given rational function
_
\
[f] = /_ A_p (t - a)^(-p),
and differing, outside R or upon the boundary of R, from [f], in the
fact that while [f] is infinite at t = a, F is infinite only at t = b.
By a succession of steps of this kind we thus have the theorem that,
given a rational function of t whose poles are outside R or upon the
boundary of R, and an arbitrary point c outside R or upon the boundary
of R, which can be reached by a finite continuous path outside R from
all the poles of the rational function, we can build another rational
function differing in R0 arbitrarily little from the former, whose poles
are all at the point c.
Now any monogenic function [f](t) whose region of definition includes
C and the interior of R can be represented at all points z in R[0] by
_
1 / [f](t)dt
[f](z) = ------ | --------,
2[pi]i _/ t - z
where the path of integration is C. This integral is the limit of a
sum
_
1 \ [f](t_i) (t_(i + 1) - t_i)
S = ------ /_ --------------------------,
2[pi]i t_i - z
where the points t_i are upon C; and the proof we have given of the
existence of the limit shows that the sum S converges to [f](z)
uniformly in regard to z, when z is in R0, so that we can suppose,
when the subdivision of C into intervals t_(i + 1) - t_i, has been
carried sufficiently far, that
| S - [f](z) | < [epsilon],
for all points z of R0, where [epsilon] is arbitrary and agreed upon
beforehand. The function S is, however, a rational function of z with
poles upon C, that is external to R0. We can thus find a rational
function differing arbitrarily little from S, and therefore
arbitrarily little from [f](z), for all points z of R0, with poles at
arbitrary positions outside R0 which can be reached by finite
continuous curves lying outside R from the points of C.
In particular, to take the simplest case, if C0, C be simple closed
polygons, and [GAMMA] be a path to which C approximates by taking the
number of sides of C continually greater, we can find a rational
function differing arbitrarily little from [f](z) for all points of R0
whose poles are at one finite point c external to [GAMMA]. By a
transformation of the form t - c = r^-1, with the appropriate change
in the rational function, we can suppose this point c to be at
infinity, in which case the rational function becomes a polynomial.
Suppose [epsilon]1, [epsilon]2, ... to be an indefinitely continued
sequence of real positive numbers, converging to zero, and P_r to be
the polynomial such that, within C0, |P_r - [f](z)| < [epsilon]_r;
then the infinite series of polynomials
P1(z) + {P2(z) - P1(z)} + {P3(z) - P2(z)} + ...,
whose sum to n terms is P_n(z), converges for all finite values of z
and represents [f](z) within C0.
When C consists of a series of disconnected polygons, some of which
may include others, and, by increasing indefinitely the number of
sides of the polygons C, the points C become the boundary points
[Gamma] of a region, we can suppose the poles of the rational
function, constructed to approximate to [f](z) within R0, to be at
points of [Gamma]. A series of rational functions of the form
H1(z) + {H2(z) - H1(z)} + {H3(z) - H2(z)} + ...
then, as before, represents [f](z) within R0. And R0 may be taken to
coincide as nearly as desired with the interior of the region bounded
by [Gamma].
S 11. _Expression of (1 - z)^(-1) by means of Polynomials.
Applications._--We pursue the ideas just cursorily explained in some
further detail.
Let c be an arbitrary real positive quantity; putting the complex
variable [zeta] = [xi] + i[eta], enclose the points [zeta] = l, [zeta]
= 1 + c by means of (i.) the straight lines [eta] = [+-]a, from [xi] =
l to [xi] = 1 + c, (ii.) a semicircle convex to [zeta] = 0 of equation
([xi] - 1)^2 + [eta]^2 = a^2, (iii.) a semicircle concave to [zeta] =
0 of equation ([xi] - 1 - c)^2 + [eta]^2 = a^2. The quantities c and a
are to remain fixed. Take a positive integer r so that 1/r (c/a) is
less than unity, and put [sigma] = 1/r (c/a). Now take
c1 = 1 + c/r, c2 = 1 + 2c/r, ... c_r = 1 + c;
if n1, n2, ... n_r, be positive integers, the rational function
_ _
1 | / c1 - 1 \^n1 |
---------- |1 - ( ----------- ) |
1 - [zeta] |_ \ c1 - [zeta] / _|
is finite at [zeta] = 1, and has a pole of order n1 at [zeta] = c1;
the rational function
_ _ _ _
1 | / c1 - 1 \^n1 | | / c2 - c1 \^n2 |^n1
---------- |1 - ( ------------ ) | |1 - ( ----------- ) |
1 - [zeta] |_ \ c1 - [zeta] / _| |_ \ c2 - [zeta]/ _|
is thus finite except for [zeta] = c2, where it has a pole of order
n1n2; finally, writing
/ c_s - c_(s-1) \^n_s
x_s = ( ------------- ),
\ c_s - [zeta] /
the rational function
U = (1 - [zeta])^(-1) (1 - x1)(1 - x2)^n1 (1 - x3)^n1n2 ...
(1 - x_r)^(n1n2 ... n_(r - 1))
has a pole only at [zeta] = 1 + c, of order n1n2 ... n_r.
The difference (1 - [zeta])^(-1) - U is of the form (1 -
[zeta])^(-1)P, where P, of the form
1 - (1 - [rho]1)(1 - [rho]2)...(1 - [rho]_k),
in which there are equalities among [rho]1, [rho]2, ... [rho]_k, is of
the form
[Sigma][rho]1 - [Sigma][rho]1[rho]2 + [Sigma][rho]1[rho]2[rho]3 - ...;
therefore, if |r_i| = |[rho]_i|, we have
|P| < [Sigma]r1 + [Sigma]r1r2 + [Sigma]r1r2r3 + ... <
(1 + r1)(1 + r2)...(1 + r_k) - 1;
now, so long as [zeta] is without the closed curve above described
round [zeta] = 1, [zeta] = 1 + c, we have
| 1 | 1 |c_m - c_(m-1)| c/r
|----------| < ---, |-------------| < --- < [sigma],
|1 - [zeta]| a |c_m - [zeta] | a
and hence
|(1 - [zeta])^(-1) - U| < a^(-1) {(1 + [sigma]^n1) (1 + [sigma]^n2)^n1
(1 + [sigma]^n3)^n1n2 ... (1 + [sigma]^n_r)^(n1n2 ... n_(r-1)) - 1}.
Take an arbitrary real positive [epsilon], and [mu], a positive
number, so that [epsilon]^[mu] - 1 < [epsilon]a, then a value of n1
such that [sigma]^n1 < [mu]/(1 + [mu]) and therefore [sigma]^n1 / (1 -
[sigma]^n1 < [mu], and values for n2, n3 ... such that[sigma]^n2 <
1/n1 [sigma]^2n1, [sigma]^n3 < 1/n1n2 [sigma]^{3n1, ... [sigma]^n_r} <
1/(n1...n_(r - 1)} [sigma]^n_rn1; then, as 1 + x < e^x, we have
|(-[zeta])^(-1) - U| < a^-1 {exp([sigma]^n1 + n1[sigma]^n2 +
n1n2[sigma]^n3 + ... + n1n2...n_(r - 1)[sigma]^n_r) - 1}, and
therefore less than
a^(-1) {exp([sigma]^n1 + [sigma]^2n1 + ... + [sigma]^n_rn1) - 1},
which is less than
_ _
1 | / [sigma]^n1 \ |
-- |exp ( -------------- ) - 1 |
a |_ \1 - [sigma]^n1/ _|
and therefore less than [epsilon].
The rational function U, with a pole at [zeta] = 1 + c, differs
therefore from (1 - [zeta])^(-1), for all points outside the closed
region put about [zeta] = 1, [zeta] = l + c, by a quantity numerically
less than [epsilon]. So long as a remains the same, r and [sigma] will
remain the same, and a less value of [epsilon] will require at most an
increase of the numbers n1, n2, ... n_r; but if a be taken smaller it
may be necessary to increase r, and with this the complexity of the
function U.
Now put
c[zeta] (c + 1)z
z = --------------, [zeta] = --------;
c + 1 - [zeta] c + z
thereby the points [zeta] = 0, 1, 1 + c become the points z = 0, 1,
[oo], the function (1 - z)^(-1) being given by (1-z)^(-1) = c(c +
1)^(-1)(1 - [zeta])^(-1) + (c + 1)^(-1); the function U becomes a
rational function of z with a pole only at z = [oo], that is, it
becomes a polynomial in z, say [(c + 1)/c]H - 1/c, where H is also a
polynomial in z, and
_ _
1 c | 1 |
----- - H = ----- | ---------- - U |;
1 - z c + 1 |_1 + [zeta] _|
the lines [eta] = [+-]a become the two circles expressed, if z = x +
iy, by
c(c + 1)
(x + c)^2 + y^2 = [+-] -------- y,
a
the points ([eta] = 0, [xi] = 1 - a), ([eta] = 0, [xi] = 1 + c + a)
become respectively the points (y = 0, x = c(1 - a)/(c + a), (y = 0, x
= -c(l + c + a)/a), whose limiting positions for a = 0 are
respectively (y = 0, x = 1), (y = 0, x = -[oo]). The circle (x + c)^2 +
y^2 = c(c + 1)y/a can be written
(x + c)^2 (x + c)^4
y = --------- + --------- {[mu] + [root][[mu]^2 - (x + c)^2]}^(-2),
2[mu] 2[mu]
where [mu] = 1/2c(c + 1)/a; its ordinate y, for a given value of x,
can therefore be supposed arbitrarily small by taking a sufficiently
small.
We have thus proved the following result; taking in the plane of z any
finite region of which every interior and boundary point is at a
finite distance, however short, from the points of the real axis for
which 1 =< x =< [oo], we can take a quantity a, and hence, with an
arbitrary c, determine a number r; then corresponding to an arbitrary
[epsilon]_s, we can determine a polynomial P_s, such that, for all
points interior to the region, we have
|(1 - z^(-1)) - P_s| < [epsilon]_s;
thus the series of polynomials
P1 + (P2 - P1) + (P3 - P2) + ...,
constructed with an arbitrary aggregate of real positive numbers
[epsilon]1, [epsilon]2, [epsilon]3, ... with zero as their limit,
converges uniformly and represents (1-z)^(-1) for the whole region
considered.
S 12. _Expansion of a Monogenic Function in Polynomials, over a Star
Region._--Now consider any monogenic function [f](z) of which the
origin is not a singular point; joining the origin to any singular
point by a straight line, let the part of this straight line, produced
beyond the singular point, lying between the singular point and z =
[oo], be regarded as a barrier in the plane, the portion of this
straight line from the origin to the singular point being erased.
Consider next any finite region of the plane, whose boundary points
constitute a path of integration, in a sense previously explained, of
which every point is at a finite distance greater than zero from each
of the barriers before explained; we suppose this region to be such
that any line joining the origin to a boundary point, when produced,
does not meet the boundary again. For every point x in this region R
we can then write
_
/ dt [f](t)
2[pi]i[f](x) = | -- -----------,
_/ t 1 - xt^(-1)
where [f](x) represents a monogenic branch of the function, in case it
be not everywhere single valued, and t is on the boundary of the
region. Describe now another region R0 lying entirely within R, and
let x be restricted to be within R0 or upon its boundary; then for any
point t on the boundary of R, the points z of the plane for which
zt^(-1) is real and positive and equal to or greater than 1, being
points for which |z| = |t| or |z| > |t|, are without the region R0,
and not infinitely near to its boundary points. Taking then an
arbitrary real positive [epsilon] we can determine a polynomial in
xt^(-1), say P(xt^(-1)), such that for all points x in R0 we have
|[1 - xt^(-1)]^(-1) - P[xt^(-1)]| < [epsilon];
the form of this polynomial may be taken the same for all points t on
the boundary of R, and hence, if E be a proper variable quantity of
modulus not greater than [epsilon],
_ _
| / dt | | / dt |
|2[pi]i[f](x) - | --[f](t)P(xt^(-1))| = | | --[f](t)E| <= [epsilon]LM,
| _/ t | | _/ t |
where L is the length of the path of integration, the boundary of R,
and M is a real positive quantity such that upon this boundary
|t^(-1)[f](t)| < M. If now
P(xt^(-1)) = c0 + c1xt^(-1) + ... + c_mx^mt^(-m),
and
_
1 /
------ | t^(-r-1)[f](t)dt = [mu]_r,
2[pi]i _/
this gives
|[f](x) - {c0[mu]0 + c1[mu]1x + ... + c_m[mu]_mx^m}| =< [epsilon]LM/2[pi],
where the quantities [mu]0, [mu]1, [mu]2, ... are the coefficients in
the expansion of [f](x) about the origin.
If then an arbitrary finite region be constructed of the kind
explained, excluding the barriers joining the singular points of
[f](x) to x = [oo], it is possible, corresponding to an arbitrary real
positive number [sigma], to determine a number m, and a polynomial
Q(x), of order m, such that for all interior points of this region
|[f](x) - Q(x)| < [sigma].
Hence as before, within this region [f](x) can be represented by a
series of polynomials, converging uniformly; when [f](x) is not a
single valued function the series represents one branch of the
function.
The same result can be obtained without the use of Cauchy's integral.
We explain briefly the character of the proof. If a monogenic function
of t, [phi](t) be capable of expression as a power series in t-x about
a point x, for |t - x| =< [rho], and for all points of this circle
|[phi](t)| < g, we know that |[phi]^(n)(x)| < g[rho]^(-n)(n!). Hence,
taking |z| < 1/3[rho], and, for any assigned positive integer [mu],
taking m so that for n > m we have ([mu] + n)^[mu] < (3/2)^n, we have
|[phi]^(([mu] + n))(x)z^n| [phi]^([mu] + n)(x)
|------------------------| < -------------------([mu] + n)^([mu])|z|^n
| n! | ([mu] + n)!
g /3 \ /[rho]\ g
< ---------------- ( -- )^n ( ----- )^n < --------------,
[rho]^([mu] + n) \2 / \ 3 / [rho]^[mu] 2^n
and therefore
_m
\ [phi]^([mu] + n)(x)
[phi]^([mu])(x + z) = /_ ------------------- z^n + [epsilon]_[mu],
n=0 n!
where
g _[oo] 1 g
|[epsilon]_([mu])| < ---------- \ --- < --------------
[rho]^[mu] /_ 2^n [rho]^[mu] 2^m
n=m+1
Now draw barriers as before, directed from the origin, joining the
singular point of [phi](z) to z = [oo], take a finite region excluding
all these barriers, let [rho] be a quantity less than the radii of
convergence of all the power series developments of [phi](z) about
interior points of this region, so chosen moreover that no circle of
radius [rho] with centre at an interior point of the region includes
any singular point of [phi](z), let g be such that |[phi](z)| < g for
all circles of radius [rho] whose centres are interior points of the
region, and, x being any interior point of the region, choose the
positive integer n so that 1/n |x| 1/3 - [rho]; then take the points
a1 = x/n, a2 = 2x/n, a3 = 3x/n, ... a_n = x; it is supposed that the
region is so taken that, whatever x may be, all these are interior
points of the region. Then by what has been said, replacing x, z
respectively by 0 and x/n, we have
_m1 [phi]^([mu] + [lambda]1)(0) /x \^[lambda]1
[phi]^([mu]) (a{1}) = \ --------------------------- ( -- ) + [alpha]_[mu]
/_ [lambda]{1}! \n /
[lambda]1=0
with
|[alpha]{[mu]}| < g/[rho]^[mu] 2^m1,
provided ([mu] + m1 + 1)^[mu] < (3/2)^(m1+1); in fact for [mu] =<
2n^(2n-2) it is sufficient to take m1 = n^2n; by another application
of the same inequality, replacing x, z respectively by a1 and x/n, we
have
_ m2 [phi]^([mu]+[lambda]2)(a1) /x \^[lambda]2
[phi]^([mu])(a2) = \ -------------------------- ( -- ) + [beta]'_[mu],
/_ [lambda]{2}! \n /
[lambda]2=0
where
|[beta]'[mu]| < g/[rho]^[mu] 2^m2
provided ([mu] + m2 + 1)^[mu] < {3/2}^(m2 + 1); we take m2 = n^(2n -
2), supposing [mu] < 2^(2n - 4). So long as [lambda]2 =< = m} =< n^(2n
- 2) and [mu] < 2n^(2n - 4) we have [mu] + [lambda]{2} < 2n^(2n - 2),
and we can use the previous inequality to substitute here for
[phi]^([mu] + [lambda]2) (a1). When this is done we find
_ m2 _ m1 [phi]^([mu] + [lambda]1 + [lambda]2)(0)
[phi]^([mu])(a2) = \ \ ---------------------------------------
/_ /_ [lambda]1! [lambda]2!
[lambda]2=0 [lambda]1=0
/x \ ^[lambda]1 + [lambda]2
( -- ) + [beta]_[mu],
\n /
where |[beta]_[mu]| < 2g/[rho]^[mu] 2^(m2), the numbers m1, m2 being
respectively n^2n and n^(2n - 2).
Applying then the original inequality to [phi]^([mu]) (a3) =
[phi]^([mu]) (a2 + x/n), and then using the series just obtained, we
find a series for [phi]^([mu]) (a3). This process being continued, we
finally obtain
_ m1 _ m2 _ m_n
[phi](x) = \ \ ... \ [phi]^(h)(0) /x \^h
/_ /_ /_ ----------- ( -- ) + [epsilon],
[lambda]1=0 [lambda]2=0 [lambda]_n=0 K \n /
where h = [lambda]1 + [lambda]2 + ... + [lambda]_n , K = [lambda]1!
[lambda]2! ... [lambda]_n!, m1 = n^(2n), m1 = n^(2n - 2), ... , m1 =
n^2, |[epsilon]| < 2g/2^(m_n).
By this formula [phi](x) is represented, with any required degree of
accuracy, by a polynomial, within the region in question; and thence
can be expressed as before by a series of polynomials converging
uniformly (and absolutely) within this region.
S 13. _Application of Cauchy's Theorem to the Determination of Definite
Integrals._--Some reference must be made to a method whereby real
definite integrals may frequently be evaluated by use of the theorem of
the vanishing of the integral of a function of a complex variable round
a contour within which the function is single valued and non singular.
We are to evaluate an integral [int][a to b] [f](x)dx; we form a
closed contour of which the portion of the real axis from x = a to x =
b forms a part, and consider the integral [int][f](z)dz round this
contour, supposing that the value of this integral can be determined
along the curve forming the completion of the contour. The contour
being supposed such that, within it, [f](z) is a single valued and
finite function of the complex variable z save at a finite number of
isolated interior points, the contour integral is equal to the sum of
the values of [int][f](z)dz taken round these points. Two instances
will suffice to explain the method. (1) The integral [int][0 to [oo]]
(tan x)/x dx is convergent if it be understood to mean the limit when
[epsilon], [zeta], [sigma], ... all vanish of the sum of the integrals
_1/2[pi]-[epsilon] _(3/2)[pi]-[zeta] _(5/2)[pi]-[sigma]
/ tan x / tan x / tan x
| ----- dx, | ----- dx, | ----- dx, ...
_/ 0 x _/1/2[pi]+[epsilon] x _/(3/2)[pi]+[zeta] x
Now draw a contour consisting in part of the whole of the positive and
negative real axis from x = -n[pi] to x = + n[pi], where n is a
positive integer, broken by semicircles of small radius whose centres
are the points x = [+-]1/2[pi], x = [+-]3/4[pi], ... , the contour
containing also the lines x = n[pi] and x = -n[pi] for values of y
between 0 and n[pi] tan [alpha], where [alpha] is a small fixed angle,
the contour being completed by the portion of a semicircle of radius
n[pi] sec [alpha] which lies in the upper half of the plane and is
terminated at the points x = [+-]n[pi], y = n[pi] tan [alpha]. Round
this contour the integral [int](tan z /z) dz has the value zero. The
contributions to this contour integral arising from the semicircles of
centres -1/2(2s - 1)[pi], + 1/2(2s - 1)[pi], supposed of the same
radius, are at once seen to have a sum which ultimately vanishes when
the radius of the semicircles diminishes to zero. The part of the
contour lying on the real axis gives what is meant by 2 [int][0 to
n[pi]](tan x / x) dx. The contribution to the contour integral from
the two straight portions at x = [+-]n[pi] is
_n[pi] tan [alpha]
/ / tan iy tan iy \
| idy ( ---------- - ----------- )
_/ 0 \n[pi] + iy -n[pi] + iy /
where i tan iy, = -[exp(y) - exp(-y)]/[exp(y) + exp(-y)], is a real
quantity which is numerically less than unity, so that the
contribution in question is numerically less than
_n[pi] tan [alpha]
/ 2n[pi]
| dy ---------------, that is than 2[alpha].
_/ 0 n^2[pi]^2 + y^2
Finally, for the remaining part of the contour, for which, with R =
n[pi] sec [alpha], we have z = R(cos [theta] + i sin [theta]) =
RE(i[theta]), we have
dz
-- = id[theta], i tan z =
z
exp(-R sin [theta]) E(iR cos [theta]) - exp(R sin [theta]) E(-iR cos [theta])
-----------------------------------------------------------------------------;
exp(-R sin [theta]) E(iR cos [theta]) + exp(R sin [theta]) E(-iR cos [theta])
when n and therefore R is very large, the limit of this contribution
to the contour integral is thus
_
/ [pi]-[alpha]
- | d[theta] = -([pi] - 2[alpha]).
_/ [alpha]
Making n very large the result obtained for the whole contour is
_
/ [oo] tan x
2 | ----- dx - ([pi] - 2[alpha]) - 2[alpha][epsilon] = 0;
_/ 0 x
where [epsilon] is numerically less than unity. Now supposing [alpha]
to diminish to zero we finally obtain
_
/ [oo] tan x [pi]
| ----- dx = ----
_/ 0 x 2
(2) For another case, to illustrate a different point, we may take the
integral
_
/ z^(a-1)
| ------- dz,
_/ 1 + z
wherein a is real quantity such that 0 < a < 1, and the contour
consists of a small circle, z = rE(i[theta]), terminated at the points
x = r cos [alpha], y = [+-] r sin [alpha], where [alpha] is small, of
the two lines y = [+-] r sin [alpha] for r cos [alpha] =< x =< R cos
[beta], where R sin [beta] = r sin [alpha], and finally of a large
circle z = RE(i[phi]), terminated at the points x = R cos [beta], y =
[+-] R sin [beta]. We suppose [alpha] and [beta] both zero, and that
the phase of z is zero for r cos a =< x =< R cos [beta], y = r sin
[alpha] = R sin [beta]. Then on r cos [alpha] =< x =< R cos [beta], y
= -r sin [alpha], the phase of z will be 2[pi], and z^([alpha] - 1)
will be equal to x^([alpha] - 1) exp (2[pi]i(a - 1)), where x is real
and positive. The two straight portions of the contour will thus
together give a contribution
_
/ R cos [beta] x^(a - 1)
[1 - exp (2[pi]i[alpha])] | --------- dx.
_/ r cos [alpha] 1 + x
It can easily be shown that if the limit of z[f](z) for z = 0 is zero,
the integral [int] [f](z)dz taken round an arc, of given angle, of a
small circle enclosing the origin is ultimately zero when the radius
of the circle diminishes to zero, and if the limit of z[f](z) for z =
[oo] is zero, the same integral taken round an arc, of given angle, of
a large circle whose centre is the origin is ultimately zero when the
radius of the circle increases indefinitely; in our case with [f](z) =
z^([alpha] - 1)/(1 + z), we have z[f](z) = z^a/(1 + z), which, for 0 <
a < 1, diminishes to zero both for z = 0 and for z = [oo]. Thus,
finally the limit of the contour integral when r = 0, R = [oo] is
_
/ [oo] x^([alpha] - 1)
[1 - exp (2[pi]i[alpha])] | --------------- dx.
_/ 0 1 + x
Within the contour [f](z) is single valued, and has a pole at z = 1;
at this point the phase of z is [pi] and z^(a - 1) is exp [i[pi](a -
1)] or - exp (i[pi]a); this is then the residue of [f](z) at z = -1;
we thus have
_
/ [oo] x^(a - 1)
[1 - exp (2[pi]ia)] | --------- dx = -2[pi]i exp (i[pi]a),
_/ 0 1 + x
that is
_
/ [oo] x^(a - 1)
| --------- dx = [pi] cosec (a[pi]).
_/ 0 1 + x
S 14. _Doubly Periodic Functions._--An excellent illustration of the
preceding principles is furnished by the theory of single valued
functions having in the finite part of the plane no singularities but
poles, which have two periods.
Before passing to this it may be convenient to make here a few remarks
as to the periodicity of (single valued) monogenic functions. To say
that [f](z) is periodic is to say that there exists a constant [omega]
such that for every point z of the interior of the region of existence
of [f](z) we have [f](z + [omega]) = [f](z). This involves,
considering all existing periods [omega] = [rho] + i[sigma], that
there exists a lower limit of [rho]^2 + [sigma]^2 other than zero; for
otherwise all the differential coefficients of [f](z) would be zero,
and [f](z) a constant; we can then suppose that not both [rho] and
[sigma] are numerically less than [epsilon], where [epsilon] >
[sigma]. Hence, if g be any real quantity, since the range (-g, ... g)
contains only a finite number of intervals of length [epsilon], and
there cannot be two periods [omega] = [rho] + i[sigma] such that
[mu][epsilon] =< [rho] < ([mu] + 1)[epsilon], [nu][epsilon] =< [sigma]
< ([nu] + 1)[epsilon], where [mu], [nu] are integers, it follows that
there is only a finite number of periods for which both [rho] and
[sigma] are in the interval (-g ... g). Considering then all the
periods of the function which are real multiples of one period
[omega], and in particular those periods [lambda][omega] wherein 0 <
[lambda] =< 1, there is a lower limit for [lambda], greater than zero,
and therefore, since there is only a finite number of such periods for
which the real and imaginary parts both lie between -g and g, a least
value of [lambda], say [lambda]0. If [Omega] = [lambda]0[omega] and
[lambda] = M[lambda]0 + [lambda]', where M is an integer and 0 [< = ]
[lambda]' < [lambda]0, any period [lambda][omega] is of the form
M[Omega] + [lambda]'[omega]; since, however, [Omega], M[Omega] and
[lambda][omega] are periods, so also is [lambda]'[omega], and hence,
by the construction of [lambda]0, we have [lambda]' = 0; thus all
periods which are real multiples of [omega] are expressible in the
form M[Omega] where M is an integer, and [Omega] a period.
If beside [omega] the functions have a period [omega]' which is not a
real multiple of [omega], consider all existing periods of the form
[mu][omega] + [nu][omega]' wherein [mu], [nu] are real, and of these
those for which 0 [< = ] [mu] =< 1, 0 < [nu] =< 1; as before there is
a least value for [nu], actually occurring in one or more periods, say
in the period [Omega]' = [mu]0[omega] + [nu]0[omega]'; now take, if
[mu][omega] + [nu][omega]' be a period, [nu] = N'[nu]0 + [nu]', where
N' is an integer, and 0 =< [nu]' < [nu]0; thence [mu][omega] +
[nu][omega]' = [mu][omega] + N'([Omega]' - [mu]0[omega]) +
[nu]'[omega]'; take then [mu] - N[mu]0 = N[lambda]0 + [lambda]', where
N is an integer and [lambda]0 is as above, and 0 =< [lambda]' <
[lambda]0; we thus have a period N[Omega] + N'[Omega]' +
[lambda]'[omega] + [nu]'[omega]', and hence a period [lambda]'[omega]
+ [nu]'[omega]', wherein [lambda]' < [lambda]0, [nu]' < [nu]0; hence
[nu]' = 0 and [lambda]' = 0. All periods of the form [mu][omega] +
[nu][omega]' are thus expressible in the form N[Omega] + N'[Omega]',
where [Omega], [Omega]' are periods and N, N' are integers. But in
fact any complex quantity, P + iQ, and in particular any other
possible period of the function, is expressible, with [mu], [nu] real,
in the form [mu][omega] + [nu][omega]'; for if [omega] = [rho] +
i[sigma], [omega]' = [rho]' + i[sigma]', this requires only P =
[mu][rho] + [nu][rho]', Q = [mu][sigma] + [nu][sigma]', equations
which, since [omega]'/[omega] is not real, always give finite values
for [mu] and [nu].
It thus appears that if a single valued monogenic function of z be
periodic, either all its periods are real multiples of one of them,
and then all are of the form M[Omega], where [Omega] is a period and M
is an integer, or else, if the function have two periods whose ratio
is not real, then all its periods are expressible in the form N[Omega]
+ N'[Omega]', where [Omega], [Omega]' are periods, and N, N' are
integers. In the former case, putting [zeta] = 2[pi]iz/[Omega], and
the function [f](z) = [phi]([zeta]), the function [phi]([zeta]) has,
like exp ([zeta]), the period 2[pi]i, and if we take t = exp([zeta])
or [zeta] = [lambda](t) the function is a single valued function of t.
If then in particular [f](z) is an integral function, regarded as a
function of t, it has singularities only for t = 0 and t = [oo], and
may be expanded in the form [Sigma](-[oo] to [oo]) a_nt^n.
Taking the case when the single valued monogenic function has two
periods [omega], [omega]' whose ratio is not real, we can form a
network of parallelograms covering the plane of z whose angular points
are the points c + m[omega] + m'[omega]', wherein c is some constant
and m, m' are all possible positive and negative integers; choosing
arbitrarily one of these parallelograms, and calling it the primary
parallelogram, all the values of which the function is at all capable
occur for points of this primary parallelogram, any point, z', of the
plane being, as it is called, _congruent_ to a definite point, z, of
the primary parallelogram, z' - z being of the form m[omega] +
m'[omega]', where m, m' are integers. Such a function cannot be an
integral function, since then, if, in the primary parallelogram
|[f](z)| < M, it would also be the case, on a circle of centre the
origin and radius R, that |[f](z)| < M, and therefore, if
[Sigma]a_nz^n be the expansion of the function, which is valid for an
integral function for all finite values of z, we should have |a_n| <
MR^(-n), which can be made arbitrarily small by taking R large enough.
The function must then have singularities for finite values of z.
We consider only functions for which these are poles. Of these there
cannot be an infinite number in the primary parallelogram, since then
those of these poles which are sufficiently near to one of the
necessarily existing limiting points of the poles would be arbitrarily
near to one another, contrary to the character of a pole. Supposing
the constant c used in naming the corners of the parallelograms so
chosen that no pole falls on the perimeter of a parallelogram, it is
clear that the integral 1/(2[pi]i) [int][f](z)dz round the perimeter
of the primary parallelogram vanishes; for the elements of the
integral corresponding to two such opposite perimeter points as z, z +
[omega] (or as z, z + [omega]') are mutually destructive. This
integral is, however, equal to the sum of the residues of [f](z) at
the poles interior to the parallelogram. Which sum is therefore zero.
There cannot therefore be such a function having only one pole of the
first order in any parallelogram; we shall see that there can be such
a function with two poles only in any parallelogram, each of the first
order, with residues whose sum is zero, and that there can be such a
function with one pole of the second order, having an expansion near
this pole of the form (z - a)^(-2) + (power series in z - a).
Considering next the function [phi](z) = [[f](z)]^(-1) d[f](z)/dz, it
is easily seen that an ordinary point of [f](z) is an ordinary point
of [phi](z), that a zero of order m for [f](z) in the neighbourhood of
which [f](z) has a form, (z - a)^m multiplied by a power series, is a
pole of [phi](z) of residue m, and that a pole of [f](z) of order n is
a pole of [phi](z) of residue -n; manifestly [phi](z) has the two
periods of [f](z). We thus infer, since the sum of the residues of
[phi](z) is zero, that for the function [f](z), the sum of the orders
of its vanishing at points belonging to one parallelogram, [Sigma]m,
is equal to the sum of the orders of its poles, [Sigma]n; which is
briefly expressed by saying that the number of its zeros is equal to
the number of its poles. Applying this theorem to the function
[f](z) - A, where A is an arbitrary constant, we have the result, that
the function [f](z) assumes the value A in one of the parallelograms
as many times as it becomes infinite. Thus, by what is proved above,
every conceivable complex value does arise as a value for the doubly
periodic function [f](z) in any one of its parallelograms, and in fact
at least twice. The number of times it arises is called the _order_ of
the function; the result suggests a property of rational functions.
Consider further the integral [int] z [f]'(z)/[f](z) dz, where [f]'(z)
= d[f](z)/dz taken round the perimeter of the primary parallelogram;
the contribution to this arising from two opposite perimeter points
such as z and z + [omega] is of the form -[omega] [int] z
[f]'(z)/[f](z) dz, which, as z increases from z0 to z0 + [omega]',
gives, if [lambda] denote the generalized logarithm, -
[omega]{[lambda][[f](z0 + [omega]')] - [lambda][[f](z0)]}, that is,
since [f](z0 + [omega]') = [f](z0), gives 2[pi]iN[omega], where N is
an integer; similarly the result of the integration along the other
two opposite sides is of the form 2[pi]iN'[omega]', where N' is an
integer. The integral, however, is equal to 2[pi]i times the sum of
the residues of z[f]'(z)/[f](z) at the poles interior to the
parallelogram. For a zero, of order m, of [f](z) at z = a, the
contribution to this sum is 2[pi]ima, for a pole of order n at z = b
the contribution is -2[pi]inb; we thus infer that [Sigma]ma -
[Sigma]nb = N[omega] + N'[omega]'; this we express in words by saying
that the sum of the values of z where [f](z) = 0 within any
parallelogram is equal to the sum of the values of z where [f](z) =
[oo] save for integral multiples of the periods. By considering
similarly the function [f](z) - A where A is an arbitrary constant, we
prove that each of these sums is equal to the sum of the values of z
where the function takes the value A in the parallelogram.
We pass now to the construction of a function having two arbitrary
periods [omega], [omega]' of unreal ratio, which has a single pole of
the second order in any one of its parallelograms.
For this consider first the network of parallelograms whose corners
are the points [Omega] = m[omega] + m'[omega]', where m, m' take all
positive and negative integer values; putting a small circle about
each corner of this network, let P be a point outside all these
circles; this will be interior to a parallelogram whose corners in
order may be denoted by z0, z0 + [omega], z0 + [omega] + [omega]', z0
+ [omega]'; we shall denote z0, z0 + [omega] by A0, B0; this
parallelogram [Pi]0 is surrounded by eight other parallelograms,
forming with [Pi]0 a larger parallelogram [Pi]1, of which one side,
for instance, contains the points z0 - [omega] - [omega]', z0 -
[omega]', z0 - [omega]' + [omega], z0 - [omega]' + 2[omega], which we
shall denote by A1, B1, C1, D1. This parallelogram [Pi]1 is surrounded
by sixteen of the original parallelograms, forming with [Pi]1 a still
larger parallelogram [Pi]2 of which one side, for instance, contains
the points z0 - 2[omega] - 2[omega]', z0 - [omega] - 2[omega]', z0 -
2[omega]', z0 + [omega] - 2[omega]', z0 + 2[omega] - 2[omega]', z0 +
3[omega] - 2[omega]', which we shall denote by A2, B2, C2, D2, E2, F2.
And so on. Now consider the sum of the inverse cubes of the distances
of the point P from the corners of all the original parallelograms.
The sum will contain the terms
1 / 1 1 1 \ / 1 1 1 \
S0 = ----- + ( ----- + ----- + ----- ) + ( ----- + ----- + ... + ----- ) + ...
PA0^3 \PA1^3 PB1^3 PC1^3/ \PA2^3 PB2^3 PE2^3/
and three other sets of terms, each infinite in number, formed in a
similar way. If the perpendiculars from P to the sides A0B0, A1B1C1,
A2B2C2D2E2, and so on, be p, p + q, p + 2q and so on, the sum S0 is at
most equal to
1 3 5 2n + 1
--- + --------- + ---------- + ... + ---------- + ...
p^3 (p + q)^3 (p + 2q)^3 (p + nq)^3
of which the general term is ultimately, when n is large, in a ratio
of equality with 2q^(-3)n^(-2), so that the series S0 is convergent,
as we know the sum [Sigma]n^(-2) to be; this assumes that p[/ = ]0; if
P be on A0B0 the proof for the convergence of S0 - 1/PA0^3, is the
same. Taking the three other sums analogous to S0 we thus reach the
result that the series
[phi](z) = -2[Sigma](z - [Omega])^(-3),
where [Omega] is m[omega] + m'[omega]', and m, m' are to take all
positive and negative integer values, and z is any point outside small
circles described with the points [Omega] as centres, is _absolutely
convergent_. Its sum is therefore independent of the order of its
terms. By the nature of the proof, which holds for all positions of z
outside the small circles spoken of, the series is also clearly
_uniformly convergent_ outside these circles. Each term of the series
being a monogenic function of z, the series may therefore be
differentiated and integrated outside these circles, and represents a
monogenic function. It is clearly periodic with the periods [omega],
[omega]'; for [phi](z + [omega]) is the same sum as [phi](z) with the
terms in a slightly different order. Thus [phi](z + [omega]) =
[phi](z) and [phi](z + [omega]') = [phi](z).
Consider now the function
_ _ _
1 / z | 2 |
[f](z) = --- + | | [phi](z) + -- | dz,
z^2 _/ 0 |_ z^3 _|
where, for the subject of integration, the area of uniform convergence
clearly includes the point z = 0; this gives
d[f](z)
------- = [phi](z)
dz
and
_ _
1 | 1 1 |
[f](z) = --- + [Sigma]' | -------------- - -------- |,
z^2 |_ (z - [Omega])^2 [Omega]^2 _|
wherein [Sigma]' is a sum excluding the term for which m = 0 and m' =
0. Hence [f](z + [omega]) - [f](z) and [f](z + [omega]') - [f](z) are
both independent of z. Noticing, however, that, by its form, [f](z) is
an even function of z, and putting z = -1/2[omega], z = -1/2[omega]'
respectively, we infer that also [f](z) has the two periods [omega]
and [omega]'. In the primary parallelogram [Pi]0, however, [f](z) is
only infinite at z = 0 in the neighbourhood of which its expansion is
of the form z^(-2) + (power series in z). Thus [f](z) is such a doubly
periodic function as was to be constructed, having in any
parallelogram of periods only one pole, of the second order.
It can be shown that any single valued meromorphic function of z with
[omega] and [omega]' as periods can be expressed rationally in terms of
[f](z) and [phi](z), and that [[phi](z)]^2 is of the form 4[[f](z)]^3 +
A[f](z) + B, where A, B are constants.
To prove the last of these results, we write, for |z| < |[Omega]|,
1 1 2z 3z^2
--------------- - --------- = --------- + --------- + ...,
(z - [Omega])^2 [Omega]^2 [Omega]^3 [Omega]^4
and hence, if [Sigma]'[Omega]^(-2n) = [sigma]_n, since
[Sigma]'[Omega]^(-(2n - 1)) = 0, we have, for sufficiently small z
greater than zero,
[f](z) = z^(-2) + 3[sigma]2.z^2 + 5[sigma]3.z^4 + ...
and
[phi](z) = -2z^(-3) + 6[sigma]2.z + 20[sigma]3.z^3 + ...;
using these series we find that the function
F(z) = [[phi](z)]^2 - 4[[f](z)]^3 + 60[sigma]2[f](z) + 140[sigma]3
contains no negative powers of z, being equal to a power series in z^2
beginning with a term in z^2. The function F(z) is, however, doubly
periodic, with periods [omega], [omega]', and can only be infinite
when either [f](z) or [phi](z) is infinite; this follows from its form
in [f](z) and [phi](z); thus in one parallelogram of periods it can be
infinite only when z = 0; we have proved, however, that it is not
infinite, but, on the contrary, vanishes, when z = 0. Being,
therefore, never infinite for finite values of z it is a constant, and
therefore necessarily always zero. Putting therefore [f](z) = [zeta]
and [phi](z) = d[zeta]/dz we see that
dz
------- = (4[zeta]^3 - 60[sigma]2[zeta] - 140[sigma]3)^(-1/2)
d[zeta]
Historically it was in the discussion of integrals such as
_
/
| d[zeta](4[zeta]^3 - 60[sigma]2.[zeta] - 140[sigma]3)^(-1/2),
_/
regarded as a branch of Integral Calculus, that the doubly periodic
functions arose. As in the familiar case
_
/ [zeta]
z = | (1 - [zeta]^2)^(-1/2) d[zeta],
_/ 0
where [zeta] = sin z, it has proved finally to be simpler to regard
[zeta] as a function of z. We shall come to the other point of view
below, under S 20, _Elliptic Integrals_.
To prove that any doubly periodic function F(z) with periods [omega],
[omega]', having poles at the points z = a1, ... z = a_m of a
parallelogram, these being, for simplicity of explanation, supposed to
be all of the first order, is rationally expressible in terms of
[phi](z) and [f](z), and we proceed as follows:--
Consider the expression
([zeta], 1)_m + [eta]([zeta], 1)_(m - 2)
[Phi](z) = ------------------------------------------
([zeta]- A1)([zeta] - A2)...([zeta] - A_m)
where A_s = [f](a_s), [zeta] is an abbreviation for [f](z) and [eta]
for [phi](z), and ([zeta], 1)_m, ([zeta], 1)_(m - 2), denote integral
polynomials in [zeta], of respective orders m and m - 2, so that there
are 2m unspecified, homogeneously entering, constants in the
numerator. It is supposed that no one of the points a1, ... a_m is one
of the points m[omega] + m'[omega]' where f(z) = [oo]. The function
[Phi](z) is a monogenic function of z with the periods [omega],
[omega]', becoming infinite (and having singularities) only when (1)
[zeta] = [oo] or (2) one of the factors [zeta] - A_s is zero. In a
period parallelogram including z = 0 the first arises only for z = 0;
since for [zeta] = [oo], [eta] is in a finite ratio to [zeta]^(3/2);
the function [Phi](z) for [zeta] = [oo] is not infinite provided the
coefficient of [zeta]^m in ([zeta], 1)_m is not zero; thus [Phi](z) is
regular about z = 0. When [zeta] - A_s = 0, that is [f](z) = f(a_s),
we have z = [+-]a_s + m[omega] + m'[omega]', and no other values of z, m
and m' being integers; suppose the unspecified coefficients in the
numerator so taken that the numerator vanished to the first order in
each of the m points -a1, -a2, ... -a_m; that is, if [phi](a_s) = B_s,
and therefore [phi](-a_s) = -B_s, so that we have the m relations
(A_s, 1)_m - B_s(A_s, 1)_(m - 2) = 0;
then the function [Phi](z) will only have the m poles a1, ... a_m.
Denoting further the m zeros of F(z) by a1', ... a_m', putting
[f](a_s') = A_s', [phi](a_s') = B_s', suppose the coefficients of the
numerator of [Phi](z) to satisfy the further m-1 conditions
(A_s', 1)_m + B_s'(A_s',1)_(m - 2) = 0
for s = 1, 2, ... (m - 1). The ratios of the 2m coefficients in the
numerator of [Phi](z) can always be chosen so that the m + (m - 1)
linear conditions are all satisfied. Consider then the ratio
F(z)/[Phi](z);
it is a doubly periodic function with no singularity other than the
one pole a_m'. It is therefore a constant, the numerator of [Phi](z)
vanishing spontaneously in a_m'. We have
F(z) = A[Phi](z),
where A is a constant; by which F(z) is expressed rationally in terms
of [f](z) and [phi](z), as was desired.
When z = 0 is a pole of F(z), say of order r, the other poles, each of
the first order, being a1, ... a_m, similar reasoning can be applied
to a function
([zeta], 1)_h + [eta]([zeta], 1)_k
----------------------------------,
([zeta] - A1)...([zeta] - A_m)
where h, k are such that the greater of 2h - 2m, 2k + 3 - 2m is equal
to r; the case where some of the poles a1, ... a_m are multiple is to
be met by introducing corresponding multiple factors in the
denominator and taking a corresponding numerator. We give a solution
of the general problem below, of a different form.
One important application of the result is the theorem that the
functions [f](z + t), [phi](z + t), which are such doubly periodic
function of z as have been discussed, can each be expressed, so far as
they depend on z, rationally in terms of [f](z) and [phi](z), and
therefore, so far as they depend on z and t, rationally in terms of
[f](z), [f](t), [phi](z) and [phi](t). It can in fact be shown, by
reasoning analogous to that given above, that
_ _
| [phi](z) - [phi](t) |^2
[f](z + t) + [f](z) + [f](t) = 1/4 | ------------------- |.
|_ [f](z) - [f](t) _|
This shows that if F(z) be any single valued monogenic function which
is doubly periodic and of meromorphic character, then F(z + t) is an
algebraic function of F(z) and F(t). Conversely any single valued
monogenic function of meromorphic character, F(z), which is such that
F(z + t) is an algebraic function of F(z) and F(t), can be shown to be
a doubly periodic function, or a function obtained from such by
degeneration (in virtue of special relations connecting the
fundamental constants).
The functions [f](z), [phi](z) above are usually denoted by RN(z),
RN'(z); further the fundamental differential equation is usually
written
(RN'z)^2 = 4(RNz)^3 - g2RNz - g3,
and the roots of the cubic on the right are denoted by e1, e2, e3; for
the odd function, RN'z, we have, for the congruent arguments
-1/2[omega]and 1/2[omega], RN'(1/2[omega]) = -RN'(-1/2[omega]) =
-RN'(1/2[omega]), and hence RN'(1/2[omega]) = 0; hence we can take e1
= RN(1/2[omega]), e2 = RN(1/2[omega] + 1/2[omega]'), e3 =
RN(1/2[omega]). It can then be proved that [RN(z) - e1][RN(z +
1/2[omega]) - e1] = (e1 - e2)(e1 - e3), with similar equations for the
other half periods. Consider more particularly the function RN(z) -
e1; like RN(z) it has a pole of the second order at z = 0, its
expansion in its neighbourhood being of the form z^(-2)(1 - e1z^2 +
Az^4 + ...); having no other pole, it has therefore either two zeros,
or a double zero in a period parallelogram ([omega], [omega]'). In
fact near its zero 1/2[omega] its expansion is (x - 1/2[omega])
RN'(1/2[omega]) + 1/2(z - 1/2[omega])^2 RN"(1/2[omega]) + ...; we have
seen that RN'(1/2[omega]) = 0; thus it has a zero of the second order
wherever it vanishes. Thus it appears that the square root [RN(z) -
e1]^1/2, if we attach a definite sign to it for some particular value
of z, is a single valued function of z; for it can at most have two
values, and the only small circuits in the plane which could lead to
an interchange of these values are those about either a pole or a
zero, neither of which, as we have seen, has this effect; the function
is therefore single valued for any circuit. Denoting the function, for
a moment, by [f]1(z), we have [f]1(z + [omega]) = [+-][f]1(z), [f]1(z
+ [omega]') = [+-][f]1(z); it can be seen by considerations of
continuity that the right sign in either of these equations does not
vary with z; not both these signs can be positive, since the function
has only one pole, of the first order, in a parallelogram ([omega],
[omega]'); from the expansion of [f]1(z) about z = 0, namely z^(-1) (1
- 1/2e1z^2 + ...), it follows that [f]1(z) is an odd function, and
hence [f]1(-1/2[omega]') = -[f]1(1/2[omega]'), which is not zero since
[[f]1(1/2[omega]')]^2 = e3 - e1, so that we have [f]1(z + [omega]') =
-[f]1(z); an equation f1(z + [omega]) = -[f]1(z) would then give
[f]1(z + [omega] + [omega]') = [f]1(z), and hence [f]1(1/2[omega] +
1/2[omega]') = [f]1(-1/2[omega] - 1/2[omega]'), of which the latter is
-[f]1(1/2[omega] + 1/2[omega]'); this would give [f]1(1/2[omega] +
1/2[omega]') = 0, while [[f]1 (1/2[omega] + 1/2[omega]')]^2 = e2 - e1.
We thus infer that [f]1(z + [omega]) = [f]1(z), [f]1(z + [omega]') =
-[f]1(z), [f]1(z + [omega] + [omega]') = -[f]1(z). The function
[f]1(z) is thus doubly periodic with the periods [omega] and
2[omega]'; in a parallelogram of which two sides are [omega] and
2[omega]' it has poles at z = 0, z = [omega]' each of the first order,
and zeros of the first order at z = 1/2[omega], z = 1/2[omega] +
[omega]'; it is thus a doubly periodic function of the second order
with two different poles of the first order in its parallelogram
([omega], 2[omega]'). We may similarly consider the functions [f]2(z)
= [RN(z) - e2]^1/2, [f]3(z) = [RN(z) - e3]^1/2; they give
[f]2(z + [omega] + [omega]') = [f]2(z), [f]2(z + [omega]) = -[f]2(z), [f]2(z + [omega]') = -[f]2(z),
[f]3(z + [omega]') = [f]3z, [f]3(z + [omega]) = -[f]3(z), [f]3(z + [omega] + [omega]') = -[f]3(z).
Taking u = z(e1 - e3)^1/2, with a definite determination of the
constant (e1 - e3)^1/2, it is usual, taking the preliminary signs so
that for z = 0 each of z[f]1(z), z[f]2(z), z[f]3(z) is equal to + 1,
to put
(e1 - e3)^1/2 [f]1(z) f2(z)
sn(u) = -------------, cn(u) = -------, dn(u) = -----,
[f]3(z) [f]3(z) f3(z)
k^2 = (e2 - e3)/(e1 - e3), K = 1/2[omega](e1 - e3)^1/2, iK' = 1/2[omega]'(e1 - e3)^1/2;
thus sn(u) is an odd doubly periodic function of the second order with
the periods 4K, 2iK, having poles of the first order at u = iK', u =
2K + iK', and zeros of the first order at u = 0, u = 2K; similarly
cn(u), dn(u) are even doubly periodic functions whose periods can be
written down, and sn^2(u) + cn^2(u) = 1, k^2sn^2(u) + dn^2(u) = 1; if
x = sn(u) we at once find, from the relations given here, that
du
-- = [(1 - x^2) (1 - k^2x^2)]^(-1/2);
dx
if we put x = sin[phi] we have
du
------ = [1 - k^2sin^2 [phi]]^(-1/2),
d[phi]
and if we call [phi] the amplitude of u, we may write [phi] = am(u), x
= sin.am(u), which explains the origin of the notation sn(u).
Similarly cn(u) is an abbreviation of cos.am(u), and dn(u) of
[Delta]am(u), where [Delta]([phi]) meant (1 - k^2sin^2 [phi])^1/2. The
addition equation for each of the functions [f]1(z), [f]2(z), [f]3(z)
is very simple, being
/(Pd) (Pd) \ [f](z) + [f](t) [f](z)[f]'(t) - [f](t)[f]'(z)
[f](z + t) = 1/2( ----- + ----- ) log --------------- = -----------------------------,
\(Pd)z (Pd)i/ [f](z) - [f](t) [f]^2(z) - [f]^2(t)
where f1'(z) means d[f]1(z)/dz, which is equal to -[f]2(z).[f]3(z),
and [f]^2(z) means [[f](z)]^2. This may be verified directly by
showing, if R denote the right side of the equation, that (Pd)R/(Pd)z
= (Pd)R/(Pd)t; this will require the use of the differential equation
[[f]1'^(z)]^2 = [[f]1^2(z) + e1 - e2] [[f]1^2(z) + e1 - e3],
and in fact we find
/ (Pd)^2 (Pd)^2\
( ------- - ------ ) log [[f](z) + [f](t)] = [f]^2(z) - [f]^2(t) =
\(Pd)z^2 dt^2 /
/ (Pd)^2 (Pd)^2\
( ------- - ------ ) log [[f](z) - [f](t)];
\(Pd)z^2 dt^2 /
hence it will follow that R is a function of z + t, and R is at once
seen to reduce to [f](z) when t = 0. From this the addition equation
for each of the functions sn(u), cn(u), dn(u) can be deduced at once;
if s1, c1, d1, s2, c2, d2 denote respectively sn(u1), cn(u1), dn(u1),
sn(u2), cn(u2), dn(u2), they can be put into the forms
sn(u1 + u2) = (s1c2d2 + s2c1d1)/D,
cn(u1 + u2) = (c1c2 - s1s2d1d2)/D,
dn(u1 + u2) = (d1d2 - k^2s1s2c1c2)/D,
where
D = 1 - k^2s1^2s2^2.
The introduction of the function [f]1(z) is equivalent to the
introduction of the function RN(z; [omega], 2[omega]') constructed
from the periods [omega], 2[omega]' as was RN(z) from [omega] and
[omega]'; denoting this function by RN1(z) and its differential
coefficient by RN'1(z), we have in fact
RN'1(z)
[f]1(z) = 1/2 ----------------------
RN1([omega]') - RN1(z)
as we see at once by considering the zeros and poles and the limit of
z[f]1(z) when z = 0. In terms of the function RN1(z) the original
function RN(z) is expressed by
RN(z) = RN1(z) + RN1(z + [omega]') - RN1([omega]'),
as a consideration of the poles and expansion near z = 0 will show.
A function having [omega], [omega]' for periods, with poles at two
arbitrary points a, b and zeros at a', b', where a' + b' = a + b save
for an expression m[omega] + m'[omega]', in which m, m' are integers,
is a constant multiple of
{RN[z - 1/2(a' + b')] - RN[a' - 1/2(a' + b')]} / {RN[z - 1/2(a + b)] - RN[a - 1/2(a + b)]};
if the expansion of this function near z = a be
_
[lambda](z - a)^(-1) + [mu] + \ [mu]_n(z - a)^n,
/_
n = 1
the expansion near z = b is
_
-[lambda](z - b)^(-1) + [mu] + \ (-1)^n [mu]_n (z - b)^n,
/_
n = 1
as we see by remarking that if z'- b = -(z - a) the function has the
same value at z and z'; hence the differential equation satisfied by
the function is easily calculated in terms of the coefficients in the
expansions.
From the function RN(z) we can obtain another function, termed the
Zeta-function; it is usually denoted by [zeta](z), and defined by
_ _ _
1 / [pi] | 1 | _ / 1 1 z \
[zeta](z) -- = | | --- - RN(z) |dz = \ ' ( ----------- + ------- + --------- ),
z _/ 0 |_ z^2 _| /_ \z - [Omega] [Omega] [Omega]^2/
for which as before we have equations
[zeta](z + [omega]) = [zeta](z) + 2[pi]i[eta],
[zeta](z + [omega]') = [zeta](z) + 2[pi]i[eta]',
where 2[eta], 2[eta]' are certain constants, which in this case do not
both vanish, since else [zeta](z) would be a doubly periodic function
with only one pole of the first order. By considering the integral
_
/
| [zeta](z)dz
_/
round the perimeter of a parallelogram of sides [omega], [omega]'
containing z = 0 in its interior, we find [eta][omega]' -
[eta]'[omega] = 1, so that neither of [eta], [eta]' is zero. We have
[zeta]'(z) = -RN(z). From [zeta](z) by means of the equation
_ _ _
[sigma](z) { / z | 1 | }
---------- = exp { | | [zeta](x) - -- |dz } =
z { _/ 0 |_ z _| }
_ _
| / 2 \ / z z^2 \ |
[Pi]' | ( 1 - ------- ) exp ( ------- + ---------- ) |,
|_ \ [Omega]/ \[Omega] 2[Omega]^2/ _|
we determine an integral function [sigma](z), termed the
Sigma-function, having a zero of the first order at each of the points
z = [Omega]; it can be seen to satisfy the equations
[sigma](z + [omega])
-------------------- = -exp [2[pi] i[eta](z + 1/2[omega])],
[sigma](z)
[sigma](z + [omega]')
--------------------- = -exp [2[pi] i[eta]'(z + 1/2[omega]')].
[sigma](z)
By means of these equations, if a1 + a2 + ... + a_m = a'1 + a'2 + ...
+ a'_m, it is readily shown that
[sigma](z - a'1)[sigma](z - a'2) ... [sigma](z - a'_m)
------------------------------------------------------
[sigma](z - a1[sigma](z - a2) ... [sigma](z - a_m)
is a doubly periodic function having a1, ... a_m as its simple poles,
and a'1, ... a'_m as its simple zeros. Thus the function [sigma](z)
has the important property of enabling us to write any meromorphic
doubly periodic function as a product of factors each having one zero
in the parallelogram of periods; these form a generalization of the
simple factors, z - a, which have the same utility for rational
functions of z. We have [zeta](z) = [sigma]'(z)/[sigma](z).
The functions [zeta](z), RN(z) may be used to write any meromorphic
doubly periodic function F(z) as a sum of terms having each only one
pole; for if in the expansion of F(z) near a pole z = a the terms with
negative powers of z-a be
A1(z - a)^(-1) + A2(z - a){-2} + ... + A_(m + 1)(z - a)^(-(m + 1)),
then the difference
A_(m + 1)
F(z) - A1[zeta](z - a) - A2[Fraktur](z - a)- ... + ---------(-1)^m RN^(m - 1)(z - a)
m!
will not be infinite at z = a. Adding to this a sum of further terms
of the same form, one for each of the poles in a parallelogram of
periods, we obtain, since the sum of the residues A is zero, a doubly
periodic function without poles, that is, a constant; this gives the
expression of F(z) referred to. The indefinite integral [int]F(z)dz
can then be expressed in terms of z, functions RN(z - a) and their
differential coefficients, functions [zeta](z - a) and functions
log[sigma](z - a).
S 15. _Potential Functions. Conformal Representation in
General._--Consider a circle of radius a lying within the region of
existence of a single valued monogenic function, u + iv, of the complex
variable z, = x + iy, the origin z = 0 being the centre of this circle.
If z = rE(i[phi]) = r(cos [phi] + i sin [phi]) be an internal point of
this circle we have
_
1 / (U + iV)
u + iv = ------ | -------- dt,
2[pi]i _/ t - z
where U + iV is the value of the function at a point of the
circumference and t = aE(i[theta]); this is the same as
_
1 / (U + iV) [1 - (r/a)E(i[theta] - i[phi])]
u + iv = ----- | ------------------------------------------ d[theta].
2[pi] _/ 1 + (r/a)^2 - 2(r/a) cos ([theta] - [phi])
If in the above formula we replace z by the external point
(a^2/r)E(i[phi]) the corresponding contour integral will vanish, so that
also
_
1 / (U + iV) [(r/a)^2 - (r/a)E(i[theta] - i[phi])]
0 = ----- | ---------------------------------------------- d[theta];
2[pi] _/ 1 + (r/a)^2 - 2(r/a) cos ([theta] - [phi])
hence by subtraction we have
_
1 / U(a^2 - r^2)
u = ----- | ------------------------------------- d[theta],
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi])
and a corresponding formula for v in terms of V. If O be the centre of
the circle, Q be the interior point z, P the point aE(i[theta]) of the
circumference, and [omega] the angle which QP makes with OQ produced,
this integral is at once found to be the same as
_ _
1 / 1 /
u = ---- | Ud[omega] - ----- | Ud[theta]
[pi] _/ 2[pi] _/
of which the second part does not depend upon the position of z, and the
equivalence of the integrals holds for every arc of integration.
Conversely, let U be any continuous real function on the
circumference, U0 being the value of it at a point P0 of the
circumference, and describe a small circle with centre at P0 cutting
the given circle in A and B, so that for all points P of the arc AP0B
we have |U - U0| < [epsilon], where [epsilon] is a given small real
quantity. Describe a further circle, centre P0 within the former,
cutting the given circle in A' and B', and let Q be restricted to lie
in the small space bounded by the arc A'P0B' and this second circle;
then for all positions of P upon the greater arc AB of the original
circle QP^2 is greater than a definite finite quantity which is not
zero, say QP^2 > D^2. Consider now the integral
_
1 / (a^2 - r^2)
u' = ----- | U ------------------------------------ d[theta], =
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi]
_ _
1 / 1 /
---- | Ud[omega] - ----- | Ud[theta],
[pi] _/ 2[pi] _/
which we evaluate as the sum of two, respectively along the small arc
AP0B and the greater arc AB. It is easy to verify that, for the whole
circumference,
_
1 / (a^2 - r^2)
U0 = ----- | U0 ------------------------------------ d[theta] =
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi]
_ _
1 / 1 /
---- | U0d[omega] - ----- | U0d[theta].
[pi] _/ 2[pi] _/
Hence we can write
_ _
1 / 1 /
u' - U0 = ----- | (U - U0) d[omega] - ----- | (U - U0) d[theta] +
2[pi] _/AP0B 2[pi] _/AP0B
_
1 / (a^2 - r^2)
----- | (U - U0) ----------- d[theta].
2[pi] _/AB QP^2
If the finite angle between QA and QB be called [Phi] and the finite
angle AOB be called [Theta], the sum of the first two components is
numerically less than
[epsilon]
--------- ([Phi] + [Theta]).
2[pi]
If the greatest value of |(U - U0)| on the greater arc AB be called H,
the last component is numerically less than
H
--- (a^2 - r^2),
D^2
of which, when the circle, of centre P0, passing through A'B' is
sufficiently small, the factor a^2 - r^2 is arbitrarily small. Thus it
appears that u' is a function of the position of Q whose limit, when
Q, interior to the original circle, approaches indefinitely near to
P0, is U0. From the form
_ _
1 / 1 /
u' = ---- | Ud[omega] - ----- | Ud[theta],
[pi] _/ 2[pi] _/
since the inclination of QP to a fixed direction is, when Q varies, P
remaining fixed, a solution of the differential equation
(Pd)^2[psi] (Pd)^2
----------- + ------- = 0,
(Pd)x^2 (Pd)y^2
where z, = x + iy, is the point Q, we infer that u' is a
differentiable function satisfying this equation; indeed, when r < a,
we can write
_
1 / (a^2 - r^2)
----- | U ------------------------------------- d[theta]
2[pi] _/ a^2 + r^2 - 2ar cos ([theta] - [phi])
_ _ _
1 / | r r^2 |
= ----- | U | 1 + 2 -- cos ([theta] - [phi]) + 2 --- cos 2([theta] - [phi]) + ...| d[theta]
2[pi] _/ |_ a a^2 _|
= a0 + a1x + b1y + a2(x^2 - y^2) + 2b2xy + ...,
where
_ _ _
1 / 1 / U cos[theta] 1 / U sin[theta]
a0 = ----- | Ud[theta], a1 = ---- | ------------ d[theta], b1 = ---- | ------------ d[theta],
2[pi] _/ [pi] _/ a [pi] _/ a
_ _
1 / U cos 2[theta] 1 / U sin 2[theta]
a2 = ---- | -------------- d[theta], b2 = ---- | -------------- d[theta].
[pi] _/ a^2 [pi] _/ a^2
In this series the terms of order n are sums, with real coefficients,
of the various integral polynomials of dimension n which satisfy the
equation (Pd)^2[psi]/(Pd)x^2 + (Pd)^2[psi]/(Pd)y^2; the series is thus
the real part of a power series in z, and is capable of
differentiation and integration within its region of convergence.
Conversely we may suppose a function, P, defined for the interior of a
finite region R of the plane of the real variables x, y, capable of
expression about any interior point x0, y0 of this region by a power
series in x - x0, y - y0, with real coefficients, these various series
being obtainable from one of them by continuation. For any region R0
interior to the region specified, the radii of convergence of these
power series will then have a lower limit greater than zero, and hence
a finite number of these power series suffice to specify the function
for all points interior to R0. Each of these series, and therefore the
function, will be differentiable; suppose that at all points of R0 the
function satisfies the equation
(Pd)^2P (Pd)P^2
------- + ------- = 0,
(Pd)x^2 (Pd)y^2
we then call it a monogenic potential function. From this, save for an
additive constant, there is defined another potential function by
means of the equation
_
/(x, y) /(Pd)P (Pd)P \
Q = | ( ----- dy - ----- dx ).
_/ \(Pd)x (Pd)y /
The functions P, Q, being given by a finite number of power series,
will be single valued in R0, and P + iQ will be a monogenic function
of z within R0. In drawing this inference it is supposed that the
region R0 is such that every closed path drawn in it is capable of
being deformed continuously to a point lying within R0, that is, is
_simply connected_.
Suppose in particular, c being any point interior to R0, that P
approaches continuously, as z approaches to the boundary of R, to the
value log r, where r is the distance of c to the points of the
perimeter of R. Then the function of z expressed by
[zeta] = (z - c) exp (-P - iQ)
will be developable by a power series in (z - z0) about every point z0
interior to R0, and will vanish at z = c; while on the boundary of R
it will be of constant modulus unity. Thus if it be plotted upon a
plane of [zeta] the boundary of R will become a circle of radius unity
with centre at [zeta] = 0, this latter point corresponding to z = c. A
closed path within R0, passing once round z = c, will lead to a closed
path passing once about [zeta] = 0. Thus every point of the interior
of R will give rise to one point of the interior of the circle. The
converse is also true, but is more difficult to prove; in fact, the
differential coefficient d[zeta]/dz does not vanish for any point
interior to R. This being assumed, we obtain a conformal
representation of the interior of the region R upon the interior of a
circle, in which the arbitrary interior point c of R corresponds to
the centre of the circle, and, by utilizing the arbitrary constant
arising in determining the function Q, an arbitrary point of the
boundary of R corresponds to an arbitrary point of the circumference
of the circle.
There thus arises the problem of the determination of a real monogenic
potential function, single valued and finite within a given arbitrary
region, with an assigned continuous value at all points of the
boundary of the region. When the region is circular this problem is
solved by the integral 1/[pi] [int] Ud[omega] - 1/[pi] [int] Ud[theta]
previously given. When the region is bounded by the outermost portions
of the circumferences of two overlapping circles, it can hence be
proved that the problem also has a solution; more generally, consider
a finite simply connected region, whose boundary we suppose to consist
of a single closed path in the sense previously explained, ABCD;
joining A to C by two non-intersecting paths AEC, AFC lying within the
region, so that the original region may be supposed to be generated by
the overlapping regions AECD, CFAB, of which the common part is AECF;
suppose now the problem of determining a single valued finite
monogenic potential function for the region AECD with a given
continuous boundary value can be solved, and also the same problem for
the region CFAB; then it can be shown that the same problem can be
solved for the original area. Taking indeed the values assigned for
the original perimeter ABCD, assume arbitrarily values for the path
AEC, continuous with one another and with the values at A and C; then
determine the potential function for the interior of AECD; this will
prescribe values for the path CFA which will be continuous at A and C
with the values originally proposed for ABC; we can then determine a
function for the interior of CFAB with the boundary values so
prescribed. This in its turn will give values for the path AEC, so
that we can determine a new function for the interior of AECD. With
the values which this assumes along CFA we can then again determine a
new function for the interior of CFAB. And so on. It can be shown that
these functions, so alternately determined, have a limit representing
such a potential function as is desired for the interior of the
original region ABCD. There cannot be two functions with the given
perimeter values, since their difference would be a monogenic
potential function with boundary value zero, which can easily be shown
to be everywhere zero. At least two other methods have been proposed
for the solution of the same problem.
A particular case of the problem is that of the conformal
representation of the interior of a closed polygon upon the upper half
of the plane of a complex variable t. It can be shown without much
difficulty that if a, b, c, ... be real values of t, and [alpha],
[beta], [gamma], ... be n real numbers, whose sum is n - 2, the
integral
_
/
z = | (t - a)^([alpha] - 1) (t - b)^([beta] - 1) ... dt,
_/
as t describes the real axis, describes in the plane of z a polygon of
n sides with internal angles equal to [alpha][pi], [beta][pi], ...,
and, a proper sign being given to the integral, points of the upper
half of the plane of t give rise to interior points of the polygon.
Herein the points a, b, ... of the real axis give rise to the corners
of the polygon; the condition [Sigma][alpha] = n - 2 ensures merely
that the point t = [oo] does not correspond to a corner; if this
condition be not regarded, an additional corner and side is introduced
in the polygon. Conversely it can be shown that the conformal
representation of a polygon upon the half plane can be effected in
this way; for a polygon of given position of more than three sides it
is necessary for this to determine the positions of all but three of
a, b, c, ...; three of them may always be supposed to be at arbitrary
positions, such as t = 0, t = 1, t = [oo].
As an illustration consider in the plane of z = x + iy, the portion of
the imaginary axis from the origin to z = ih, where h is positive and
less than unity; let C be this point z = ih; let BA be of length unity
along the positive real axis, B being the origin and A the point z =
1; let DE be of length unity along the negative real axis, D being
also the origin and E the point z = -1; let EFA be a semicircle of
radius unity, F being the point z = i. If we put [zeta] = [(z^2 +
h^2)/(1 + h^2z^2)]^1/2, with [zeta] = 1 when z = 1, the function is
single valued within the semicircle, in the plane of z, which is slit
along the imaginary axis from the origin to z = ih; if we plot the
value of [zeta] upon another plane, as z describes the continuous
curve ABCDE, [zeta] will describe the real axis from [zeta] = 1 to
[zeta] = -1, the point C giving [zeta] = 0, and the points B, D giving
the points [zeta] = [+-]h. Near z = 0 the expansion of [zeta] is
[zeta] - h = z^2 1 - h^4 / 2h + ..., or [zeta] + h = -z^2 (1 - h^4)/2h
+ ...; in either case an increase of 1/2[pi] in the phase of z gives
an increase of [pi] in the phase of [zeta] - h or [zeta] + h. Near z =
ih the expansion of [zeta] is [zeta] = (z - ih)^1/2 [2ih/(1 -
h^4)]^1/2 + ..., and an increase of 2[pi] in the phase of z - ih also
leads to an increase of [pi] in the phase of [zeta]. Then as z
describes the semicircle EFA, [zeta] also describes a semicircle of
radius unity, the point z = i becoming [zeta] = i. There is thus a
conformal representation of the interior of the slit semicircle in the
z-plane, upon the interior of the whole semicircle in the
[zeta]-plane, the function
z = [([zeta]^2 - h^2) / (1 - h^2[zeta]^2)]^1/2
being single valued in the latter semicircle. By means of a
transformation t = ([zeta] + 1)^2/([zeta] - 1)^2, the semicircle in
the plane of [zeta] can further be conformably represented upon the
upper half of the whole plane of t.
As another illustration we may take the conformal representation of an
equilateral triangle upon a half plane. Taking the elliptic function
RN(u) for which RN'^2(u) = 4RN^3(u) - 4, so that, with [epsilon] = exp
(2/3[pi]i), we have e1 = 1, e2 = [epsilon]^2, e3 = [epsilon], the half
periods may be taken to be
_ _
/ [oo] dt / [oo] dt
1/2[omega] = | --------------, 1/2[omega]' = | -------------- = 1/2[epsilon][omega];
_/ 1 2(t^3 - 1)^1/2 _/ e3 2(t^3 - 1)^1/2
drawing the equilateral triangle whose vertices are O, of argument O,
A of argument [omega], and B of argument [omega] + [omega]' =
-[epsilon]^2[omega], and the equilateral triangle whose angular points
are O, B and C, of argument [omega]', let E, of argument
{1/3}(2[omega] + [omega]'), and D, of argument 1/3([omega] +
2[omega]'), be the centroids of these triangles respectively, and let
BE, OE, AE cut OA, AB, BO in K, L, H respectively, and BD, OD, CD cut
OC, BC, OB in F, G, H respectively; then if u = [xi] + i[eta] be any
point of the interior of the triangle OEH and v = [epsilon]u0 =
[epsilon]([xi] - i[eta]) be any point of the interior of the triangle
OHD, the points respectively of the ten triangles OEK, EKA, EAL, ELB,
EBH, DHB, DBG, DGC, DCF, DFO are at once seen to be given by
-[epsilon]v, [omega] + [epsilon]u, [omega] - [eta]^2v, [omega] +
[omega]' + [epsilon]^2u, [omega] + [omega]' - v, [omega] + [omega]' -
u, [omega] + [omega]' + [epsilon]v, [omega]' - [epsilon]u, [omega]' +
[epsilon]^2v, -[epsilon]^2u. Further, when u is real, since the term
-2(u + m[omega] + m'[epsilon]^2[omega])^(-3), which is the conjugate
complex of -2(u + m[omega] + m'[epsilon]^2[omega])^3, arises in the
infinite sum which expresses RN'(u), namely as -2(u + [mu][omega] +
[mu]'[epsilon][omega])^(-3), where [mu] = m - m', [mu]' = -m', it
follows that RN'(u) is real; in a similar way we prove that RN'(u) is
pure imaginary when u is pure imaginary, and that RN'(u) =
RN'([epsilon]u) = RN'([epsilon]^2u), as also that for v = [epsilon]u0,
RN'(v) is the conjugate complex of RN'(u). Hence it follows that the
variable
t = 1/2iRN'(u)
takes each real value once as u passes along the perimeter of the
triangle ODE, being as can be shown respectively [oo], 1, 0, -1 at O,
D, H, E, and takes every complex value of imaginary part positive once
in the interior of this triangle. This leads to
_
/ [oo]
u = 1/3i | (t^2 - 1)^(-2/3) dt
_/ t
in accordance with the general theory.
It can be deduced that [tau] = t^2 represents the triangle ODH on the
upper half plane of [tau], and [zeta] = {i-[tau]^(-1)}^(1/2)
represents similarly the triangle OBD.
S 16. _Multiple valued Functions. Algebraic Functions._--The
explanations and definitions of a monogenic function hitherto given have
been framed for the most part with a view to single valued functions.
But starting from a power series, say in z - c, which represents a
single value at all points of its circle of convergence, suppose that,
by means of a derived series in z - c', where c' is interior to the
circle of convergence, we can continue the function beyond this, and
then by means of a series derived from the first derived series we can
make a further continuation, and so on; it may well be that when, after
a closed circuit, we again consider points in the first circle of
convergence, the value represented may not agree with the original
value. One example is the case z^(1/2), for which two values exist for
any value of z; another is the generalized logarithm [Lambda] (z), for
which there is an infinite number of values. In such cases, as before,
the region of existence of the function consists of all points which can
be reached by such continuations with power series, and the singular
points, which are the limiting points of the point-aggregate
constituting the region of existence, are those points in whose
neighbourhood the radii of convergence of derived series have zero for
limit. In this description the point z = [oo] does not occupy an
exceptional position, a power series in z - c being transformed to a
series in 1/z when z is near enough to c by means of z - c = c(1 -
cz^(-1)) [1 - (1 - cz^(-1))]^(-1), and a series in 1/z to a series in z
- c, when z is near enough to c, by means of
1 1 / z - c \^(-1)
-- = -- ( 1 + ----- ).
z c \ c /
The commonest case of the occurrence of multiple valued functions is
that in which the function s satisfies an algebraic equation [f](s, z)
= p_0s^n + p1s^(n - 1) + ... + p_n = 0, wherein p0, p1, ... p_n are
integral polynomials in z. Assuming [f](s, z) incapable of being
written as a product of polynomials rational in s and z, and excepting
values of z for which the polynomial coefficient of s^n vanishes, as
also the values of z for which beside [f](s, z) = 0 we have also
(Pd)f(s, z)/(Pd)s = 0, and also in general the point z = [oo], the
roots of this equation about any point z = c are given by n power
series in z-c. About a finite point z = c for which the equation
(Pd)f(s, z)/(Pd)s = 0 is satisfied by one or more of the roots s of
[f](s, z) = 0, the n roots break up into a certain number of cycles,
the r roots of a cycle being given by a set of power series in a
radical (z - c)^(1/r), these series of the cycle being obtainable from
one another by replacing (z - c)^(1/r) by [omega](z - r)^(1/r), where
[omega], equal to exp (2[pi]ih/r), is one of the rth roots of unity.
Putting then z - c = t^r we may say that the r roots of a cycle are
given by a single power series in t, an increase of 2[pi] in the phase
of t giving an increase of 2[pi]r in the phase of z - c. This single
series in t, giving the values of s belonging to one cycle in the
neighbourhood of z = c when the phase of z-c varies through 2[pi]r, is
to be looked upon as defining a single _place_ among the aggregate of
values of z and s which satisfy [f](s, z) = 0; two such places may be
at the same _point_ (z = c, s = d) without coinciding, the
corresponding power series for the neighbouring points being
different. Thus for an ordinary value of z, z = c, there are n places
for which the neighbouring values of s are given by n power series in
z-c; for a value of z for which (Pd)f(s, z)/(Pd)s = 0 there are less
than n places. Similar remarks hold for the neighbourhood of z = [oo];
there may be n places whose neighbourhood is given by n power series
in z^(-1) or fewer, one of these being associated with a series in t,
where t = (z^(-1))^(1/r); the sum of the values of r which thus arise
is always n. In general, then, we may say, with t of one of the forms
(z-c), (z-c)^(1/r), z^(-1), (z^(-1))^(1/r). that the neighbourhood of
any place (c, d) for which [f](c, d) = 0 is given by a pair of
expressions z = c + P(t), s = d + Q(t), where P(t) is a (particular
case of a) power series vanishing for t = 0, and Q(t) is a power
series vanishing for t = 0, and t vanishes at (c, d), the expression
z-c being replaced by z^(-1) when c is infinite, and similarly the
expression s-d by s^(-1) when d is infinite. The last case arises when
we consider the finite values of z for which the polynomial
coefficient of s^n vanishes. Of such a pair of expressions we may
obtain a continuation by writing t = t0 + [lambda]1[tau] +
[lambda]2[tau]^2 + .. , where [tau] is a new variable and [lambda]1 is
not zero; in particular for an ordinary finite place this equation
simply becomes t = t0 + [tau]. It can be shown that all the pairs of
power series z = c + P(t), s = d + Q(t) which are necessary to
represent all pairs of values of z, s satisfying the equation [f](s,
z) = 0 can be obtained from one of them by this process of
continuation, a fact which we express by saying that the equation
[f](s, z) = 0 defines a _monogenic algebraic construct_. With less
accuracy we may say that an irreducible algebraic equation [f](s, z) =
0 determines a single monogenic function s of z.
Any rational function of z and s, where [f](s, z) = 0, may be
considered in the neighbourhood of any place (c, d) by substituting
therein z = c + P(t), s = d + Q(t); the result is necessarily of the
form t^m H(t), where H(t) is a power series in t not vanishing for t =
0 and m is an integer. If this integer is positive, the function is
said to vanish to order m at the place; if this integer is negative, =
-[mu], the function is infinite to order [mu] at the place. More
generally, if A be an arbitrary constant, and, near (c, d), R(s, z) -
A is of the form t^mH(t), where m is positive, we say that R(s, z)
becomes m times equal to A at the place; if R(s, z) is infinite of
order [mu] at the place, so also is R(s, z) - A. It can be shown that
the sum of the values of m at all the places, including the places z =
[oo], where R(s, z) vanishes, which we call the number of zeros of
R(s, z) on the algebraic construct, is finite, and equal to the sum of
the values of [mu] where R (s, z) is infinite, and more generally
equal to the sum of the values of m where R(s, z) = A; this we express
by saying that a rational function R(s, z) takes any value (including
[oo]) the same number of times on the algebraic construct; this number
is called the _order_ of the rational function.
That the total number of zeros of R (s, z) is finite is at once
obvious, these values being obtainable by rational elimination of s
between [f](s, z) = 0, R(s, z) = 0. That the number is equal to the
total number of infinities is best deduced by means of a theorem which
is also of more general utility. Let R(s, z) be any rational function
of s, z, which are connected by [f](s, z) = 0; about any place (c, d)
for which z = c + P(t), s = d + Q(t), expand the product
dz
R(s, z) --
dt
in powers of t and pick out the coefficient of t^(-1). There is only a
finite number of places of this kind. The theorem is that the sum of
these coefficients of t^(-1) is zero. This we express by
_ _
| dz |
|R(s, z) -- | = 0.
|_ dt _|t^(-1)
The theorem holds for the case n = 1, that is, for rational functions
of one variable z; in that case, about any finite point we have z - c
= t, and about z = [oo] we have z^(-1) = t, and therefore dz/dt =
-t^(-2); in that case, then, the theorem is that in any rational
function of z,
_ / A1 A2 A_m \
\ ( ----- + --------- + ... + --------- ) + Pz^h + Qz^(h - 1) + ... + R,
/_ \z - a (z - a)^2 (z - a)^m /
the sum [Sigma]A1 of the sum of the residues at the finite poles is
equal to the coefficient of 1/z in the expansion, in ascending powers
of 1/z, about z = [oo]; an obvious result. In general, if for a finite
place of the algebraic construct associated with [f](s, z) = 0, whose
neighbourhood is given by z = c + t^r, s = d + Q(t), there be a
coefficient of t^(-1) in R(s, z)dz/dt, this will be r times the
coefficient of t^(-r) in R(s, z) or R[d + Q(t), c + t^r], namely will
be the coefficient of t^(-r) in the sum of the r series obtainable
from R[d + Q(t), c + t^r] by replacing t by [omega]t, where [omega] is
an rth root of unity; thus the sum of the coefficients of t^(-1) in
R(s, z)dz/dt for all the places which arise for z = c, and the
corresponding values of s, is equal to the coefficient of (z - c)^(-1)
in R(s1, z) + R(s2z) + ... + R(s_n, z), where s1, ... s_n are the n
values of s for a value of z near to z = c; this latter sum [Sigma]
R(s_i, z) is, however, a rational function of z only. Similarly, near
z = [oo], for a place given by z^(-1) = t^r, s = d + Q(t), or s^(-1) =
Q(t), the coefficient of t^(-1) in R(s, z)dz/dt is equal to -r times
the coefficient of t^r in R[d + Q(t), t^(-r)], that is equal to the
negative coefficient of z^(-l) in the sum of the r series R[d +
Q([omega]t), t^(-r)], so that, as before, the sum of the coefficients
of t^(-1) in R(s, z)dz/dt at the various places which arise for z =
[oo] is equal to the negative coefficient of z^(-1) in the same
rational function of z, [Sigma] R(s_i, z). Thus, from the
corresponding theorem for rational functions of one variable, the
general theorem now being proved is seen to follow.
Apply this theorem now to the rational function of s and z,
1 dR(s, z)
------- -------;
R(s, z) dz
at a zero of R(s, z) near which R(s, z) = t^mH(t), we have
1 dR(s, z) dz d
------- ------- -- = -- {[lambda] [R(s, z)]}
R(s, z) dz dt dt
where [lambda] denotes the generalized logarithmic function, that is
equal to
mt^(-1) + power series in t;
similarly at a place for which R(s, z) = t^(-[mu]) K(t); the theorem
_ _
| 1 dR(s, z) dz |
| ------- -------- -- | t^(-1) = 0
|_ R(s, z) dz dt _|
thus gives [Sigma]m = [Sigma][mu], or, in words, the total number of
zeros of R(s, z) on the algebraic construct is equal to the total
number of its poles. The same is therefore true of the function R(s,
z) - A, where A is an arbitrary constant; thus the number in question,
being equal to the number of poles of R(s, z) - A, is equal also to
the number of times that R(s, z) = A on the algebraic construct.
We have seen above that all single valued doubly periodic meromorphic
functions, with the same periods, are rational functions of two
variables s, z connected by an equation of the form s^2 = 4z^3 + Az +
B. Taking account of the relation connecting these variables s, z with
the argument of the doubly periodic functions (which was above denoted
by z), it can then easily be seen that the theorem now proved is a
generalization of the theorem proved previously establishing for a
doubly periodic function a definite _order_. There exists a
generalization of another theorem also proved above for doubly
periodic functions, namely, that the sum of the values of the argument
in one parallelogram of periods for which a doubly periodic function
takes a given value is independent of that value; this generalization,
known as Abel's Theorem, is given S 17 below.
S 17. _Integrals of Algebraic Functions._--In treatises on Integral
Calculus it is proved that if R(z) denote any rational function, an
indefinite integral [int]R(z)dz can be evaluated in terms of rational
and logarithmic functions, including the inverse trigonometrical
functions. In generalization of this it was long ago discovered that if
s^2 = az^2 + bz + c and R(s, z) be any rational function of s, z any
integral [int]R(s, z)dz can be evaluated in terms of rational functions
of s, z and logarithms of such functions; the simplest case is
[int]s^(-1)dz or [int](az^2 + bz + c)^(-1/2)dz. More generally if f(s,
z) = 0 be such a relation connecting s, z that when [theta] is an
appropriate rational function of s and z both s and z are rationally
expressible, in virtue of [f](s, z) = 0 in terms of [theta], the
integral [int]R(s, z)dz is reducible to a form [int]H ([theta])d[theta],
where H([theta]) is rational in [theta], and can therefore also be
evaluated by rational functions and logarithms of rational functions of
s and z. It was natural to inquire whether a similar theorem holds for
integrals [int]R(s, z)dz wherein s^2 is a cubic polynomial in z. The
answer is in the negative. For instance, no one of the three integrals
_ _ _
/ dz / zdz / dz
| --, | ---, | --------
_/ s _/ s _/ (z - c)s
can be expressed by rational and logarithms of rational functions of s
and z; but it can be shown that every integral [int]R(s, z)dz can be
expressed by means of integrals of these three types together with
rational and logarithms of rational functions of s and z (see below
under S 20, Elliptic Integrals). A similar theorem is true when s^2 =
quartic polynomial in z; in fact when s^2 = A(z - a)(z - b)(z - c)(z -
d), putting y = s(z - a)^(-2), x = (z - a)^(-1), we obtain y2 = cubic
polynomial in x. Much less is the theorem true when the fundamental
relation [f](s, z) = 0 is of more general type. There exists then,
however, a very general theorem, known as _Abel's Theorem_, which may be
enunciated as follows: Beside the rational function R(s, z) occurring in
the integral [int]R(s, z)dz, consider another rational function H(s, z);
let (a1), ... (a_m) denote the places of the construct associated with
the fundamental equation [f](s, z) = 0, for which H(s, z) is equal to
one value A, each taken with its proper multiplicity, and let (b1), ...
(b_m) denote the places for which H(s, z) = B, where B is another value;
then the sum of the m integrals [int] [(b_i) to (a_i)] R(s, z)dz is
equal to the sum of the coefficients of t^(-1) in the expansions of the
function
dz / H(s, z) - B \
R(s, z) -- [lambda] ( ----------- ),
dt \ H(s, z) - A /
where [lambda] denotes the generalized logarithmic function, at the
various places where the expansion of R(s, z)dz/dt contains negative
powers of t. This fact may be obtained at once from the equation
_ _
| 1 dz |
| -------------- R(s, z) -- | = 0,
|_ H(s, z) - [mu] dt _|t^(-1)
wherein [mu] is a constant. (For illustrations see below, under S 20,
Elliptic Integrals.)
S 18. _Indeterminateness of Algebraic Integrals._--The theorem that the
integral [int][a to x] [f](z)dz is independent of the path from a to z,
holds only on the hypothesis that any two such paths are equivalent,
that is, taken together from the complete boundary of a region of the
plane within which [f](z) is finite and single valued, besides being
differentiable. Suppose that these conditions fail only at a finite
number of isolated points in the finite part of the plane. Then any path
from a to z is equivalent, in the sense explained, to any other path
together with closed ~~ paths beginning and ending at the arbitrary
point a each enclosing one or more of the exceptional points, these
closed paths being chosen, when [f](z) is not a single valued function,
so that the final value of [f](z) at a is equal to its initial value. It
is necessary for the statement that this condition may be capable of
being satisfied.
For instance, the integral [int][1 to z] z^(-1)dz is liable to an
additive indeterminateness equal to the value obtained by a closed
path about z = 0, which is equal to 2[pi]i; if we put u = [int][1 to
z] z^(-1)dz and consider z as a function of u, then we must regard
this function as unaffected by the addition of 2[pi]i to its argument
u; we know in fact that z = exp (u) and is a single valued function of
u, with the period 2[pi]i. Or again the integral [int][0 to z] (1 +
z^2)^(-1)dz is liable to an additive indeterminateness equal to the
value obtained by a closed path about either of the points z = [+-]i;
thus if we put u = [int][0 to z] (1 + z^2)^(-1)dz, the function z of u
is periodic with period [pi], this being the function tan (u). Next we
take the integral u = [int][(0) to (z)] (1 - z^2)^(-1/2)dz, agreeing
that the upper and lower limits refer not only to definite values of
z, but to definite values of z each associated with a definite
determination of the sign of the associated radical (1 - z^2)^(-1/2).
We suppose 1 + z, 1 - z each to have phase zero for z = 0; then a
single closed circuit of z = -1 will lead back to z = 0 with (l -
z^2)^1/2 = -1; the additive indeterminateness of the integral,
obtained by a closed path which restores the initial value of the
subject of integration, may be obtained by a closed circuit containing
both the points [+-]1 in its interior; this gives, since the integral
taken about a vanishing circle whose centre is either of the points z
= [+-] 1 has ultimately the value zero, the sum
_ _ _ _
/ -1 dz / 0 dz / 1 dz / 0 dz
| ----------- + | -------------- + | -------------- + | -------------,
_/ 0 (1-z^2)^1/2 _/-1 -(1 - z^2)^1/2 _/ 0 -(1 - z^2)^1/2 _/ 1 (1 - z^2)^1/2
where, in each case, (1 - z^2)^1/2 is real and positive; that is, it
gives
_
/ 1 dz
-4 | -------------
_/ 0 (1 - z^2)^1/2
or 2[pi]. Thus the additive indeterminateness of the integral is of
the form 2k[pi], where k is an integer, and the function z of u, which
is sin (u), has 2[pi] for period. Take now the case
_
/ (z) dz
u = | ------------------------------------,
_/ (z0) [root]{(z - a)(z - b)(z - c)(z - d)}
adopting a definite determination for the phase of each of the factors
z - a, z - b, z - c, z - d at the arbitrary point z0, and supposing
the upper limit to refer, not only to a definite value of z, but also
to a definite determination of the radical under the sign of
integration. From z0 describe a closed loop about the point z = a,
consisting, suppose, of a straight path from z0 to a, followed by a
vanishing circle whose centre is at a, completed by the straight path
from a to z0. Let similar loops be imagined for each of the points b,
c, d, no two of these having a point in common. Let A denote the value
obtained by the positive circuit of the first loop; this will be in
fact equal to twice the integral taken from z0 along the straight path
to a; for the contribution due to the vanishing circle is ultimately
zero, and the effect of the circuit of this circle is to change the
sign of the subject of integration. After the circuit about a, we
arrive back at z0 with the subject of integration changed in sign; let
B, C, D denote the values of the integral taken by the loops enclosing
respectively b, c and d when in each case the initial determination of
the subject of integration is that adopted in calculating A. If then
we take a circuit from z0 enclosing both a and b but not either c or
d, the value obtained will be A - B, and on returning to z0 the
subject of integration will have its initial value. It appears thus
that the integral is subject to an additive indeterminateness equal to
any one of the six differences such as A - B. Of these there are only
two linearly independent; for clearly only A - B, A - C, A - D are
linearly independent, and in fact, as we see by taking a closed
circuit enclosing all of a, b, c, d, we have A - B + C - D = 0; for
there is no other point in the plane beside a, b, c, d about which the
subject of integration suffers a change of sign, and a circuit
enclosing all of a, b, c, d may by putting z = 1/[zeta] be reduced to
a circuit about [zeta] = 0 about which the value of the integral is
zero. The general value of the integral for any position of z and the
associated sign of the radical, when we start with a definite
determination of the subject of integration, is thus seen to be of the
form u0 + m(A - B) + n(A - C), where m and n are integers. The value
of A - B is independent of the position of z0, being obtainable by a
single closed positive circuit about a and b only; it is thus equal to
twice the integral taken once from a to b, with a proper initial
determination of the radical under the sign of integration. Similar
remarks to the above apply to any integral [int]H(z)dz, in which H(z)
is an algebraic function of z; in any such case H(z) is a rational
function of z and a quantity s connected therewith by an irreducible
rational algebraic equation [f](s, z) = 0. Such an integral [f]K(z,
s)dz is called an Abelian Integral.
S 19. _Reversion of an Algebraic Integral._--In a limited number of
cases the equation u = [int] [z0 to z] H(z)dz, in which H(z) is an
algebraic function of z, defines z as a single valued function of u.
Several cases of this have been mentioned in the previous section; from
what was previously proved under S 14, _Doubly Periodic Functions_, it
appears that it is necessary for this that the integral should have at
most two linearly independent additive constants of indeterminateness;
for instance, for an integral
_
/ z
u = | [(z - a)(z - b)(z - c)(z - d)(z - e)(z - f)]^(-1/2) dz,
_/ z0
there are three such constants, of the form A - B, A - C, A - D, which
are not connected by any linear equation with integral coefficients, and
z is not a single valued function of u.
S 20. _Elliptic Integrals._--An integral of the form [int] R(z, s)dz,
where s denotes the square root of a quartic polynomial in z, which may
reduce to a cubic polynomial, and R denotes a rational function of z and
s, is called an _elliptic integral_.
To each value of z belong two values of s, of opposite sign; starting,
for some particular value of z, with a definite one of these two
values, the sign to be attached to s for any other value of z will be
determined by the path of integration for z. When z is in the
neighbourhood of any finite value z0 for which the radical s is not
zero, if we put z - z0 = t, we can find s - s0 = a power series in t,
say s = s0 + Q(t); when z is in the neighbourhood of a value, a, for
which s vanishes, if we put z = a + t^2, we shall obtain s = tQ(t),
where Q(t) is a power series in t; when z is very large and s^2 is a
quartic polynomial in z, if we put z^(-1) = t, we shall find s^(-1) =
t^2Q(t); when z is very large and s^2 is a cubic polynomial in z, if we
put z^(-1) = t^2, we shall find s^(-l) = t^3Q(t). By means of
substitutions of these forms the character of the integral [int] R(z,
s)dz may be investigated for any position of z; in any case it takes a
form [int] [Ht^(-m) + Kt^(-m + 1) + ... + Pt^(-1) + R + St + ...]dt
involving only a finite number of negative powers of t in the subject
of integration. Consider first the particular case [int] s^(-1)dz; it
is easily seen that neither for any finite nor for infinite values of
z can negative powers of t enter; the integral is _everywhere finite_,
and is said to be of _the first kind_; it can, moreover, be shown
without difficulty that no integral [int] R(z, s)dz, save a constant
multiple of [int] s^(-1)dz, has this property. Consider next, s^2 being
of the form a0z^4 + 4a1z^3 + ..., wherein a0 may be zero, the integral
[int] {a0z^2 + 2a1z) s^(-1)dz; for any finite value of z this integral
is easily proved to be everywhere finite; but for infinite values of z
its value is of the form At^(-1) + Q(t), where Q(t) is a power series;
denoting by [root]a0 a particular square root of a0 when a0 is not
zero, the integral becomes infinite for z = [oo] for both signs of s,
the value of A being + [root]a0 or - [root]a0 according as s is
[root]a0.z^2 (1 + [2a1/a0] z^(-1) + ...) or is the negative of this;
hence the integral J1 = [int] ([a0z^2 + 2a1z / s] + [root]a0)dz becomes
infinite when z is infinite, for the former sign of s, its infinite
term being 2[root]a0 t^(-1) or 2a0.z, but does not become infinite for
z infinite for the other sign of s. When a0 = 0 the signs of s for z =
[oo] are not separated, being obtained one from the other by a circuit
of z about an infinitely large circle, and the form obtained
represents an integral becoming infinite as before for z = [oo], its
infinite part being 2[root]a1.t^(-1) or 2[root]a1.[root]z. Similarly
if z0 be any finite value of z which is not a root of the polynomial
[f](z) to which s^2 is equal, and s0 denotes a particular one of the
determinations of s for z = z0, the integral
_
/ / s0^2 + 1/2(z - z0) [f]'(z0) s0 \
J2 = | ( --------------------------- + --------- ) dz,
_/ \ (z - z0)^2s (z - z0)^2 /
wherein [f]'(z) = d[f](z)/dz, becomes infinite for z = z0, s = s0, but
not for z = z0, s = -s0. its infinite term in the former case being
the negative of 2s0(z - z0). For no other finite or infinite value of
z is the integral infinite. If z = [theta] be a root of [f](z), in
which case the corresponding value of s is zero, the integral
_
/ dz
J3 = 1/2[f]'([theta]) | --------------
_/ (z - [theta])s
becomes infinite for z = 0, its infinite part being, if z - [theta] =
t^2, equal to -[[f]'([theta])] 1/2t^(-1): and this integral is not
elsewhere infinite. In each of these cases, of the integrals J1, J2,
J3, the subject of integration has been chosen so that when the
integral is written near its point of infinity in the form
[int][At^(-2) + Bt^(-1) + Q(t)]dt, the coefficient B is zero, so that
the infinity is of algebraic kind, and so that, when there are two
signs distinguishable for the critical value of z, the integral
becomes infinite for only one of these. An integral having only
algebraic infinities, for finite or infinite values of z, is called an
integral of the _second kind_, and it appears that such an integral
can be formed with only one such infinity, that is, for an infinity
arising only for one particular, and arbitrary, pair of values (s, z)
satisfying the equation s^2 = [f](z), this infinity being of the first
order. A function having an algebraic infinity of the mth order (m >
1), only for one sign of s when these signs are separable, at (1) z =
[oo], (2) z = z0, (3) z = a, is given respectively by (s d/dz)^(m -
1)J1, (s d/dz)^(m - 1) J2, (s d/dz )^(m - 1) J3, as we easily see. If
then we have any elliptic integral having algebraic infinities we can,
by subtraction from it of an appropriate sum of constant multiples of
J1, J2, J3 and their differential coefficients just written down,
obtain, as the result, an integral without algebraic infinities. But,
in fact, if J, J^1 denote any two of the three integrals J1, J2, J3,
there exists an equation AJ + BJ' + C[f]s^(-1)dz = rational function
of s, z, where A, B, C are properly chosen constants. For the rational
function
s + s0
------ + z [root]a0
z - z0
is at once found to become infinite for (z0, s0), not for (z0, -s0),
its infinite part for the first point being 2s/(z - z0), and to become
infinite for z infinitely large, and one sign of s only when these are
separable, its infinite part there being 2z [root] a0 or 2 [root] a1
[root] z when a0 = 0. It does not become infinite for any other pair
(z, s) satisfying the relation s^2 = [f](z); this is in accordance
with the easily verified equation
_
s + s0 / dz
------- + z [root]a0 - J1 + J2 + (a0z0^2 + 2a1z0) | -- = 0;
z - z^0 _/ s
and there exists the analogous equation
_
s / dz
----------- + z [root]a0 - J1 + J3 + (a0[theta]^2 + 2a1[theta]) | -- = 0.
z - [theta] _/ s
Consider now the integral
_
/ /s + s0 \ dz
P = | ( --------- + z [root]a0 ) --;
_/ \z - z0 / 2s
this is at once found to be infinite, for finite values of z, only for
(z0, s0), its infinite part being log (z - z0), and for z = [oo], for
one sign of s only when these are separable, its infinite part being
-log t, that is -log z when a0 /= 0, and -log (z^1/2) when a0 = 0.
And, if [f]([theta]) = 0, the integral
_
/ / s \ dz
P1 = | ( ----------- + z [root]a0 ) --
_/ \z - [theta] / 2s
is infinite at z = [theta], s = 0 with an infinite part log t, that is
log (z - [theta])^1/2, is not infinite for any other finite value of z,
and is infinite like P for z = [oo]. An integral possessing such
logarithmic infinities is said to be of the third kind.
Hence it appears that any elliptic integral, by subtraction from it of
an appropriate sum formed with constant multiples of the integral J3
and the rational functions of the form (s d/dz)^(m - 1) J1 with
constant multiples of integrals such as P or P1, with constant
multiples of the integral u = [int]s^(-1)dz, and with rational
functions, can be reduced to an integral H becoming infinite only for
z = [oo], for one sign of s only when these are separable, its
infinite part being of the form A log t, that is, A log z or A log
(z^1/2). Such an integral H = [int]R(z, s)dz does not exist, however,
as we at once find by writing R(z, s) = P(z) + sQ(z), where P(z), Q(z)
are rational functions of z, and examining the forms possible for
these in order that the integral may have only the specified infinity.
An analogous theorem holds for rational functions of z and s; there
exists no rational function which is finite for finite values of z and
is infinite only for z = [oo] for one sign of s and to the first order
only; but there exists a rational function infinite in all to the
first order for each of two or more pairs (z, s), however they may be
situated, or infinite to the second order for an arbitrary pair (z,
s); and any rational function may be formed by a sum of constant
multiples of functions such as
s + s0 s
------ + z [root]a0 or ----------- + z [root]a0
z - z0 z - [theta]
and their differential coefficients.
The consideration of elliptic integrals is therefore reducible to that
of the three
_ _ _
/ dz / /a0z^2 + 2a1z \ / /s + s0 \ dz
u = | --, J = | ( ------------ + z [root]a0 ) dz, P = | ( ------ + z [root]a0 )--
_/ s _/ \ s / _/ \z - z0 / 2s
respectively of the first, second and third kind. Now the equation s^2
= a0z^4 + ... = a0(z - [theta]) (z - [phi]) (z - [psi]) (z - [chi]),
by putting
y = 2s(z - [theta])^(-2) [a0([theta] - [phi]) ([theta] - [psi]) ([theta] - [chi])]^(-1/2)
1 1 / 1 1 1 \
x = ----------- + -- ( --------------- + -------------- + --------------- )
z - [theta] 3 \[theta] - [phi] [theta] - [psi] [theta] - [chi]/
is at once reduced to the form y^2 = 4x^3- g2x - g3 = 4(x - e1)(x -
e2(x - e3), say; and these equations enable us to express s and z
rationally in terms of x and y. It is therefore sufficient to consider
three elliptic integrals
_ _ _
/ dx / xdx / y + y0 dx
u = | --, J = | ---, P = | ------ --.
_/ y _/ y _/ x - x0 2y
Of these consider the first, putting
_
/ ([oo]) dx
u = | --,
_/ (x) y
where the limits involve not only a value for x, but a definite sign
for the radical y. When x is very large, if we put x^(-1) = t^2, y^(-1)
= 2t^3(1 - 1/4 g2t^4 - 1/4 g3t^6)^(-1/2), we have
_
/ t / 1 \ 1
u = | ( 1 + -- g2t^4 + ... )dt = t + -- g2t^5 + ...,
_/ 0 \ 8 / 40
whereby a definite power series in u, valid for sufficiently small
value of u, is found for t, and hence a definite power series for x,
of the form
x = u^(-2) + (1/20)g2u^2 + ...
Let this expression be valid for 0 < |u| < R, and the function defined
thereby, which has a pole of the second order for u = 0, be denoted by
[phi](u). In the range in question it is single valued and satisfies
the differential equation
[[phi]'(u)]^2 = 4[[phi](u)]^3 - g2[phi](u) - g3;
in terms of it we can write x = [phi](u), y = -[phi]'(u), and,
[phi]'(u) being an odd function, the sign attached to y in the
original integral for x = [oo] is immaterial. Now for any two values
u, v in the range in question consider the function
_ _
| [phi]'(u) - [phi]'(v) |^2
F(u, v) = 1/4 | --------------------- | - [phi](u) - [phi](v);
|_ [phi](u) - [phi](v) _|
it is at once seen, from the differential equation, to be such that
(Pd)F/(Pd)u = (Pd)F/(Pd)v; it is therefore a function of u + v;
supposing |u + v| < R we infer therefore, by putting v = 0, that
_ _
| [phi]'(u) - [phi]'(v)] |^2
[phi](u + v) = 1/4 | --------------------- | - [phi](u) - [phi](v).
|_ [phi](u) - [phi](v) _|
By repetition of this equation we infer that if u1, ... u_n be any
arguments each of which is in absolute value less than R, whose sum is
also in absolute value less than R, then [phi](u1 + ... + u_n) is a
rational function of the 2n functions [phi](u_s), [phi]'(u_s); and
hence, if |u| < R, that
[phi](u) = H [[phi](u/n), [phi]'(u/n)],
where H is some rational function of the arguments [phi](u/n),
[phi]'(u/n). In fact, however, so long as |u/n| < R, each of the
functions [phi](u/n), [phi]'(u/n) is single valued and without
singularity save for the pole at u = 0; and a rational function of
single valued functions, each of which has no singularities other than
poles in a certain region, is also a single valued function without
singularities other than poles in this region. We infer, therefore,
that the function of u expressed by H[[phi](u/n), [phi]'(u/n)] is
single valued and without singularities other than poles so long as
|u| < nR; it agrees with [phi](u) when |u| < R, and hence furnishes a
continuation of this function over the extended range |u| < nR.
Moreover, from the method of its derivation, it satisfies the
differential equation [[phi]'(u)]^2 = 4[[phi](u)]^3 - g2[phi](u) - g3.
This equation has therefore one solution which is a single valued
monogenic function with no singularities other than poles for any
finite part of the plane, having in particular for u = 0, a pole of
the second order; and the method adopted for obtaining this near u = 0
shows that the differential equation has no other such solution. This,
however, is not the only solution which is a single valued meromorphic
function, a the functions [phi](u + [alpha]), wherein [alpha] is
arbitrary, being such. Taking now any range of values of u, from u =
0, and putting for any value of u, x = [phi](u), y = -[phi]'(u), so
that y^2 = 4x^3 - g2x - g3, we clearly have
_
/ ([oo]) dx
u = | --;
_/ (x, y) y
conversely if x0 = [phi](u0), y0 = -[phi]'(u0) and [xi], [eta] be any
values satisfying [eta]2 = 4[xi]^2 - g2[xi] - g3, which are
sufficiently near respectively to x0, y0, while v is defined by
_
/ ([xi], [eta]) d[xi]
v - u0 = -| -----,
_/ (x0, y0) [eta]
then [xi], [eta] are respectively [phi](v) and -[phi]'(v); for this
equation leads to an expansion for [xi]-x0 in terms of v = u0 and only
one such expansion, and this is obtained by the same work as would be
necessary to expand [phi](v) when v is near to u0; the function
[phi](u) can therefore be continued by the help of this equation, from
v = u0, provided the lower limit of |[xi] - x0| necessary for the
expansions is not zero in the neighbourhood of any value (x0, y0). In
fact the function [phi](u) can have only a finite number of poles in
any finite part of the plane of u; each of these can be surrounded by
a small circle, and in the portion of the finite part of the plane of
u which is outside these circles, the lower limit of the radii of
convergence of the expansions of [phi](u) is greater than zero; the
same will therefore be the case for the lower limit of the radii |[xi]
- x0| necessary for the continuations spoken of above provided that
the values of ([xi], [eta]) considered do not lead to infinitely
increasing values of v; there does not exist, however, any definite
point ([xi]0, [eta]0) in the neighbourhood of which the integral [int]
[([xi], [eta]) to (x0, y0)] d[xi]/[eta] increases indefinitely, it is
only by a path of infinite length that the integral can so increase.
We infer therefore that if ([xi],[eta]) be any point, where [eta]2 =
4[xi]^3 - g2[xi] - g3, and v be defined by
_
/ ([oo]) dx
v = | -- ,
_/ ([xi], [eta]) y
then [xi] = [phi](v) and [eta] = -[phi]'(v). Thus this equation
determines ([xi], [eta]) without ambiguity. In particular the additive
indeterminatenesses of the integral obtained by closed circuits of the
point of integration are periods of the function [phi](u); by
considerations advanced above it appears that these periods are sums
of integral multiples of two which may be taken to be
_ _
/ [oo] dx / [oo] dx
[omega] = 2| --, [omega]' = 2 | --;
_/ e1 y _/ e3 y
these quantities cannot therefore have a real ratio, for else, being
periods of a monogenic function, they would, as we have previously
seen, be each integral multiples of another period; there would then
be a closed path for (x, y), starting from an arbitrary point (x0,
y0), other than one enclosing two of the points (e1, 0), (e2, 0), (e3,
0), ([oo], [oo]), which leads back to the initial point (x0, y0),
which is impossible. On the whole, therefore, it appears that the
function [phi](u) agrees with the function RN(u) previously discussed,
and the discussion of the elliptic integrals can be continued in the
manner given under S 14, _Doubly Periodic Functions_.
S 21. _Modular Functions._--One result of the previous theory is the
remarkable fact that if
_ _
/ [oo] dx / [oo] dx
[omega] = 2| --, [omega]' = 2 | --,
_/ e1 y _/ e3 y
where y^2 = 4(x - e1) (x - e2) (x - e3), then we have
e1 = (1/2[omega])^(-2) + [Sigma]' {[(m + 1/2)[omega] + m'[omega]']^(-2) -
[m[omega] + m'[omega]']^(-2)},
and a similar equation for e3, where the summation refers to all integer
values of m and m' other than the one pair m = 0, m' = 0. This, with
similar results, has led to the consideration of functions of the
complex ratio [omega]'/[omega].
It is easy to see that the series for RN(u), u^(-2) + [Sigma] [(u +
m[omega] + m'[omega]')^2-(m[omega] + m'[omega]')^2], is unaffected by
replacing [omega], [omega]' by two quantities [Omega], [Omega]' equal
respectively to p[omega] + q[omega]', p'[omega]' + q'[omega]', where
p, q, p', q' are any integers for which pq' - p'q = [+-]1; further it
can be proved that all substitutions with integer coefficients [Omega]
= p[omega] + q[omega]', [Omega]' = p'[omega] + q'[omega]', wherein pq'
- p'q = 1, can be built up by repetitions of the two particular
substitutions ([Omega] = -[omega]', [Omega]' = [omega]), ([Omega] =
[omega], [Omega]' = [omega] + [omega]'). Consider the function of the
ratio [omega]'/[omega] expressed by
h = -RN (1/2[omega]') / RN(1/2[omega]);
it is at once seen from the properties of the function RN(u) that by
the two particular substitutions referred to we obtain the
corresponding substitutions for h expressed by
h' = 1/h, h' = 1 - h;
thus, by all the integer substitutions [Omega] = p[omega] + q[omega]',
[Omega]' = p'[omega] + q'[omega]', in which pq' - p'q = 1, the
function h can only take one of the six values h, 1/h, 1 - h, 1/(1 -
h), h/(h - 1), (h - 1)/h, which are the roots of an equation in
[theta],
(1 - [theta] + [theta]^2)^3 (1 - h + h^2)^3
--------------------------- = ---------------;
[theta]^2(1 - [theta])^2 h^2(1 - h)^2
the function of [tau], = [omega]'/[omega], expressed by the right
side, is thus unaltered by every one of the substitutions [tau]' = (p'
+ q'[tau] / p + q[tau]), wherein p, q, p', q' are integers having pq'
- p'q = 1. If the imaginary part [sigma], of [tau], which we may write
[tau] = [rho] + i[sigma], is positive, the imaginary part of [tau]',
which is equal to [sigma](pq' - p'q)/[(p + q[rho])^2 + q^2[sigma]^2],
is also positive; suppose [sigma] to be positive; it can be shown that
the upper half of the infinite plane of the complex variable [tau] can
be divided into regions, all bounded by arcs of circles (or straight
lines), no two of these regions overlapping, such that any
substitution of the kind under consideration, [tau]' = (p' +
q'[tau])/(p + q[tau]) leads from an arbitrary point [tau], of one of
these regions, to a point [tau]' of another; taking [tau] = [rho] +
i[sigma], one of these regions may be taken to be that for which -1/2
< [rho] < 1/2, [rho]^2 + [sigma]^2 > 1, together with the points for
which [rho] is negative on the curves limiting this region; then every
other region is obtained from this so-called fundamental region by one
and only one of the substitutions [tau] = (p' + q'[tau])/(p + q[tau]),
and hence by a definite combination of the substitutions [tau]' =
-1/[tau], [tau]' = 1 + [tau]. Upon the infinite half plane of [tau],
the function considered above,
4 [RN^2(1/2[omega]) + RN(z(1/2[omega]) RN(1/2[omega]') + RN^2 (1/2[omega]')]^3
z([tau]) = -- ----------------------------------------------------------------------------
27 RN^2(1/2[omega]) RN^2(1/2[omega]') [RN(1/2[omega]) + RN(1/2[omega]')]^2
is a single valued monogenic function, whose only essential
singularities are the points [tau]' = (p' + q'[tau])/(p + q[tau]) for
which [tau] = [oo], namely those for which [tau]' is any real rational
value; the real axis is thus a line over which the function z([tau])
cannot be continued, having an essential singularity in every arc of
it, however short; in the fundamental region, z([tau]) has thus only
the single essential singularity, r = [rho] + i[sigma], where [sigma]
= [oo]; in this fundamental region z([tau]) takes any assigned complex
value just once, the relation z([tau]') = z([tau]) requiring, as can
be shown, that [tau]' is of the form (p' + q'[tau])/(p + q[tau]), in
which p, q, p', q' are integers with pq' - p'q = 1; the function
z([tau]) has thus a similar behaviour in every other of the regions.
The division of the plane into regions is analogous to the division of
the plane, in the case of doubly periodic functions, into
parallelograms; in that case we considered only functions without
essential singularities, and in each of the regions the function
assumed every complex value twice, at least. Putting, as another
function of [tau], J([tau]) = z([tau])[z([tau]) - 1], it can be shown
that J([tau]) = 0 for [tau] = exp (2/3[pi]i), that J([tau]) = 1 for
[tau] = i, these being values of [tau] on the boundary of the
fundamental region; like z([tau]) it has an essential singularity for
[tau] = [rho] + i[sigma], [sigma] = + [oo]. In the theory of linear
differential equations it is important to consider the inverse
function [tau](J); this is infinitely many valued, having a cycle of
three values for circulation of J about J = 0 (the circuit of this
point leading to a linear substitution for [tau] of period 3, such as
[tau]' = -(1 + [tau])^(-1)), having a cycle of two values about J = 1
(the circuit leading to a linear substitution for [tau] of period 2,
such as [tau]' = -[tau]^(-1)), and having a cycle of infinitely many
values about J = [oo] (the circuit leading to a linear substitution
for [tau] which is not periodic, such as [tau]' = 1 + [tau]). These
are the only singularities for the function [tau](J). Each of the
functions
_ _
| RN(1/2[omega]) + 2RN(1/2[omega]') |^(1/8)
[J([tau])]^(1/3), [J([tau])-1]^1/2, | - --------------------------------- | ,
|_ RN(1/2[omega]) - RN(1/2)[omega]') _|
beside many others (see below), is a single valued function of [tau],
and is expressible without ambiguity in terms of the single valued
function of [tau],
/i[pi][tau]\ [oo]
[eta]([tau]) = exp( ---------- ) [Pi] [1-exp (2i[pi]n[tau])],
\ 12 / n=1
/i[pi][tau]\ _[oo]
= exp(----------- ) \ (-1)^m exp [(3m^2 + m) i[pi][tau]].
\ 12 / /_
m = -[oo]
It should be remarked, however, that [eta]([tau]) is not unaltered by
all the substitutions we have considered; in fact
[eta](-[tau]^(-1)) = (-i[tau])1/2[eta]([tau]), [eta](1 + [tau]) =
exp (1/12 i[pi]) [eta]([tau]).
The aggregate of the substitutions [tau]' = (p' + q'[tau])/(p +
q[tau]), wherein p, q, p', q' are integers with pq' - p'q = 1,
represents a _Group_; the function J([tau]), unaltered by all these
substitutions, is called a _Modular Function_. More generally any
function unaltered by all the substitutions of a group of linear
substitutions of its variable is called an _Automorphic Function_. A
rational function, of its variable h, of this character, is the
function (1 - h + h^2)^3 h^(-2)(1 - h)^(-2) presenting itself
incidentally above; and there are other rational functions with a
similar property, the group of substitutions belonging to any one of
these being, what is a very curious fact, associable with that of the
rotations of one of the regular solids, about an axis through its
centre, which bring the solid into coincidence with itself. Other
automorphic functions are the double periodic functions already
discussed; these, as we have seen, enable us to solve the algebraic
equation y^2 = 4x^3 - g2x - g3 (and in fact many other algebraic
equations, see below, under S 23, _Geometrical Applications of
Elliptic Functions_) in terms of single valued functions x = RN(u), y
= -RN'(u). A similar utility, of a more extended kind, belongs to
automorphic functions in general; but it can be shown that such
functions necessarily have an infinite number of essential
singularities except for the simplest cases.
The modular function J([tau]) considered above, unaltered by the group
of linear substitutions [tau]' = (p' + q'[tau]) / (p + q[tau]), where
p, q, p', q' are integers with pq' - p'q = 1, may be taken as the
independent variable x of a differential equation of the third order,
of the form
s''' 3 /s''\^2 1 - a^2 1 - [beta]^2 [alpha]^2 + [beta]^2 - [gamma]^2 - 1
---- - -- ( --- ) = ---------- + ------------ + ------------------------------------,
s' 2 \ s'/ 2(x - 1)^2 2x^2 2x(x - 1)
where s' = ds/dx, &c., of which the dependent variable s is equal to
[tau]. A differential equation of this form is satisfied by the
quotient of two independent integrals of the linear differential
equation of the second order satisfied by the hypergeometric
functions. If the solution of the differential equation for s be
written s([alpha], [beta], [gamma], x), we have in fact [tau] = s(1/2,
1/3, 0, J). If we introduce also the function of [tau] given by
2RN (1/2[omega]') + f V(1/2[omega])
[lambda] = -----------------------------------,
RN(1/2[omega]') - RN(1/2[omega])
we similarly have [tau] = s(0, 0, 0, [lambda]); this function [lambda]
is a single valued function of [tau], which is also a modular
function, being unaltered by a group of integral substitutions also of
the form [tau]' = (p' + q'[tau])/(p + q[tau]), with pq' - p'q = 1, but
with the restriction that p' and q are even integers, and therefore p
and q' are odd integers. This group is thus a subgroup of the general
modular group, and is in fact of the kind called a self-conjugate
subgroup. As in the general case this subgroup is associated with a
subdivision of the plane into regions of which any one is obtained
from a particular region, called the fundamental region, by a
particular one of the substitutions of the subgroup. This fundamental
region, putting [tau] = [rho] + i[sigma], may be taken to be that
given by -1 < [rho] < 1, ([rho] + 1/2)^2 + [sigma]^2 > 1/4, ([rho] -
1/2)^2 + [sigma]^2 > 1/4, and is built up of six of the regions which
arose for the general modular group associated with J([tau]). Within
this fundamental region, [lambda] takes every complex value just once,
except the values [lambda] = 0, 1, [oo], which arise only at the
angular points [tau] = 0, [tau] = [oo], [tau] = -1 and the equivalent
point [tau] = 1; these angular points are essential singularities for
the function [lambda]([tau]). For [lambda]([tau]) as for J([tau]), the
region of existence is the upper half plane of [tau], there being an
essential singularity in every length of the real axis, however short.
If, beside the plane of [tau], we take a plane to represent the values
of [lambda], the function [tau] = s(0, 0, 0, [lambda]) being
considered thereon, the values of [tau] belonging to the interior of
the fundamental region of the [tau]-plane considered above, will
require the consideration of the whole of the [lambda]-plane taken
once with the exception of the portions of the real axis lying between
-[oo] and 0 and between 1 and + [oo], the two sides of the first
portion corresponding to the circumferences of the [tau]-plane
expressed by ([rho] + 1/2)^2 + [sigma]^2 = 1/4, ([rho] - 1/2)^2 +
[sigma]^2 = 1/4, while the two sides of the latter portion, for which
[lambda] is real and > 1, correspond to the lines of the [tau]-plane
expressed by [rho] = [+-]1. The line for which [lambda] is real,
positive and less than unity corresponds to the imaginary axis of the
[tau]-plane, lying in the interior of the fundamental region. All the
values of [tau] = s(0, 0, 0, [lambda]) may then be derived from those
belonging to the fundamental region of the [tau]-plane by making
[lambda] describe a proper succession of circuits about the points
[lambda] = 0, [lambda] = 1; any such circuit subjects [tau] to a
linear substitution of the subgroup of [tau] considered, and
corresponds to a change of [tau] from a point of the fundamental
region to a corresponding point of one of the other regions.
S 22. _A Property of Integral Functions deduced from the Theory of
Modular Functions._--Consider now the function exp(z), for finite values
of z; for such values of z, exp(z) never vanishes, and it is impossible
to assign a closed circuit for z in the finite part of the plane of z
which will make the function [lambda] = exp(z) pass through a closed
succession of values in the plane of [lambda] having [lambda] = 0 in its
interior; the function s[0, 0, 0, exp(z)], however z vary in the finite
part of the plane, will therefore never be subjected to those linear
substitutions imposed upon s(0, 0, 0, [lambda]) by a circuit of [lambda]
about [lambda] = 0; more generally, if [phi](z) be an integral function
of z, never becoming either zero or unity for finite values of z, the
function [lambda] = [phi](z), however z vary in the finite part of the
plane, will never make, in the plane of [lambda], a circuit about either
[lambda] = 0 or [lambda] = 1, and s(0, 0, 0, [lambda]), that is s[0, 0,
0, [phi](z)], will be single valued for all finite values of z; it will
moreover remain finite, and be monogenic. In other words, s[0, 0, 0,
[phi](z)] is also an integral function--whose imaginary part, moreover,
by the property of s(0, 0, 0, [lambda]), remains positive for all finite
values of z. In that case, however, exp{is[0, 0, 0, [phi](z)]} would
also be an integral function of z with modulus less than unity for all
finite values of z. If, however, we describe a circle of radius R in the
z plane, and consider the greatest value of the modulus of an integral
function upon this circle, this certainly increases indefinitely as R
increases. We can infer therefore that _an integral function [phi](z)
which does not vanish for any finite value of z, takes the value unity
and hence_ (by considering the function A^(-1)[phi](z)) _takes every
other value for some definite value of z_; or, an integral function for
which both the equations [phi](z) = A, [phi](z) = B are unsatisfied by
definite values of z, does not exist, A and B being arbitrary constants.
A similar theorem can be proved in regard to the values assumed by the
function [phi](z) for points z of modulus greater than R, however
great R may be, also with the help of modular functions. In general
terms it may be stated that it is a very exceptional thing for an
integral function not to assume every complex value an infinite number
of times.
Another application of modular functions is to prove that the function
s([alpha], [beta], [gamma], [lambda]) is a single valued function of
[tau] = s(0, 0, 0, [lambda]); for, putting [tau]' = ([tau] - i)/([tau]
+ i), the values of [tau]' which correspond to the singular points
[lambda] = 0, 1, [oo] of s([alpha], [beta], [gamma], [lambda]), though
infinite in number, all lie on the circumference of the circle
|[tau]'| = 1, within which therefore s([alpha], [beta], [gamma], x) is
expressible in a form [Sigma] [n = 0 to [oo]] a_n[tau]'^n. More
generally any monogenic function of [lambda] which is single valued
save for circuits of the points [lambda] = 0, 1, [oo], is a single
valued function of [tau] = s(0, 0, 0, [lambda]). Identifying [lambda]
with the square of the modulus in Legendre's form of the elliptical
integral, we have [tau] = iK'/K, where
_ _
/1 dt /1 dt
K = | ----------------------------------, K' = | ---------------------------------------;
_/0 [root][1 - t^2] [1 - [lambda]t^2] _/0 [root][1 - t^2] [1 - (1 - [lambda])t^2]
functions such as [lambda]^1/4, (1 - [lambda])^1/4, [[lambda](1 -
[lambda])]^1/4, which have only [lambda] = 0, 1, [oo] as singular
points, were expressed by Jacobi as power series in q =
e^(i[pi][tau]), and therefore, at least for a limited range of values
of [tau], as single valued functions of [tau]; it follows by the
theorem given that any product of a root of [lambda] and a root of 1 -
[lambda] is a single valued function of [tau]. More generally the
differential equation
d^2y dy
x(1 - x) ---- + [[gamma] - ([alpha] + [beta] + 1)x] -- -[alpha][beta][gamma] = 0
dx^2 dx
may be solved by expressing both the independent and dependent
variables as single valued functions of a single variable [tau], the
expression for the independent variable being x = [lambda]([tau]).
S 23. _Geometrical Applications of Elliptic Functions._--Consider any
irreducible algebraic equation rational in x, y, f(x, y) = 0, of such a
form that the equation represents a plane curve of order n with 1/2n(n
-3) double points; taking upon this curve n-3 arbitrary fixed points,
draw through these and the double points the most general curve of order
n -2; this will intersect [f] in n(n - 2) - n(n - 3) - (n - 3) = 3
other points, and will contain homogeneously at least 1/2(n - 1)n -
1/2n(n -3) - (n - 3) = 3 arbitrary constants, and so will be of the form
[lambda][phi] + [lambda]1[phi]1 + [lambda]2[phi]2 + ... = 0, wherein
[lambda]3, [lambda]4, ... are in general zero. Put now [xi] =
[phi]1/[phi], [eta] = [phi]2/[phi] and eliminate x, y between these
equations and [f](x, y) = 0, so obtaining a rational irreducible
equation F([xi], [eta]) = 0, representing a further plane curve. To any
point (x, y) of [f] will then correspond a definite point ([xi], [eta])
of F.
For a general position of (x, y) upon [f] the equations [phi]1(x',
y')/[phi](x', y') = [phi]1(x, y)/[phi](x, y), [phi]2(x', y')/[phi](x',
y') = [phi]2(x, y)/[phi](x, y), subject to [f](x', y') = 0, will have
the same number of solutions (x', y'); if their only solution is x' =
x, y' = y, then to any position ([xi],[eta]) of F will conversely
correspond only one position (x, y) of [f]. If these equations have
another solution beside (x, y), then any curve [lambda][phi] +
[lambda]1[phi]1 + [lambda]2[phi]2 = 0 which passes (through the double
points of [f] and) through the n - 2 points of [f] constituted by the
fixed n-3 points and a point (x0, y0), will necessarily pass through a
further point, say (x0', y0'), and will have only one further
intersection with [f]; such a curve, with the n - 2 assigned points,
beside the double points, of [f], will be of the form [mu][psi] +
[mu]1[psi]1 + ... = 0, where [mu]2, [mu]3, ... are generally zero;
considering the curves [psi] + t[psi]1 = 0, for variable t, one of
these passes through a further arbitrary point of [f], by choosing t
properly, and conversely an arbitrary value of t determines a single
further point of [f]; the co-ordinates of the points of [f] are thus
rational functions of a parameter t, which is itself expressible
rationally by the co-ordinates of the point; it can be shown
algebraically that such a curve has not 1/2(n - 3)n but 1/2(n - 3)n +
1 double points. We may therefore assume that to every point of F
corresponds only one point of [f], and there is a birational
transformation between these curves; the coefficients in this
transformation will involve rationally the co-ordinates of the n-3
fixed points taken upon [f], that is, at the least, by taking these to
be consecutive points, will involve the co-ordinates of one point of
[f], and will not be rational in the coefficients of [f] unless we can
specify a point of [f] whose co-ordinates are rational in these. The
curve F is intersected by a straight line a[xi] + b[eta] + c = 0 in as
many points as the number of unspecified intersections of [f] with
a[phi] + b[phi]1 + c[phi]2 = 0, that is, 3; or F will be a cubic
curve, without double points.
Such a cubic curve has at least one point of inflection Y, and if a
variable line YPQ be drawn through Y to cut the curve again in P and
Q, the locus of a point R such that YR is the harmonic mean of YP and
YQ, is easily proved to be a straight line. Take now a triangle of
reference for homogeneous co-ordinates XYZ, of which this straight
line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation
of the cubic curve will then be of the form
ZY^2 = aX^3 + bX^2Z + cXZ^2 + dZ^3;
by putting X equal to [lambda]X + [mu]Z, that is, choosing a suitable
line through Y to be X = 0, and choosing [lambda] properly, this is
reduced to the form
ZY^2 = 4X^3 - g2XZ^2 -g3Z^3,
of which a representation is given, valid for every point, in terms of
the elliptic functions RN(u), RN'(u), by taking X = ZRN(u), Y =
ZRN'(u). The value of u belonging to any point is definite save for
sums of integral multiples of the periods of the elliptic functions,
being given by
_
/ (x) ZdX - XdZ
u = | ---------,
_/ ([oo]) ZY
where ([oo]) denotes the point of inflection.
It thus appears that the co-ordinates of any point of a plane curve,
[f], of order n with 1/2(n - 3)n double points are expressible as
elliptic functions, there being, save for periods, a definite value of
the argument u belonging to every point of the curve. It can then be
shown that if a variable curve, [phi], of order m be drawn, passing
through the double points of the curve, the values of the argument u
at the remaining intersections of [phi] with [f], have a sum which is
unaffected by variation of the coefficients of [phi], save for
additive aggregates of the periods. In virtue of the birational
transformation this theorem can be deduced from the theorem that if
any straight line cut the cubic y^2 = 4x^3 - g2x - g3, in points (u1),
(u2), (u3), the sum u1 + u2 + u3 is zero, or a period; or the general
theorem is a corollary from Abel's theorem proved under S 17,
_Integrals of Algebraic Functions_. To prove the result directly for
the cubic we remark that the variation of one of the intersections (x,
y) of the cubic with the straight line y = mx + n, due to a variation
[delta]m, [delta]n in m and n, is obtained by differentiation of the
equation for the three abscissae, namely the equation
F(x) = 4x^3 - g2x - g3 - (mx + n)^2 = 0,
and is thus given by
dx x[delta]m + [delta]n
-- = 2 --------------------,
y F'(x)
and the sum of three such fractions as that on the right for the three
roots of F(x) = 0 is zero; hence u1 + u2 + u3 is independent of the
straight line considered; if in particular this become the inflexional
tangent each of u1, u2, u3 vanishes. It may be remarked in passing
that x1 + x2 + x3 = 1/4m^2, and hence is 1/4{(y1 - y2)/(x1 - x2)}^2; so
that we have another proof of the addition equation for the function
RN(u). From this theorem for the cubic curve many of its geometrical
properties, as for example those of its inflections, the properties of
inscribed polygons, of the three kinds of corresponding points, and
the theory of residuation, are at once obvious. And similar results
hold for the curve of order n with 1/2(n - 3)n double points.
S 24. _Integrals of Algebraic Functions in Connexion with the Theory of
Plane Curves._--The developments which have been explained in connexion
with elliptic functions may enable the reader to appreciate the vastly
more extensive theory similarly arising for any algebraical
irrationality, [f](x, y) = o.
The algebraical integrals [int] R(x, y)dx associated with this may as
before be divided into those of the _first kind_, which have no
infinities, those of the _second kind_, possessing only algebraical
infinities, and those of the _third kind_, for which logarithmic
infinities enter. Here there is a certain number, p, greater than
unity, of linearly independent integrals of the first kind; and this
number p is unaltered by any birational transformation of the
fundamental equation [f](x, y) = 0; a rational function can be
constructed with poles of the first order at p + 1 arbitrary positions
(x, y), satisfying [f](x, y) = 0, but not with a fewer number unless
their positions are chosen properly, a property we found for the case
p = 1; and p is the number of linearly independent curves of order n-3
passing through the double points of the curve of order n expressed by
[f](x, y) = 0. Again any integral of the second kind can be expressed
as a sum of p integrals of this kind, with poles of the first order at
arbitrary positions, together with rational functions and integrals of
the first kind; and an integral of the second kind can be found with
one pole of the first order of arbitrary position, and an integral of
the third kind with two logarithmic infinities, also of arbitrary
position; the corresponding properties for p = 1 are proved above.
There is, however, a difference of essential kind in regard to the
inversion of integrals of the first kind; if u = [int] R(x, y)dx be
such an integral, it can be shown, in common with all algebraic
integrals associated with [f](x, y) = 0, to have 2p linearly
independent additive constants of indeterminateness; the upper limit
of the integral cannot therefore, as we have shown, be a single valued
function of the value of the integral. The corresponding theorem, if
[int] R_i(x, y)dx denote one of the integrals of the first kind, is
that the p equations
_ _
/ /
| R_i (x1, y1)dx1 + ... + | Ri (x_p, y_p) dx_p = u_i,
_/ _/
determine the rational symmetric functions of the p positions (x1,
y1), ... (x_p, y_p) as single valued functions of the p variables, u1,
... u_p. It is thus necessary to enter into the theory of functions of
several independent variables; and the equation [f](x, y) = 0 is thus
not, in this way, capable of solution by single valued functions of
one variable. That solution in fact is to be sought with the help of
automorphic functions, which, however, as has been remarked, have, for
p > 1, an infinite number of essential singularities.
S 25. _Monogenic Functions of Several Independent Variables._--A
monogenic function of several independent complex variables u_i, ... u_p
is to be regarded as given by an aggregate of power series all
obtainable by continuation from any one of them in a manner analogous to
that before explained in the case of one independent variable. The
singular points, defined as the limiting points of the range over which
such continuation is possible, may either be _poles_, or _polar points
of indetermination_, or _essential singularities_.
A pole is a point (u1^(0), ... u_p^(0)) in the neighbourhood of which
the function is expressible as a quotient of converging power series
in u1 - u1^(0) ... u_p - u_p^(0); of these the denominator series D
must vanish at (u1^(0), ... u_p^(0)), since else the fraction is
expressible as a power series and the point is not a singular point,
but the numerator series N must not also vanish at (u1^(0), ...
u_p^(0)), or if it does, it must be possible to write D = M0, N = MN0,
where M is a converging power series vanishing at (u1^(0), ...
u_p^(0)), and N0 is a converging power series, in (u1 - u1^(0) ... u_p
- u_p^(0)), not so vanishing. A polar point of indetermination is a
point about which the function can be expressed as a quotient of two
converging power series, both of which vanish at the point. As in such
a simple case as (Ax + By) / (ax + by), about x = 0, y = 0, it can be
proved that then the function can be made to approach to any
arbitrarily assigned value by making the variables u1, ... u_p
approach to u1^(0), ... u_p^(0) by a proper path. It is the necessary
existence of such polar points of indetermination, which in case p > 2
are not merely isolated points, which renders the theory essentially
more difficult than that of functions of one variable. An essential
singularity is any which does not come under one of the two former
descriptions and includes very various possibilities. A point at
infinity in this theory is one for which any one of the variables u1,
... u_p is indefinitely great; such points are brought under the
preceding definitions by means of the convention that for u_i^(0) =
[oo], the difference u_i - u_i^(0) is to be understood to stand for
u_i^(-1) . This being so, a single valued function of u1, ... u_p
without essential singularities for infinite or finite values of the
variables can be shown, by induction, to be, as in the case of p = 1,
necessarily a rational function of the variables. A function having no
singularities for finite values of all the variables is as before
called an integral function; it is expressible by a power series
converging for all finite values of the variables; a single valued
function having for finite values of the variables no singularities
other than poles or polar points of indetermination is called a
meromorphic function; as for p = 1 such a function can be expressed as
a quotient of two integral functions having no common zero point other
than the points of indetermination of the function; but the proof of
this theorem is difficult.
The single valued functions which occur, as explained above, in the
inversion of algebraic integrals of the first kind, for p > 1, are
meromorphic. They must also be periodic, unaffected that is when the
variables u1, ... u_p are _simultaneously_ increased each by a proper
constant, these being the additive constants of indeterminateness for
the p integrals [int] R_i(x, y)dx arising when (x, y) makes a closed
circuit, the same for each integral. The theory of such single valued
meromorphic periodic functions is simpler than that of meromorphic
functions of several variables in general, as it is sufficient to
consider only finite values of the variables; it is the natural
extension of the theory of doubly periodic functions previously
discussed. It can be shown to reduce, though the proof of this
requires considerable developments of which we cannot speak, to the
theory of a single integral function of u1, ... u_p, called the _Theta
Function_. This is expressible as a series of positive and negative
integral powers of quantities exp (c1u1), exp (c2u2), ... exp (c_p
u_p), wherein c1, ... c_p are proper constants; for p = 1 this theta
function is essentially the same as that above given under a different
form (see S 14, _Doubly Periodic Functions_), the function [sigma](u).
In the case of p = 1, all meromorphic functions periodic with the same
two periods have been shown to be rational functions of two of them
connected by a single algebraic equation; in the same way all
meromorphic functions of p variables, periodic with the same sets of
simultaneous periods, 2p sets in all, can be shown to be expressible
rationally in terms of p + 1 such periodic functions connected by a
single algebraic equation. Let x1, ... x_p, y denote p + 1 such
functions; then each of the partial derivatives dx_i/(Pd)u_i will
equally be a meromorphic function of the same periods, and so
expressible rationally in terms of x1, ... x_p, y; thus there will
exist p equations of the form
dx_i = R1 du1 + ... + R_p du_p,
and hence p equations of the form
du_i = H_(i, 1)dx1 + ... + H_(i, p)dx_p,
wherein H_(i, j) are rational functions of x1, ... x_p, y, these being
connected by a fundamental algebraic (rational) equation, say [f](x1,
... x_p, y) = 0. This then is the generalized form of the
corresponding equation for p = 1.
S 26. _Multiply-Periodic Functions and the Theory of Surfaces._--The
theory of algebraic integrals [int] R(x, y)dx, wherein x, y are
connected by a rational equation [f](x, y) = 0, has developed
concurrently with the theory of algebraic curves; in particular the
existence of the number p invariant by all birational transformations is
one result of an extensive theory in which curves capable of birational
correspondence are regarded as equivalent; this point of view has made
possible a general theory of what might otherwise have remained a
collection of isolated theorems.
In recent years developments have been made which point to a similar
unity of conception as possible for surfaces, or indeed for algebraic
constructs of any number of dimensions. These developments have been
in two directions, at first followed independently, but now happily
brought into the most intimate connexion. On the analytical side, E.
Picard has considered the possibility of classifying integrals of the
form [int](Rds + Sdy), belonging to a surface [f](x, y, z) = 0,
wherein R and S are rational functions of x, y, z, according as they
are (1) everywhere finite, (2) have poles, which then lie along curves
upon the surface, or (3) have logarithmic infinities, also then lying
along curves, and has brought the theory to a high degree of
perfection. On the geometrical side A. Clebsch and M. Noether, and
more recently the Italian school, have considered the geometrical
characteristics of a surface which are unaltered by birational
transformation. It was first remarked that for surfaces of order n
there are associated surfaces of order n-4, having properties in
relation thereto analogous to those of curves of order n-3 for a plane
curve of order n; if such a surface [f](x, y, z) = 0 have a double
curve with triple points triple also for the surface, and [phi](x, y,
z) = 0 be a surface of order n - 4 passing through the double curve,
the double integral
_ _
/ / [phi] dx dy
| | -----------
_/ _/ (Pd)f/(Pd)z
is everywhere finite; and, the most general everywhere finite integral
of this form remains invariant in a birational transformation of the
surface [f], the theorem being capable of generalization to algebraic
constructs of any number of dimensions. The number of linearly
independent surfaces of order n - 4, possessing the requisite
particularity in regard to the singular lines and points of the
surface, is thus a number invariant by birational transformation, and
the equality of these numbers for two surfaces is a necessary
condition of their being capable of such transformation. The number of
surfaces of order m having the assigned particularity in regard to the
singular points and lines of the fundamental surface can be given by a
formula for a surface of given singularity; but the value of this
formula for m = n - 4 is not in all cases equal to the actual number
of surfaces of order n - 4 with the assigned particularity, and for a
cone (or ruled surface) is in fact negative, being the negative of the
deficiency of the plane section of the cone. Nevertheless this number
for m = n - 4 is also found to be invariant for birational
transformation. This number, now denoted by p_a, is then a second
invariant of birational transformation. The former number, of actual
surfaces of order n - 4 with the assigned particularity in regard to
the singularities of the surface, is now denoted by p_g. The
difference p_g - p_a, which is never negative, is a most important
characteristic of a surface. When it is zero, as in the case of the
general surface of order n, and in a vast number of other ordinary
cases, the surface is called regular.
On a plane algebraical curve we may consider linear series of sets of
points, obtained by the intersection with it of curves [lambda][phi] +
[lambda]1[phi]1 + ... = 0, wherein [lambda], [lambda]1, ... are
variable coefficients; such a series consists of the sets of points
where a rational function of given poles, belonging to the construct
[f](x, y) = 0, has constant values. And we may consider series of sets
of points determined by variable curves whose coefficients are
algebraical functions, not necessarily rational functions, of
parameters. Similarly on a surface we may consider linear systems of
curves, obtained by the intersection with the given surface of
variable surfaces [lambda][phi] + [lambda]1[phi]1 + ... = 0, and may
consider algebraic systems, of which the individual curve is given by
variable surfaces whose coefficients are algebraical, not necessarily
rational, functions of parameters. Of a linear series upon a plane
curve there are two numbers manifestly invariant in birational
transformation, the _order_, which is the number of points forming a
set of the series, and the _dimension_, which is the number of
parameters [lambda]1/[lambda], [lambda]2/[lambda], ... entering
linearly in the equation of the series. The series is _complete_ when
it is not contained in a series of the same order but of higher
dimension. So for a linear system of curves upon a surface, we have
three invariants for birational transformation; the _order_, being in
the number of variable intersections of two curves of the system, the
_dimension_, being the number of linear parameters [lambda]1/[lambda],
[lambda]2/[lambda], ... in the equation for the system, and the
_deficiency_ of the individual curves of the system. Upon any curve of
the linear system the other curves of the system define a linear
series, called the _characteristic_ series; but even when the linear
system is complete, that is, not contained in another linear system of
the same order and higher dimension, it does not follow that the
characteristic series is complete; it may be contained in a series
whose dimension is greater by p_g - p_a than its own dimension. When
this is so it can be shown that the linear system of curves is
contained in an algebraic system whose dimension is greater by p_g -
p_a than the dimension of the linear system. The extra p = p_g - p_a
variable parameters so entering may be regarded as the independent
co-ordinates of an algebraic construct [f](y, x1, ... x_p) = 0; this
construct has the property that its co-ordinates are single valued
meromorphic functions of p variables, which are periodic, possessing
2p systems of periods; the p variables are expressible in the forms
_
/
u_i = | R1(x, y) dx1 + ... + R_p(x, y) dx_p,
_/
wherein R_i(x, y) denotes a rational function of x1, ... x_p and y.
The original surface has correspondingly p integrals of the form
[int](R dx + S dy), wherein R, S are rational in x, y, z, which are
everywhere finite; and it can be shown that it has no other such
integrals. From this point of view, then, the number p, = p_g - p_a
is, for a surface, analogous to the deficiency of a plane curve;
another analogy arises in the comparison of the theorems: for a plane
curve of zero deficiency there exists no algebraic series of sets of
points which does not consist of sets belonging to a linear series;
for a surface for which p_g - p_a = 0 there exists no algebraic system
of curves not contained in a linear system.
But whereas for a plane curve of deficiency zero, the co-ordinates of
the points of the curve are rational functions of a single parameter,
it is not necessarily the case that for a surface having p_g - p_a = 0
the co-ordinates of the points are rational functions of two
parameters; it is necessary that p_g - p_a = 0, but this is not
sufficient. For surfaces, beside the p_g linearly independent surfaces
of order n - 4 having a definite particularity at the singularities of
the surface, it is useful to consider surfaces of order k(n - 4), also
having each a definite particularity at the singularities, the number
of these, not containing the original surface as component, which are
linearly independent, is denoted by P_k. It can then be stated that a
sufficient condition for a surface to be rational consists of the two
conditions p_a = 0, P2 = 0. More generally it becomes a problem to
classify surfaces according to the values of the various numbers which
are invariant under birational transformation, and to determine for
each the simplest form of surface to which it is birationally
equivalent. Thus, for example, the hyperelliptic surface discussed by
Humbert, of which the co-ordinates are meromorphic functions of two
variables of the simplest kind, with four sets of periods, is
characterized by p_g = 1, p_a = -1; or again, any surface possessing a
linear system of curves of which the order exceeds twice the
deficiency of the individual curves diminished by two, is reducible by
birational transformation to a ruled surface or is a rational surface.
But beyond the general statement that much progress has already been
made in this direction, of great interest to the student of the theory
of functions, nothing further can be added here.
BIBLIOGRAPHY.--The learner will find a lucid introduction to the
theory in E. Goursat, _Cours d'analyse mathematique_, t. ii. (Paris,
1905), or, with much greater detail, in A.R. Forsyth, _Theory of
Functions of a Complex Variable_ (2nd ed., Cambridge, 1900); for
logical rigour in the more difficult theorems, he should consult W.F.
Osgood, _Lehrbuch der Functionentheorie_, Bd. i. (Leipzig, 1906-1907);
for greater precision in regard to the necessary quasi-geometrical
axioms, beside the indications attempted here, he should consult W.H.
Young, _The Theory of Sets of Points_ (Cambridge, 1906), chs.
viii.-xiii., and C. Jordan, _Cours d'analyse_, t. i. (Paris, 1893),
chs. i., ii.; a comprehensive account of the _Theory of Functions of
Real Variables_ is by E.W. Hobson (Cambridge, 1907). Of the theory
regarded as based after Weierstrass upon the theory of power series,
there is J. Harkness and F. Morley, _Introduction to the Theory of
Analytic Functions_ (London, 1898), an elementary treatise; for the
theory of the convergence of series there is also T.J. I'A. Bromwich,
_An Introduction to the Theory of Infinite Series_ (London, 1908); but
the student should consult the collected works of Weierstrass (Berlin,
1894 ff.), and the writings of Mittag-Leffler in the early volumes of
the _Acta mathematica_; earlier expositions of the theory of functions
on the basis of power series are in C. Meray, _Lecons nouvelles sur
l'analyse infinitesimale_ (Paris, 1894), and in Lagrange's books on
the Theory of Functions. An account of the theory of potential in its
applications to the present theory is found in most treatises; in
particular consult E. Picard, _Traite d'analyse_, t. ii. (Paris,
1893). For elliptic functions there is an introductory book, P. Appell
and E. Lacour, _Principes de la theorie des fonctions elliptiques et
applications_ (Paris, 1897), beside the treatises of G.H. Halphen,
_Traite des fonctions elliptiques et de leurs applications_ (three
parts, Paris, 1886 ff.), and J. Tannery et J. Molk, _Elements de la
theorie des fonctions elliptiques_ (Paris, 1893 ff.); a book, A.G.
Greenhill, _The Applications of Elliptic Functions_ (London, 1892),
shows how the functions enter in problems of many kinds. For modular
functions there is an extensive treatise, F. Klein and R. Fricke,
_Theorie der elliptischen Modulfunctionen_ (Leipzig, 1890); see also
the most interesting smaller volume, F. Klein, _Uber das Ikosaeder_
(Leipzig, 1884) (also obtainable in English). For the theory of
Riemann's surface, and algebraic integrals, an interesting
introduction is P. Appeil and E. Goursat, _Theorie des fonctions
algebriques et de leurs integrales_; for Abelian functions see also H.
Stahl, _Theorie der Abel'schen Functionen_ (Leipzig, 1896), and H.F.
Baker, _An Introduction to the Theory of Multiply Periodic Functions_
(Cambridge, 1907), and H.F. Baker, _Abel's Theorem and the Allied
Theory, including the Theory of the Theta Functions_ (Cambridge,
1897); for theta functions of one variable a standard work is C.G.
Jacobi, _Fundamenta nova, &c._ (Konigsberg, 1828); for the general
theory of theta functions, consult W. Wirtinger, _Untersuchungen uber
Theta-Functionen_ (Leipzig, 1895). For a history of the theory of
algebraic functions consult A. Brill and M. Noether, _Die Entwicklung
der Theorie der algebraischen Functionen in alterer und neuerer Zeit,
Bericht der deutschen Mathematiker-Vereinigung_ (1894); and for a
special theory of algebraic functions, K. Hensel and G. Landsberg,
_Theorie der algebraischen Function u.s.w._ (Leipzig, 1902). The
student will, of course, consult also Riemann's and Weierstrass's
_Ges. Werke_. For the applications to geometry in general an important
contribution, of permanent value, is E. Picard and G. Simart, _Theorie
des fonctions algebriques de deux variables independantes_ (Paris,
1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable
summary by MM. Castelnuovo and Enriques, _Sur quelques resultats
nouveaux dans la theorie des surfaces algebriques_, containing many
references to the numerous memoirs to be found, for the most part, in
the transactions of scientific societies and the mathematical journals
of Italy.
Beside the books above enumerated there exists an unlimited number of
individual memoirs, often of permanent importance and only
imperfectly, or too elaborately, reproduced in the pages of the
volumes in which the student will find references to them. The German
_Encyclopaedia of Mathematics_, and the Royal Society's _Reference
Catalogue of Current Scientific Literature, Pure Mathematics_,
published yearly, should also be consulted. (H. F. Ba.)
FOOTNOTE:
[1] The word "function" (from Lat. _fungi_, to perform) has many
uses, with the fundamental sense of an activity special or proper to
an office, business or profession, or to an organ of an animal or
plant, the definite work for which the organ is an apparatus. From
the use of the word, as in the Italian _funzione_, for a ceremony of
the Roman Church, "function" is often employed for a public ceremony
of any kind, and loosely of a social entertainment or gathering.
FUNDY, BAY OF, an inlet of the North Atlantic, separating New Brunswick
from Nova Scotia. It is 145 m. long and 48 m. wide at the mouth, but
gradually narrows towards the head, where it divides into Chignecto Bay
to the north, which subdivides into Shepody Bay and Cumberland Basin
(the French Beaubassin), and Minas Channel, leading into Minas Basin, to
the east and south. Off its western shore opens Passamaquoddy Bay, a
magnificent sheet of deep water with good anchorage, receiving the
waters of the St Croix river and forming part of the boundary between
New Brunswick and the state of Maine, The Bay of Fundy is remarkable for
the great rise and fall of the tide, which at the head of the bay has
been known to reach 62 ft. In Passamaquoddy Bay the rise and fall is
about 25 ft., which gradually increases toward the narrow upper reaches.
At spring tides the water in the Bay of Fundy is 19 ft. higher than it
is in Bay Verte, in Northumberland Strait, only 15 m. distant. Though
the bay is deep, navigation is rendered dangerous by the violence and
rapidity of the tide, and in summer by frequent fogs. At low tide, at
such points as Moncton or Amherst, only an expanse of red mud can be
seen, and the tide rushes in a bore or crest from 3 to 6 ft. in height.
Large areas of fertile marshes are situated at the head of the bay, and
the remains of a submerged forest show that the land has subsided in the
latest geological period at least 40 ft. The bay receives the waters of
the St Croix and St John rivers, and has numerous harbours, of which the
chief are St Andrews (on Passamaquoddy Bay) and St John in New
Brunswick, and Digby and Annapolis (on an inlet known as Annapolis
Basin) in Nova Scotia. It was first explored by the Sieur de Monts (d.
c. 1628) in 1604 and named by him La Baye Francaise.
FUNERAL RITES, the ceremonies associated with different methods of
disposing of the dead. (See also BURIAL AND BURIAL ACTS; CEMETERY; and
CREMATION.) In general we have little record, except in their tombs, of
races which, in a past measured not merely by hundreds but by thousands
of years, occupied the earth; and exploration of these often furnishes
our only clue to the religions, opinions, customs, institutions and arts
of long vanished societies. In the case of the great culture folks of
antiquity, the Babylonians, Egyptians, Hindus, Persians, Greeks and
Romans, we have, besides their monuments, the evidence of their
literatures, and so can know nearly as much of their rites as we do of
our own. The rites of modern savages not only help us to interpret
prehistoric monuments, but explain peculiarities in our own rituals and
in those of the culture folks of the past of which the significance was
lost or buried under etiological myths. We must not then confine
ourselves to the rites of a few leading races, neglecting their less
fortunate brethren who have never achieved civilization. It is better to
try to classify the rites of all races alike according as they embody
certain leading conceptions of death, certain fears, hopes, beliefs
entertained about the dead, about their future, and their relations with
the living.
The main ideas, then, underlying funeral rites may roughly be
enumerated as follows:
1. The pollution or taboo attaching to a corpse.
2. Mourning.
3. The continued life of the dead as evinced in the housing and
equipment of the dead, in the furnishing of food for them, and in the
orientation and posture assigned to the body.
4. Communion with the dead in a funeral feast and otherwise.
5. Sacrifice for the dead and expiation of their sins.
6. Death witchery.
7. Protection of the dead from ghouls.
8. Fear of ghosts.
1. A dead body is unclean, and the uncleanness extends to things and
persons which touch it. Hence the Jewish law (Num. v. 2) enacted that
"whoever is unclean by the dead shall be put outside the camp, that they
defile not the camp in the midst whereof the Lord dwells." Such persons
were unclean until the even, and might not eat of the holy things unless
they bathed their flesh in water. A high priest might on no account "go
in to any dead body" (Lev. xxi. 11). Why a corpse is so widely tabooed
is not certain; but it is natural to see one reason in the corruption
which in warm climates soon sets in. The common experience that where
one has died another is likely to do so may also have contributed,
though, of course, there was no scientific idea of infection. The old
Persian scriptures are full of this taboo. He who has touched a corpse
is "powerless in mind, tongue and hand" (_Zend Avesta_ in _Sacred Books
of the East_, pt. i. p. 120), and the paralysis is inflicted by the
innumerable _drugs_ or evil spirits which invest a corpse. Fire and
earth, being alike creations of the good and pure god Ahuramazda, a
body must not be burned or buried; and so the ancient Persians and their
descendants the Parsees build Dakmas or "towers of silence" on hill-tops
far from human habitations. Inside these the corpses are laid on a
flagged terrace which drains into a central pit. Twice a year the bones,
picked clean by dogs and birds of prey, are collected in the pit, and
when it is full another tower is built. In ancient times perhaps the
bodies of the magi or priests alone were exposed at such expense; the
common folk were covered with wax and laid in the earth, the wax saving
the earth from pollution. In Rome and Greece the corpse was buried by
night, lest it should pollute the sunlight; and a trough of water was
set at the door of the house of death that men might purify themselves
when they came out, before mixing in general society. Priests and
magistrates in Rome might not meet or look on a corpse, for they were
thereby rendered unclean and incapable of fulfilling their official
duties without undergoing troublesome rites of purification. At a Roman
funeral, when the remains had been laid in the tomb, all present were
sprinkled with lustral water from a branch of olive or laurel called
_aspergillum_; and when they had gone home they were asperged afresh and
stepped over a fire. The house was also swept out with a broom, probably
lest the ghost of the dead should be lying about the floor. Many races,
to avoid pollution, destroy the house and property of the deceased. Thus
the Navahos pull down the hut in which he died, leaving its ruins on the
ground; but if it be an expensive hut, a shanty is extemporized
alongside, into which the dying man is transferred before death. No one
will use the timbers of a hut so ruined. A burial custom of the Solomon
Islands, noted by R.H. Codrington (_The Melanesians_, p. 255), may be
dictated by the same scruple. There "the mourners having hung up a dead
man's arms on his house make great lamentations; all remains afterwards
untouched, the house goes to ruin, mantled, as time goes on, with the
vines of the growing yams, a picturesque and indeed, perhaps, a touching
sight; for these things are not set up that they may in a ghostly manner
accompany their former owner." H. Oldenberg (_Religion des Veda_, p.
426) describes how Hindus shave themselves and cut off their nails after
a death, at the same time that they wash, renew the hearth fire, and
furnish themselves with new vessels. For the hair and nails may harbour
pollution, just as the medieval Greeks believed that evil spirits could
lurk in a man's beard (Leo Allatius, _De opinionibus quorundam
Graecorum_). The dead man's body is shorn and the nails cut for a
kindred reason; for it must be purified as much as can be before it is
burned as an offering on the pyre and before he enters on a new sphere
of existence.
2. We are accustomed to regard mourning costume as primarily an outward
sign of our grief. Originally, however, the special garb seems to have
been intended to warn the general public that persons so attired were
unclean. In ancient Rome mourners stayed at home and avoided all feasts
and amusements; laying aside gold, purple and jewels, they wore black
dresses called _lugubria_ or even skins. They cut neither hair nor
beard, nor lighted fire. Under the emperors women began to wear white.
On the west coast of Africa negroes wear white, on the Gold Coast red.
The Chinese wear hemp, which is cheap, for mourning dress must as a rule
be destroyed when the season of grief is past to get rid of the taboo.
Among the Aruntas of Australia the wives of a dead man smear themselves
with white pipe-clay until the last ceremonies are finished, sometimes
adding ashes--this not to conceal themselves from the ghost (which may
partly be the aim of some mourning costumes), but to show the ghost that
they are duly sorrowing for their loss. These widows must not talk
except on their hands for a whole year. "Among the Maoris," says Frazer
(_Golden Bough_, i. 323), "anyone who had handled a corpse, helped to
convey it to the grave, or touched a dead man's bones; was cut off from
all intercourse and almost all communication with mankind. He could not
enter any house, or come into contact with any person or thing, without
utterly bedevilling them. He might not even touch food with his hands,
which had become so frightfully tabooed or unclean as to be quite
useless. Food would be set for him on the ground, and he would then sit
or kneel down, and, with his hands carefully held behind his back, would
gnaw at it as best he could." Often a degraded outcast was kept in a
village to feed mourners. Such a taboo is strictly similar to those
which surround a sacred chief or his property, a menstruous woman or a
homicide, rendering them dangerous to themselves and to all who approach
them.
3. Primitive folk cannot conceive of a man's soul surviving apart from
his body, nor of another life as differing from this, and the dead must
continue to enjoy what they had here. Accordingly the Patagonians kill
horses at the grave that the dead may ride to _Alhuemapu_, or country of
the dead. After a year they collect a chief's bones, arrange them, tie
them together and dress them in his best garments with beads and
feathers. Then they lay him with his weapons in a square pit, round
which dead horses are placed set upright on their feet by stakes. As
late as 1781 in Poland F. Casimir's horse was slain and buried with him.
In the Caucasus a Christian lady's jewels are buried with her. The
Hindus used to burn a man's widow on his pyre, because he could not do
without her; and St Boniface commends the self-sacrifice of the Wend
widows who in his day burned themselves alive on their husbands' pyres.
The tumuli met with all over the north of Europe (in the Orkneys alone
2000 remain) are regular houses of the dead, models of those they
occupied in life. The greater the dignity of the deceased, the loftier
was his barrow. Silbury hill is 170 ft. high; the tomb of Alyattes,
father of Croesus, was a fourth of a league round; the Pyramids are
still the largest buildings in existence; at Oberea in Tahiti is a
barrow 267 ft. long, 87 wide and 44 high. Some Eskimo just leave a dead
man's body in his house, and shut it up, often leaving by his side a
dog's head to guide him on his last journey, along with his tools and
kayak. The Sea Dyaks set a chief adrift in his war canoe with his
weapons. So in Norse story Hake "was laid wounded on a ship with the
dead men and arms; the ship was taken out to sea and set on fire." The
Viking was regularly buried in his ship or boat under a great mound. He
sailed after death to Valhalla. In the ship was laid a stone as anchor
and the tools, clothes, weapons and treasures of the dead. The
Egyptians, whose land was the gift of the river Nile, equally believed
that the dead crossed over water, and fashioned the hearse in the form
of a boat. Hence perhaps was derived the Greek myth of Charon and the
Styx, and the custom, which still survives in parts of Europe, of
placing a coin in the mouth of the dead with which to pay the ferryman.
The Egyptians placed in the tomb books of a kind to guide the dead to
the next world. The Copts in a later age did the same, and to this
custom we owe the recovery in Egypt of much ancient literature. The
Armenians till lately buried with a priest his missal or gospel.
In Egyptian entombments of the XIIth to the XIVth dynasties were added
above the sepulchres what Professor Petrie terms soul-houses, viz. small
models of houses furnished with couch and table, &c., for the use of the
_ka_ or double whenever it might wish to come above ground and partake
of meats and drinks. They recall, in point of size, the hut-urns of the
Etruscans, but the latter had another use, for they contain incinerated
remains. Etruscan tombs, like those of Egypt and Asia Minor, were made
to resemble the dwelling-houses of the living, and furnished with
coffered ceilings, panelled walls, couches, stools, easy chairs with
footstools attached, all hewn out of the living rock (Dennis, _Cities
and Cemeteries of Etruria_, vol i. p. lxx.).
Of the old Peruvian mummies in the Kircherian Museum at Rome, several
are of women with babies in their arms, whence it is evident that a
mother had her suckling buried with her; it would console her in the
next world and could hardly survive her in this. The practice of burying
ornaments, tools and weapons with the dead characterizes the inhumations
of the Quaternary epoch, as if in that dim and remote age death was
already regarded as the portal of another life closely resembling this.
The cups, tools, weapons, ornaments and other articles deposited with
the dead are often carefully broken or turned upside down and inside
out; for the soul or _manes_ of objects is liberated by such fracture or
inversion and so passes into the dead man's use and possession. For the
same reason where the dead are burned, their properties are committed to
the flames. The ghost of the warrior has a ghostly sword and buckler to
fight with and a ghostly cup to drink from, and he is also nourished by
the impalpable odour and reek of the animal victims sacrificed over his
grave. Instead of valuable objects cheap images and models are often
substituted; and why not, if the mere ghosts of the things are all that
the wraith can enjoy? Thus Marco Polo (ii. 76) describes how in the land
of Kinsay (Hang-chau) "the friends and relations make a great mourning
for the deceased, and clothe themselves in hempen garments, and follow
the corpse, playing on a variety of instruments and singing hymns to
their idols. And when they come to the burning place they take
representations of things cut out of parchment, such as caparisoned
horses, male and female slaves, camels, armour, suits of cloth of gold
(and money), in great quantities, and these things they put on the fire
along with the corpse so that they are all burned with it. And they tell
you that the dead man shall have all these slaves and animals of which
the effigies are burned, alive in flesh and blood, and the money in
gold, at his disposal in the next world; and that the instruments which
they have caused to be played at his funeral, and the idol hymns that
have been chaunted shall also be produced again to welcome him in the
next world." The manufacture of such paper _simulacra_ for consumption
at funerals is still an important industry in Chinese cities. The
ancient Egyptians, assured that a man's _ka_ or double shall revivify
his body, took pains to guard the flesh from corruption, steeping the
corpse in natron and stuffing it with spices. A body so prepared is
called a mummy (q.v.), and the custom was already of a hoary antiquity
in 3200 B.C., when the oldest dated mummy we have was made. The bowels,
removed in the process, were placed in jars over the corpse in the tomb,
together with writing tablets, books, musical instruments, &c., of the
dead. Cemeteries also remain full of mummies of crocodiles, cats, fish,
cows and other sacred animals. The Greeks settled in Egypt learned to
mummify their dead, but the custom was abhorrent to the Jews, although
the Christian belief in the resurrection of the flesh must have been
formed to a large extent under Egyptian influence. Half the superiority
of the Jewish to other ancient religions lay in this, that it prescribed
no funeral rites other than the simplest inhumation.
The dead all over the world and from remote antiquity have been laid not
anyhow in the earth, but with the feet and face towards the region in
which their future will be spent; the Samoans and Fijians towards the
far west whither their souls have preceded them; the Guarayos with head
turned eastwards because their god Tamoi has in that quarter "his happy
hunting grounds where the dead will meet again" (Tylor, _Prim. Cult._
ii. 422). The legend is that Christ was buried with His head to the
west, and the church follows the custom, more ancient than itself, of
laying the dead looking to the East, because that is the attitude of
prayer, and because at the last trump they will hurry eastwards. So in
Eusebius (_Hist. Eccl._ 430.19) a martyr explains to his pagan judge
that the heavenly Jerusalem, the fatherland of the pious, lay exactly in
the east at the rising place of the sun. Where the body is laid out
straight it is difficult to discern the presence of any other idea than
that it is at rest. In Scandinavian barrows, e.g. in the one opened at
Goldhavn in 1830, the skeletons have been found seated on a low stone
bench round the wall of the grave chamber facing its opening, which
always looks south or east, never north. Here the dead were continuing
the drinking bouts they enjoyed on earth.
The Peruvians mummified their dead and placed them jointed and huddled
up with knees to chin, looking toward the sunset, with the hands held
before the face. In the oldest prehistoric tombs along the Nile the
bodies are doubled up in the same position. It would seem as if in these
and numerous other similar cases the dead were deliberately given in
their graves the attitude of a foetus in the womb, and, as Dr Budge
remarks (_Egyptian Ideas of the Future Life_, London, 1899, p. 162), "we
may perhaps be justified in seeing in this custom the symbol of a hope
that, as the child is born from this position into the world, so might
the deceased be born into the life beyond the grave." The late
Quaternary skeletons of the Mentone cave were laid in a layer of
ferrugineous earth specially laid down for them, and have contracted a
red colour therefrom. Many other prehistoric skeletons found in Italy
have a reddish colour, perhaps for the same reason, or because, as often
to-day, the bones were stripped of flesh and painted. Ambrose relates
that the skeletons of the martyrs Gervasius and Protasius, which he
found and deposited A.D. 386 under the altar of his new basilica in
Milan, were _mirae magnitudinis ut prisca aetas ferebat_, and were also
coloured red. He imagined the red to be the remains of the martyrs'
blood! _Hic sanguis clamat coloris indicio._ Salomon Reinach has rightly
divined that what Ambrose really hit upon was a prehistoric tomb. Red
earth was probably chosen as a medium in which to lay a corpse because
demons flee from red. Sacred trees and stones are painted red, and for
the most solemn of their rites savages bedaub themselves with red clay.
It is a favourite taboo colour.
4. A feast is an essential feature of every primitive funeral, and in
the Irish "wake" it still survives. A dead man's soul or double has to
be fed at the tomb itself, perhaps to keep it from prowling about the
homes of the survivors in search of victuals; and such food must also be
supplied to the dead at stated intervals for months or years. Many races
leave a narrow passage or tube open down to the cavity in which the
corpse lies, and through it pour down drinks for the dead. Traces of
such tubes are visible in the prehistoric tombs of the British Isles.
However, such provision of food is not properly a funeral feast unless
the survivors participate. In the Eastern churches and in Russia the
departed are thus fed on the ninth, twelfth and fortieth days from
death. "Ye appease the shades of the dead with wine and meals," was the
charge levelled at the Catholics by the 4th-century Manichaeans, and it
has hardly ceased to be true even now after the lapse of sixteen
centuries. The funeral feast proper, however, is either a meal of
communion with or in the dead, which accompanies interment, or a banquet
off the flesh of victims slain in atonement of the dead man's sins. Some
anthropologists see in the common meal held at the grave "the pledge and
witness of the unity of the kin, the chief means, if not of making, at
least of repairing and renewing it."[1] The flesh provided at these
banquets is occasionally that of the dead man himself; Herodotus and
Strabo in antiquity relate this of several half-civilized races in the
East and West, and a similar story is told by Marco Polo of certain
Tatars. Nor among modern savages are funeral feasts off the flesh of the
dead unknown, and they seem to be intended to effect and renew a
sacramental union or kinship of the living with the dead. The Uaupes in
the Amazons incinerate a corpse a month after death, pound up the ashes,
and mix them with their fermented drink. They believe that the virtues
of the dead will thus be passed on to his survivors. The life of the
tribe is kept inside the tribe and not lost. Such cannibal sacraments,
however, are rare, and, except in a very few cases, the evidence for
them weak. The slaying and eating of animal victims, however, at the
tomb is universal and bears several meanings, separately or all at once.
The animals may be slain in order that their ghosts may accompany the
deceased in his new life. This significance we have already dwelt upon.
Or it is believed that the shade feeds upon them, as the shades came up
from Hades and lapped up out of a trench the blood of the animals slain
by Ulysses. The survivors by eating the flesh of a victim, whose blood
and soul the dead thus consume, sacramentally confirm the mystic tie of
blood kinship with the dead. Or lastly, the victim may be offered for
the sins of the dead. His sins are even supposed to be transferred into
it and eaten by the priest. Such expiatory sacrifices of animals for the
dead survive in the Christian churches of Armenia, Syria and of the East
generally. Their vicarious character is emphasized in the prayers which
accompany them, but the popular understanding of them probably combines
all the meanings above enumerated. It has been suggested by Robertson
Smith (_Religion of the Semites_, 336) that the world-wide customs of
tearing the hair, rending the garments, and cutting and wounding the
body were originally intended to establish a life-bond between the dead
and the living. The survivors, he argues, in leaving portions of their
hair and garments, and yet more by causing their own blood to stream
over the corpse from self-inflicted wounds, by cutting off a finger and
throwing it into the grave, leave what is eminently their own with the
dead, so drawing closer their tie with him. Conversely, many savages
daub themselves with the blood and other effluences of their dead
kinsmen, and explain their custom by saying that in this way a portion
of the dead is incorporated in themselves. Often the survivors,
especially the widows, attach the bones or part of them to their persons
and wear them, or at least keep them in their houses. The retention of
the locks of the deceased and of parts of his dress is equally common.
There is also another side to such customs. Having in their possession
bits of the dead, and being so far in communion with him, the survivors
are surer of his friendship. They have ensured themselves against ghosts
who are apt to be by nature envious and mischievous. But whatever their
original significance, the tearing of cheeks and hair and garments and
cutting with knives are mostly expressions of real sorrow, and, as
Robertson Smith remarks, of deprecation and supplication to an angry god
or spirit. It must not be supposed that the savage or ancient man feels
less than ourselves the poignancy of loss.
6. Death-witchery has close parallels in the witch and heretic hunts of
the Christians, but, happily for us, only flourishes to-day among
savages. Sixty % of the deaths which occur in West Africa are, according
to Miss Mary Kingsley--a credible witness--believed to be due to
witchcraft and sorcery. The blacks regard old age or effusion of blood
as the sole legitimate causes of death. All ordinary diseases are in
their opinion due to private magic on the part of neighbours, just as a
widespread epidemic marks the active hatred "of some great outraged
nature spirit, not of a mere human dabbler in devils."[2] Similarly in
Christian countries an epidemic is set down to the wrath of a God
offended by the presence of Jews, Arians and other heretics. The duty of
an African witch-doctor is to find out who bewitched the deceased, just
as it was of an inquisitor to discover the heretic. Every African
post-mortem accordingly involves the murder of the person or persons who
bewitched the dead man and caused him to die. The death-rate by these
means is nearly doubled; but, since the use of poison against an
obnoxious neighbour is common, the right person is occasionally
executed. It is also well for neighbours not to quarrel, for, if they do
and one of them dies of smallpox, the other is likely to be slain as a
witch, and his lungs, liver and spleen impaled on a pole at the entrance
of the village. It is the same case with the Australian blacks: "no such
thing as natural death is realized by the native; a man who dies has of
necessity been killed by some other man, or perhaps even by a woman, and
sooner or later that man or woman will be attacked. In the normal
condition of the tribe every death meant the killing of another
individual."[3]
7. Lastly, a primitive interment guards against the double risk of the
ghost haunting the living and of ghouls or vampires taking possession of
the corpse. The latter end is likely to be achieved if the body is
cremated, for then there is no nidus to harbour the demon; but whether,
in the remote antiquity to which belong many barrows containing
incinerated remains, this motive worked, cannot be ascertained. The
Indo-European race seems to have cremated at an early epoch, perhaps
before the several races of East and West separated. In Christian
funeral rites many prayers are for the protection of the body from
violation by vampires, and it would seem as if such a motive dictated
the architectural solidity of some ancient tombs. Christian graves were
for protection regularly sealed with the cross; and the following is a
characteristic prayer from the old Armenian rite for the burial of a
layman:
"Preserve, Almighty Lord, this man's spirit with all saints and with
all lovers of Thy holy name. And do Thou seal and guard the sepulchre
of Thy servant, Thou who shuttest up the depths and sealest them with
Thy almighty right hand ... so let the seal of Thy Lordship abide
unmoved upon this man's dwelling-place and upon the shrine which
guards Thy servant. And _let not any filthy and unclean devil dare to
approach him, such as assail the body and souls of the heathen_, who
possess not the birth of the holy font, and have not the dread seal
laid upon their graves."
A terrible and revolting picture of the superstitious belief in ghouls
which violate Christian tombs is given by Leo Allatius (who held it) in
his tract _De opinionibus quorundam Graecorum_ (Paris, 1646). It was
probably the fear of such demonic assaults on the dead that inspired the
insanitary custom of burying the dead under the floors of churches, and
as near as possible to the altar. In the Greek Church this practice was
happily forbidden by the code of Justinian as well as by the older law
in the case of churches consecrated with _Encaenia_ and deposition of
relics. In the Armenian Church the same rule holds, and Ephrem Syrus in
his testament particularly forbade his body to be laid within a church.
Such prohibitions, however, are a witness to the tendency in question.
The custom of lighting candles round a dead body and watching at its
side all night was originally due to the belief that a corpse, like a
person asleep, is specially liable to the assaults of demons. The
practice of tolling a bell at death must have had a similar origin, for
it was a common medieval belief that the sound of a consecrated bell
drives off the demons which when a man dies gather near in the air to
waylay his fleeting soul. For a like reason the consecrated bread of the
Eucharist was often buried with believers, and St Basil is said to have
specially consecrated a Host to be placed in his coffin.
8. Some of the rites described under the previous heads may be really
inspired by the fear of the dead haunting the living, but it must be
kept in mind that the taboo attaching to a dead body is one thing and
fear of a ghost another. A corpse is buried or burned, or scaffolded on
a tree, a tower or a house-top, in order to get it out of the way and
shield society from the dangerous infection of its taboo; but ghosts
_qua_ ghosts need not be feared and a kinsman's ghost usually is not. On
the contrary, it is fed and consoled with everything it needs, is asked
not to go away but to stay, is in a thousand ways assured of the sorrow
and sympathy of the survivors. Even if the body be eaten, it is merely
to keep the soul of the deceased inside the circle of kinsmen, and
Strabo asserts that the ancient Irish and Massagetae regarded it as a
high honour to be so consumed by relatives. In Santa Cruz in Melanesia
they keep the bones for arrow heads and store a skull in a box and set
food before it "saying that this is the man himself" (R.H. Codrington,
_The Melanesians_, p. 264), or the skull and jaw bone are kept and "are
called _mangite_, which are _saka_, hot with spiritual power, and by
means of which the help of the _lio'a_, the powerful ghost of the man
whose relics these are, can be obtained" (ibid. p. 262). Here we have
the savage analogue to Christian relics. So the Australian natives make
pointing sticks out of the small bones of the arm, with which to bewitch
enemies.
We may conclude then that in the most primitive societies, where
blood-kinship is the only social tie and root of social custom it is the
shades, not of kinsmen, but of strangers, who as such are enemies, that
are dangerous and uncanny. In more developed societies, however, all
ghosts alike are held to be so; and if a ghost walks it is because its
body has not been properly interred or because its owner was a
malefactor. Still, even allowing for this, it remains true that for a
friendly ghost the proper place is the grave and not the homes of the
living, and accordingly the Aruntas with cries of _Wah! Wah!_ with
wearing of fantastic head-dresses, wild dancing and beating of the air
with hands and weapons "drive the spirit away from the old camp which it
is supposed to haunt," and which has been set fire to, and hunt it at a
run into the grave prepared, and there stamp it down into the earth.
"The loud shouting of the men and women shows him that they do not wish
to be frightened by him in his present state, and that they will be
angry with him if he does not rest." (Spencer and Gillen, _Native
Tribes of Central Australia_, p. 508). In Mesopotamia cemeteries have
been discovered where the sepulchral jars were set upside down, clearly
by way of hindering the ghosts from escaping into the upper world. In
the Dublin museum we see specimens of ancient Celtic tombs showing the
same peculiarity. For a like reason perhaps the name of the dead must
among the Aruntas not be uttered, nor the grave approached, by certain
classes of kinsmen. The same repugnance to naming the dead exists all
over the world, and leads survivors who share the dead man's name to
adopt another, at least for a time. If the dead man's name was that of a
plant, tree, animal or stream, that too is changed. Here is a potent
cause of linguistic change, that also renders any historical tradition
impossible. The survivors seem to fear that the ghost will come when he
hears his name called; but it also hangs together with the taboo which
hedges round the dead as it does kings, chieftains and priests.
AUTHORITIES.--B. Spencer and F.J. Gillen, _The Native Tribes of
Central Australia_ (London, 1899); F.B. Jevons, _Introduction to
History of Religion_ (London, 1896); E.S. Hartland, _The Legend of
Perseus_, vol. ii.; J.G. Frazer, _The Golden Bough_ (London, 1900);
L.W. Faraday, "Custom and Belief in the Icelandic Sagas," in
_Folk-lore_, vol. xvii. No. 4; E.B. Tylor, _Primitive Culture_
(London, 1903); E.A. W. Budge, _The Mummy_ (Cambridge, 1893); C.
Royer, "Les Rites funeraires aux epoques prehistoriques," _Revue
d'anthropologie_ (1876); Forrer, _Uber die Totenbestattung bei den
Pfahlbauern_ (Ausland, 1885); J. Lubbock, _Origin of Civilization_
(London, 1875) and _Prehistoric Times_ (London, 1865); L.A. Muratori,
"De antiquis Christianorum sepulchris," _Anecd. Graeca_ (Padua, 1709);
Onaphr. Panvinius, _De ritu sepeliendi mortuos apua veteres
Christianos_, reprinted in Volbeding's _Thesaurus_ (Leipzig, 1841).
(F. C. C.)
FOOTNOTES:
[1] E.S. Hartland, _Legend of Perseus_ (1895), ii. 278.
[2] Mary Kingsley, _West African Studies_ (1901), p. 178.
[3] B. Spencer and F.J. Gillen, _The Native Tribes of Central
Australia_ (1899), p. 48.
FUNGI (pl. of Lat. _fungus_, a mushroom), the botanical name covering in
the broad sense all the lower cellular Cryptogams devoid of chlorophyll,
which arise from spores, and the thallus of which is either unicellular
or composed of branched or unbranched tubes or cell-filaments (hyphae)
with apical growth, or of more or less complex wefted sheets or
tissue-like masses of such (mycelium). The latter may in certain cases
attain large dimensions, and even undergo cell-divisions in their
interior, resulting in the development of true tissues. The spores,
which may be uni- or multicellular, are either abstricted free from the
ends of hyphae (acrogenous), or formed from segments in their course
(_chlamydospores_) or from protoplasm in their interior (endogenous).
The want of chlorophyll restricts their mode of life--which is rarely
aquatic--since they are therefore unable to decompose the carbon dioxide
of the atmosphere, and renders them dependent on other plants or
(rarely) animals for their carbonaceous food-materials. These they
obtain usually in the form of carbohydrates from the dead remains of
other organisms, or in this or other forms from the living cells of
their hosts; in the former case they are termed saprophytes, in the
latter parasites. While some moulds (_Penicillium_, _Aspergillus_) can
utilize almost any organic food-materials, other fungi are more
restricted in their choice--e.g. insect-parasites, horn- and
feather-destroying fungi and parasites generally. It was formerly the
custom to include with the Fungi the Schizomycetes or Bacteria, and the
Myxomycetes or Mycetozoa; but the peculiar mode of growth and division,
the cilia, spores and other peculiarities of the former, and the
emission of naked amoeboid masses of protoplasm, which creep and fuse to
streaming plasmodia, with special modes of nutrition and spore-formation
of the latter, have led to their separation as groups of organisms
independent of the true Fungi. On the other hand, lichens, previously
regarded as autonomous plants, are now known to be dual organisms--fungi
symbiotic with algae.
The number of species in 1889 was estimated by Saccardo at about 32,000,
but of these 8500 were so-called _Fungi imperfecti_--i.e. forms of which
we only know certain stages, such as conidia, pycnidia, &c., and which
there are reasons for regarding as merely the corresponding stages of
higher forms. Saccardo also included about 400 species of Myxomycetes
and 650 of Schizomycetes. Allowing for these and for the cases,
undoubtedly not few, where one and the same fungus has been described
under different names, we obtain Schroeter's estimate (in 1892) of
20,000 species. In illustration of the very different estimates that
have been made, however, may be mentioned that of De Bary in 1872 of
150,000 species, and that of Cooke in 1895 of 40,000, and Massee in 1899
of over 50,000 species, the fact being that no sufficient data are as
yet to hand for any accurate census. As regards their geographical
distribution, fungi, like flowering plants, have no doubt their centres
of origin and of dispersal; but we must not forget that every exchange
of wood, wheat, fruits, plants, animals, or other commodities involves
transmission of fungi from one country to another; while the migrations
of birds and other animals, currents of air and water, and so forth, are
particularly efficacious in transmitting these minute organisms. Against
this, of course, it may be argued that parasitic forms can only go where
their hosts grow, as is proved to be the case by records concerning the
introduction of _Puccinia malvacearum_, _Peronospora viticola_,
_Hemileia vastatrix_, &c. Some fungi--e.g. moulds and yeasts--appear to
be distributed all over the earth. That the north temperate regions
appear richest in fungi may be due only to the fact that North America
and Europe have been much more thoroughly investigated than other
countries; it is certain that the tropics are the home of very numerous
species. Again, the accuracy of the statement that the fleshy Agaricini,
Polyporei, _Pezizae_, &c., are relatively rarer in the tropics may
depend on the fact that they are more difficult to collect and remit for
identification than the abundantly recorded woody and coriaceous forms
of these regions. When we remember that many parts of the world are
practically unexplored as regards fungi, and that new species are
constantly being discovered in the United States, Australia and northern
Europe--the best explored of all--it is clear that no very accurate
census of fungi can as yet be made, and no generalizations of value as
to their geographical distribution are possible.
The existence of fossil fungi is undoubted, though very few of the
identifications can be relied on as regards species or genera. They
extend back beyond the Carboniferous, where they occur as hyphae, &c.,
preserved in the fossil woods, but the best specimens are probably those
in amber and in siliceous petrifactions of more recent origin.
[Illustration: FIG. 1.--1, _Peronospora parasitica_ (De Bary). Mycelium
with haustoria (h); 2, _Erysiphe_; A and B, mycelium (m), with haustoria
(h). (After De Bary.)]
_Organs._--Individual hyphae or their branches often exhibit
specializations of form. In many Basidiomycetes minute branches arise
below the septa; their tips curve over the outside of the latter, and
fuse with the cell above just beyond it, forming a _clamp-connexion_.
Many parasitic hyphae put out minute lateral branches, which pierce
the cell-wall of the host and form a peg-like (_Trichosphaeria_),
sessile (_Cystopus_), or stalked (_Hemileia_), knot-like, or a more or
less branched (_Peronospora_) or coiled (_Protomyces_) haustorium. In
_Rhizopus_ certain hyphae creep horizontally on the surface of the
substratum, and then anchor their tips to it by means of a tuft of
short branches (_appressorium_), the walls of which soften and gum
themselves to it, then another branch shoots out from the tuft and
repeats the process, like a strawberry-runner. Appressoria are also
formed by some parasitic fungi, as a minute flattening of the tip of a
very short branch (_Erysiphe_), or the swollen end of any hypha which
comes in contact with the surface of the host (_Piptocephalis_,
_Syncephalis_), haustoria piercing in each case the cell-wall below.
In _Botrytis_ the appressoria assume the form of dense tassels of
short branches. In _Arthrobotrys_ side-branches of the mycelium sling
themselves around the host (_Tylenchus_) much as tendrils round a
support.
Many fungi (_Phallus_, _Agaricus_, _Fumago_, &c.) when strongly
growing put out ribbon-like or cylindrical cords, or sheet-like
mycelial plates of numerous parallel hyphae, all growing together
equally, and fusing by anastomoses, and in this way extend long
distances in the soil, or over the surfaces of leaves, branches, &c.
These mycelial strands may be white and tender, or the outer hyphae
may be hard and black, and very often the resemblance of the
subterranean forms to a root is so marked that they are termed
rhizomorphs. The outermost hyphae may even put forth thinner hyphae,
radiating into the soil like root-hairs, and the convergent tips may
be closely appressed and so divided by septa as to resemble the
root-apex of a higher plant (_Armillaria mellea_).
_Sclerotia._--Fungi, like other plants, are often found to store up
large quantities of reserve materials (oil, glycogen, carbohydrates,
&c.) in special parts of their vegetative tissues, where they lie
accumulated between a period of active assimilation and one of renewed
activity, forming reserves to be consumed particularly during the
formation of large fructifications. These reserve stores may be packed
away in single hyphae or in swollen cells, but the hyphae containing
them are often gathered into thick cords or mycelial strands
(_Phallus_, mushroom, &c.), or flattened and anastomosing ribbons and
plates, often containing several kinds of hyphae (_Merulius
lacrymans_). In other cases the strands undergo differentiation into
an outer layer with blackened, hardened cell-walls and a core of
ordinary hyphae, and are then termed rhizomorphs (_Armillaria
mellea_), capable not only of extending the fungus in the soil, like
roots, but also of lying dormant, protected by the outer casing. Such
aggregations of hyphae frequently become knotted up into dense masses
of interwoven and closely packed hyphae, varying in size from that of
a pin's head or a pea (_Peziza_, _Coprinus_) to that of a man's fist
or head, and weighing 10 to 25 lb. or more (_Polyporus Mylittae_, _P.
tumulosus_, _Lentinus Woermanni_, _P. Sapurema_, &c.). The interwoven
hyphae fuse and branch copiously, filling up all interstices. They
also undergo cutting up by numerous septa into short cells, and these
often divide again in all planes, so that a pseudoparenchyma results,
the walls of which may be thickened and swollen internally, or
hardened and black on the exterior. In many cases the swollen
cell-walls serve as reserves, and sometimes the substance is so
thickly deposited in strata as to obliterate the lumen, and the hyphae
become nodular (_Polyporus sacer_, _P. rhinoceros_, _Lentinus
Woermanni_). The various sclerotia, if kept moist, give rise to the
fructifications of the fungi concerned, much as a potato tuber does to
a potato plant, and in the same way the reserve materials are
consumed. They are principally Polyporei, Agaricini, Pezizae; none are
known among the Phycomycetes, Uredineae or Ustilagineae. The functions
of mycelial strands, rhizomorphs and sclerotia are not only to collect
and store materials, but also to extend the fungus, and in many cases
similar strands act as organs of attack. The same functions of storage
in advance of fructification are also exercised by the stromata so
common in Ascomycetes.
_Tissue Differentiations._--The simpler mycelia consist of hyphae all
alike and thin-walled, or merely differing in the diameter of the
branches of various orders, or in their relations to the environment,
some plunging into the substratum like roots, others remaining on its
surface, and others (aerial hyphae) rising into the air. Such hyphae
may be multicellular, or they may consist of simple tubes with
numerous nuclei and no septa (_Phycomycetes_), and are then
non-cellular. In the more complex tissue-bodies of higher fungi,
however, we find considerable differences in the various layers or
strands of hyphae.
An epidermis-like or cortical protective outer layer is very common,
and is usually characterized by the close septation of the densely
interwoven hyphae and the thickening and dark colour of their outer
walls (sclerotia, _Xylaria_, &c.). Fibre-like hyphae with the lumen
almost obliterated by the thick walls occur in mycelial cords
(_Merulius_). Latex-tubes abound in the tissues of _Lactarius_,
_Stereum_, _Mycena_, _Fistulina_, filled with white or coloured milky
fluids, and Istvanffvi has shown that similar tubes with fluid or oily
contents are widely spread in other Hymenomycetes. Sometimes fatty oil
or watery sap is found in swollen hyphal ends, or such tubes contain
coloured sap. Cystidia and paraphyses may be also classed here. In
_Merulius lacrymans_ Hartig has observed thin-walled hyphae with large
lumina, the septa of which are perforated like those of sieve-tubes.
As regards its composition, the cell-wall of fungi exhibits variations
of the same kind as those met with in higher plants. While the
fundamental constituent is a cellulose in many Mucorini and other
Phycomycetes, in others bodies like pectose, callose, &c., commonly
occur, and Wisselingh's researches show that chitin, a gluco-proteid
common in animals, forms the main constituent in many cases, and is
probably deposited directly as such, though, like the other
substances, it may be mixed with cellulose. As in other cell-walls, so
here the older membranes may be altered by deposits of various
substances, such as resin, calcium oxalate, colouring matters; or more
profoundly altered throughout, or in definite layers, by
lignification, suberization (_Trametes_, _Daedalea_), or swelling to a
gelatinous mucilage (_Tremella_, _Gymnosporangium_), while
cutinization of the outer layers is common. One of the most striking
alterations of cell-walls is that termed _carbonization_, in which the
substance gradually turns black, hard and brittle, as if charred--e.g.
_Xylaria_, _Ustulina_, some sclerotia. At the other extreme the
cell-walls of many lichen-fungi are soft and colourless, but turn blue
in iodine, as does starch. The young cell-wall is always tenuous and
flexible, and may remain so throughout, but in many cases thickenings
and structural differentiations, as well as the changes referred to
above, alter the primary wall considerably. Such thickening may be
localized, and _pits_ (e.g. _Uredospores_, septa of Basidiomycetes),
_spirals_, _reticulations_, _rings_, &c. (capillitium fibres of
_Podaxon_, _Calostoma_, _Battarrea_), occur as in the vessels of
higher plants, while sculptured networks, pittings and so forth are as
common on fungus-spores as they are on pollen grains.
_Cell-Contents._--The cells of fungi, in addition to protoplasm,
nuclei and sap-vacuoles, like other vegetable cells, contain formed
and amorphous bodies of various kinds. Among those directly visible to
the microscope are oil drops, often coloured (_Uredineae_) crystals of
calcium oxalate (_Phallus_, _Russula_), proteid crystals (_Mucor_,
_Pilobolus_, &c.) and resin (Polyporei). The oidia of Erysipheae
contain fibrosin bodies and the hyphae of Saprolegnieae cellulin
bodies, but starch apparently never occurs. Invisible to the
microscope, but rendered visible by reagents, are glycogen, _Mucor_,
Ascomycetes, yeast, &c. In addition to these cell-contents we have
good indirect evidence of the existence of large series of other
bodies, such as proteids, carbohydrates, organic acids, alkaloids,
enzymes, &c. These must not be confounded with the numerous substances
obtained by chemical analysis of masses of the fungus, as there is
often no proof of the manner of occurrence of such bodies, though we
may conclude with a good show of probability that some of them also
exist preformed in the living cell. Such are sugars (glucose, mannite,
&c.), acids (acetic, citric and a whole series of lichen-acids),
ethereal oils and resinous bodies, often combined with the intense
colours of fungi and lichens, and a number of powerful alkaloid
poisons, such as muscarin (_Amanita_), ergotin (_Claviceps_), &c.
Among the enzymes already extracted from fungi are _invertases_
(yeasts, moulds, &c.), which split cane-sugar and other complex sugars
with hydrolysis into simpler sugars such as dextrose and levulose;
_diastases_, which convert starches into sugars (_Aspergillus_, &c.);
_cytases_, which dissolve cellulose similarly (_Botrytis_, &c.);
_peptases_, using the term as a general one for all enzymes which
convert proteids into peptones and other bodies (_Penicillium_, &c.);
lipases, which break up fatty oils (_Empusa_, _Phycomyces_, &c.);
oxydases, which bring about the oxidations and changes of colour
observed in _Boletus_, and _zymase_, extracted by Buchner from yeast,
which brings about the conversion of sugar into alcohol and
carbon-dioxide. That such enzymes are formed in the protoplasm is
evident from the behaviour of hyphae, which have been observed to
pierce cell-membranes, the chitinous coats of insects, artificial
collodion films and layers of wax, &c. That a fungus can secrete more
than one enzyme, according to the materials its hyphae have to attack,
has been shown by the extraction of diastase, inulase, trehalase,
invertase, maltase, raffinase, malizitase, emulsin, trypsin and lipase
from _Aspergillus_ by Bourquelot, and similar events occur in other
fungi. The same fact is indicated by the wide range of organic
substances which can be utilized by _Penicillium_ and other moulds,
and by the behaviour of parasitic fungi which destroy various
cell-contents and tissues. Many of the coloured pigments of fungi are
fixed in the cell-walls or excreted to the outside (_Peziza
aeruginosa_). Matruchot has used them for staining the living
protoplasm of other fungi by growing the two together. Striking
instances of coloured mycella are afforded by _Corticium sanguineum_,
blood-red; _Elaphomyces Leveillei_, yellow-green; _Chlorosplenium
aeruginosum_, verdigris green; and the _Dematei_, brown or black.
_Nuclei._--Although many fungi have been regarded as devoid of nuclei,
and all have not as yet been proved to contain them, the numerous
investigations of recent years have revealed them in the cells of all
forms thoroughly examined, and we are justified in concluding that the
nucleus is as essential to the cell of a fungus as to that of other
organisms. The hyphae of many contain numerous, even hundreds of
nuclei (Phycomycetes); those of others have several (_Aspergillus_) in
each segment, or only two (_Exoascus_) or one (_Erysiphe_) in each
cell. Even the isolated cells of the yeast plant have each one
nucleus. As a rule the nuclei of the mycelium are very minute (1.5-2
[mu] in _Phycomyces_), but those of many asci and spores are large and
easily rendered visible. As with other plants, so in fungi the
essential process of fertilization consists in the fusion of two
nuclei, but owing to the absence of well-marked sexual organs from
many fungi, a peculiar interest attaches to certain nuclear fusions in
the vegetative cells or in young spores of many forms. Thus in
Ustilagineae the chlamydospores, and in Uredineae the teleutospores,
each contain two nuclei when young, which fuse as the spores mature.
In young asci a similar fusion of two nuclei occurs, and also in
basidia, in each case the nucleus of the ascus or of the basidium
resulting from the fusion subsequently giving rise by division to the
nuclei of the ascospores and basidiospores respectively. The
significance of these fusions will be discussed under the various
groups. Nuclear division is usually accompanied by all the essential
features of karyokinesis.
_Spores._--No agreement has ever been arrived at regarding the
consistent use of the term spore. This is apparently owing to the
facts that too much has been attempted in the definition, and that
differences arise according as we aim at a morphological or a
physiological definition. Physiologically, any cell or group of cells
separated off from a hypha or unicellular fungus, and capable of
itself growing out--germinating--to reproduce the fungus, is a spore;
but it is evident that so wide a definition does not exclude the
ordinary vegetative cells of sprouting fungi, such as yeasts, or small
sclerotium like cell-aggregates of forms like _Coniothecium_.
Morphologically considered, spores are marked by peculiarities of
form, size, colour, place of origin, definiteness in number, mode of
preparation, and so forth, such that they can be distinguished more or
less sharply from the hyphae which produce them. The only
physiological peculiarity exhibited in common by all spores is that
they germinate and initiate the production of a new fungus-plant.
Whether a spore results from the sexual union of two similar gametes
(zygospore) or from the fertilization of an egg-cell by the protoplasm
of a male organ (oospore); or is developed asexually as a motile
(zoospore) or a quiescent body cut off from a hypha (conidium) or
developed along its course (oidium or chlamydospore), or in its
protoplasm (endospore), are matters of importance which have their
uses in the classification and terminology of spores, though in many
respects they are largely of academic interest.
[Illustration: FIG. 2.--_Peronospora parasitica_ (De Bary).
Conidiophore with conidia.]
Klebs has attempted to divide spores into three categories as follows:
(1) kinospores, arising by relatively simple cell-divisions and
subserving rapid dissemination and propagation, e.g. zoospores,
conidia, endogonidia, stylospores, &c.; (2) paulospores, due to simple
rearrangement of cell-contents, and subserving the persistence of the
fungus through periods of exigency, e.g. gemmae, chlamydospores,
resting-cells, cysts, &c.; (3) carpospores, produced by a more or less
complex formative process, often in special fructifications, and
subserving either or both multiplication and persistence, e.g.
zygospores, oospores, brand-spores, aecidiospores, ascospores,
basidiospores, &c. Little or nothing is gained by these definitions,
however, which are especially physiological. In practice these various
kinds of spores of fungi receive further special names in the separate
groups, and names, moreover, which will appear, to those unacquainted
with the history, to have been given without any consistency or regard
to general principles; nevertheless, for ordinary purposes these names
are far more useful in most cases, owing to their descriptive
character, than the proposed new names, which have been only partially
accepted.
_Sporophores._--In some of the simpler fungi the spores are not borne
on or in hyphae which can be distinguished from the vegetative parts
or mycelium, but in the vast majority of cases the sporogenous hyphae
either ascend free into the air or radiate into the surrounding water
as distinct branches, or are grouped into special columns, cushions,
layers or complex masses obviously different in colour, consistency,
shape and other characters from the parts which gather up and
assimilate the food-materials. The term "receptacle" sometimes applied
to these spore-bearing hyphae is better replaced by sporophore. The
sporophore is obsolete when the spore-bearing hyphae are not sharply
distinct from the mycelium, simple when the constituent hyphae are
isolated, and compound when the latter are conjoined. The chief
distinctive characters of the sporogenous hyphae are their
orientation, usually vertical; their limited apical growth; their
peculiar branching, form, colour, contents, consistency; and their
spore-production. According to the characters of the last, we might
theoretically divide them into conidiophores, sporangiophores,
gametophores, oidiophores, &c.; but since the two latter rarely occur,
and more than one kind of spore or spore-case may occur on a
sporophore, it is impossible to carry such a scheme fully into
practice.
A simple sporophore may be merely a single short hypha, the end of
which stops growing and becomes cut off as a conidium by the formation
of a septum, which then splits and allows the conidium to fall. More
generally the hypha below the septum grows forwards again, and repeats
this process several times before the terminal conidium falls, and so
a chain of conidia results, the oldest of which terminates the series
(_Erysiphe_); when the primary branch has thus formed a basipetal
series, branches may arise from below and again repeat this process,
thus forming a tuft (_Penicillium_). Or the primary hypha may first
swell at its apex, and put forth a series of short peg-like branches
(_sterigmata_) from the increased surface thus provided, each of which
develops a similar basipetal chain of conidia (_Aspergillus_), and
various combinations of these processes result in the development of
numerous varieties of exquisitely branched sporophores of this type
(_Botrytis_, _Botryosporium_, _Verticillium_, &c.).
[Illustration: FIG. 3.--_Cystopus candidus_.
A. a, Conidia.
b, Conidiophores.
c, Conidium emitting zoospores.
d, Free zoospore.
B.og, Oogonium.
os, Oosphere.
an, Antheridium.
C. Formation of zoospores by oospores.
z, Free zoospores. (After De Bary.)]
A second type is developed as follows: the primary hypha forms a
septum below its apex as before, and the terminal conidium, thus
abstricted, puts out a branch at its apex, which starts as a mere
point and rapidly swells to a second conidium; this repeats the
process, and so on, so that we now have a chain of conidia developed
in acropetal succession, the oldest being below, and, as in
_Penicillium_, &c., branches put forth lower down may repeat the
process (_Hormodendron_). In all these cases we may speak of simple
conidiophores. The simple sporophore does not necessarily terminate in
conidia, however. In _Mucor_, for example, the end of the primary
hypha swells into a spheroidal head (sporangium), the protoplasm of
which undergoes segmentation into more or less numerous globular
masses, each of which secretes an enveloping cell-wall and becomes a
spore (endospore), and branched systems of sporangia may arise as
before (_Thamnidium_). Such may be termed sporangiophores. In
_Sporodinia_ the branches give rise also to short branches, which meet
and fuse their contents to form zygospores. In Peronospora,
Saprolegnia, &c., the ends of the branches swell up into sporangia,
which develop zoospores in their interior (zoosporangia), or their
contents become oospheres, which may be fertilized by the contents of
other branches (antheridia) and so form egg-cases (oogonia). Since in
such cases the sporophore bears sexual cells, they may be conveniently
termed gametophores.
Compound sporophores arise when any of the branched or unbranched
types of spore-bearing hyphae described above ascend into the air in
consort, and are more or less crowded into definite layers, cushions,
columns or other complex masses. The same laws apply to the individual
hyphae and their branches as to simple sporophores, and as long as the
conidia, sporangia, gametes, &c., are borne on their external
surfaces, it is quite consistent to speak of these as compound
sporophores, &c., in the sense described, however complex they may
become. Among the simplest cases are the sheet-like aggregates of
sporogenous hyphae in _Puccinia_, _Uromyces_, &c., or of basidia in
_Exobasidium_, _Corticium_, &c., or of asci in _Exoascus_,
_Ascocorticium_, &c. In the former, where the layer is small, it is
often termed a sorus, but where, as in the latter, the sporogenous
layer is extensive, and spread out more or less sheet-like on the
supporting tissues, it is more frequently termed a hymenium. Another
simple case is that of the columnar aggregates of sporogenous hyphae
in forms like _Stilbum_, _Coremium_, &c. These lead us to cases where
the main mass of the sporophore forms a supporting tissue of closely
crowded or interwoven hyphae, the sporogenous terminal parts of the
hyphae being found at the periphery or apical regions only. Here we
have the cushion-like type (stroma) of _Nectria_ and many
Pyrenomycetes, the clavate "receptacle" of _Clavaria_, &c., passing
into the complex forms met with in _Sparassis_, _Xylaria_,
_Polyporei_, and _Agaricini_, &c. In these cases the compound
sporophore is often termed the hymenophore, and its various parts
demand special names (pileus, stipes, gills, pores, &c.) to denote
peculiarities of distribution of the hymenium over the surface.
Other series of modifications arise in which the tissues corresponding
to the stroma invest the sporogenous hyphal ends, and thus enclose the
spores, asci, basidia, &c., in a cavity. In the simplest case the
stroma, after bearing its crop of conidia or oidia, develops
ascogenous branches in the loosened meshes of its interior (e.g.
_Onygena_). Another simple case is where the plane or slightly convex
surface of the stroma rises at its margins and overgrows the
sporogenous hyphal ends, so that the spores, asci, &c., come to lie in
the depression of a cavity--e.g. _Solenia_, _Cyphella_--and even
simpler cases are met with in _Mortierella_, where the zygospore is
invested by the overgrowth of a dense mat of closely branching hyphae,
and in _Gymnoascus_, where a loose mat of similarly barren hyphae
covers in the tufts of asci as they develop.
In such examples as the above we may regard the hymenium (_Solenia_,
_Cyphella_), zygospores, or asci as truly invested by later growth,
but in the vast majority of cases the processes which result in the
enclosure of the spores, asci, &c., in a "fructification" are much
more involved, inasmuch as the latter is developed in the interior of
hyphal tissues, which are by no means obviously homologous with a
stroma. Thus in _Penicillium_, _Eurotium_, _Erysiphe_, &c., hyphal
ends which are the initials of ascogenous branches, are invested by
closely packed branches at an early stage of development, and the asci
develop inside what has by that time become a complete investment.
Whether a true sexual process precedes these processes or not does not
affect the present question, the point being that the resulting
spheroidal "fructification" (cleistocarp, perithecium) has a definite
wall of its own not directly comparable with a stroma. In other cases
(_Hypomyces_, _Nectria_) the perithecia arise on an already mature
stroma, while yet more numerous examples can be given (_Poronia_,
_Hypoxylon_, _Claviceps_, &c.) where the perithecia originate below
the surface of a stroma formed long before. Similarly with the various
types of conidial or oidial "fructifications," termed pycnidia,
spermogonia, aecidia, &c. In the simplest of these cases--e.g.
_Fumago_--a single mycelial cell divides by septa in all three planes
until a more or less solid clump results. Then a hollow appears in the
centre owing to the more rapid extension of the outer parts, and into
this hollow the cells lining it put forth short sporogenous branches,
from the tips of which the spores (stylospores, conidia, spermatia)
are abstricted. In a similar way are developed the pycnidia of
_Cicinnobolus_, _Pleospora_, _Cucurbitaria_, _Leptosphaeria_ and
others. In other cases (_Diplodia_, _Aecidium_, &c.) conidial or
oidial "fructifications" arise by a number of hyphae interweaving
themselves into a knot, as if they were forming a Sclerotium. The
outer parts of the mass then differentiate as a wall or investment,
and the interior becomes a hollow, into which hyphal ends grow and
abstrict the spores. Much more complicated are the processes in a
large series of "fructifications," where the mycelium first develops a
densely packed mass of hyphae, all alike, in which labyrinths of
cavities subsequently form by separation of hyphae in the previously
homogeneous mass, and the hymenium covers the walls of these cavities
and passages as with a lining layer. Meanwhile differences in
consistency appear in various strata, and a dense outer protective
layer (peridium), soft gelatinous layers, and so on are formed, the
whole eventually attaining great complexity--e.g. puff-balls,
earth-stars and various _Phalloideae_.
_Spore-Distribution._--Ordinary conidia and similarly abstricted dry
spores are so minute, light and numerous that their dispersal is
ensured by any current of air or water, and we also know that rats and
other burrowing animals often carry them on their fur; similarly with
birds, insects, slugs, worms, &c., on claws, feathers, proboscides,
&c., or merely adherent to the slimy body. In addition to these
accidental modes of dispersal, however, there is a series of
interesting adaptations on the part of the fungus itself. Passing over
the locomotor activity of zoospores (_Pythium_, _Peronospora_,
_Saprolegnia_) we often find spores held under tension in sporangia
(_Pilobolus_) or in asci (_Peziza_) until ripe, and then forcibly shot
out by the sudden rupture of the sporangial wall under the pressure of
liquid behind--mechanism comparable to that of a pop-gun, if we
suppose air replaced by watery sap. Even a single conidium, held tense
to the last moment by the elastic cell-wall, may be thus shot forward
by a spurt of liquid under pressure in the hypha abstricting it (e.g.
_Empusa_), and similarly with _basidiospores_ (_Coprinus_, _Agaricus_,
&c.). A more complicated case is illustrated by _Sphaerobolus_, where
the entire mass of spores, enclosed in its own peridium, is suddenly
shot up into the air like a bomb from a mortar by the elastic
retroversion of a peculiar layer which, up to the last moment,
surrounded the bomb, and then suddenly splits above, turns inside out,
and drives the former as a projectile from a gun. Gelatinous or
mucilaginous degenerations of cell-walls are frequently employed in
the interests of spore dispersal. The mucilage surrounding endospores
of _Mucor_, conidia of _Empusa_, &c., serves to gum the spore to
animals. Such gums are formed abundantly in pycnidia, and, absorbing
water, swell and carry out the spores in long tendrils, which emerge
for days and dry as they reach the air, the glued spores gradually
being set free by rain, wind, &c. In oidial chains (_Sclerotinia_) a
minute double wedge of wall-substance arises in the middle lamella
between each pair of contiguous oidia, and by its enlargement splits
the separating lamella. These disjunctors serve as points of
application for the elastic push of the swelling spore-ends, and as
the connecting outer lamella of cell-wall suddenly gives way, the
spores are jerked asunder. In many cases the slimy masses of spermatia
(_Uredineae_), conidia (_Claviceps_), basidiospores (_Phallus_,
_Coprinus_), &c., emit more or less powerful odours, which attract
flies or other insects, and it has been shown that bees carry the
fragrant oidia of _Sclerotinia_ to the stigma of _Vaccinium_ and
infect it, and that flies carry away the foetid spores of _Phallus_,
just as pollen is dispersed by such insects. Whether the strong odour
of trimethylamine evolved by the spores of _Tilletia_ attracts insects
is not known.
The recent observations and exceedingly ingenious experiments of Falck
have shown that the sporophores of the Basidiomycetes--especially the
large sporophores of such forms as _Boletus_, _Polyporus_--contain
quantities of reserve combustible material which are burnt up by the
active metabolism occurring when the fruit-body is ripe. By this means
the temperature of the sporophore is raised and the difference between
it and the surrounding air may be one of several degrees. As a result
convection currents are produced in the air which are sufficient to
catch the basidiospores in their fall and carry them, away from the
regions of comparative atmospheric stillness near the ground, to the
upper air where more powerful air-currents can bring about their wide
distribution.
_Classification._--It has been accepted for some time now that the
majority of the fungi proper fall into three main groups, the
Phycomycetes, Ascomycetes and Basidiomycetes, the Schizomycetes and
Myxomycetes (Mycetozoa) being considered as independent groups not
coming under the true fungi.
The chief schemes of classification put forward in detail have been
those of P.A. Saccardo (1882-1892), of Oskar Brefeld and Von Tavel
(1892), of P.E.L. Van Tieghem (1893) and of J. Schroeter (1892). The
scheme of Brefeld, which was based on the view that the Ascomycetes and
Basidiomycetes were completely asexual and that these two groups had
been derived from one division (Zygomycetes) of the Phycomycetes, has
been very widely accepted. The recent work of the last twelve years has
shown, however, that the two higher groups of fungi exhibit distinct
sexuality, of either a normal or reduced type, and has also rendered
very doubtful the view of the origin of these two groups from the
Phycomycetes. The real difficulty of classification of the fungi lies in
the polyphyletic nature of the group. There is very little doubt that
the primitive fungi have been derived by degradation from the lower
algae. It appears, however, that such a degradation has occurred not
only once in evolution but on several occasions, so that we have in the
Phycomycetes not a series of naturally related forms, but groups which
have arisen perfectly independently of one another from various groups
of the algae. It is also possible in the absence of satisfactory
intermediate forms that the Ascomycetes and Basidiomycetes have also
been derived from the algae independently of the Phycomycetes, and
perhaps of one another.
A natural classification on these lines would obviously be very
complicated, so that in the present state of our knowledge it will be
best to retain the three main groups mentioned above, bearing in mind
that the Phycomycetes especially are far from being a natural group. The
following gives a tabular survey of the scheme adopted in the present
article:
A. PHYCOMYCETES. Alga-like fungi with unicellular thallus and
well-marked sexual organs.
CLASS I.--Oomycetes. Mycelium usually well developed, but sometimes
poor or absent. Sexual reproduction by oogonia and antheridia; asexual
reproduction by zoospores or conidia.
1. Monoblepharidineae. Mycelium present, antheridia with
antherozoids, oogonium with single oosphere: Monoblepharidaceae.
2. Peronosporineae. Mycelium present; antheridia but no
antherozoids; oogonia with one or more oospheres: Peronosporaceae,
Saprolegniaceae.
3. Chytridineae. Mycelium poorly developed or absent; oogonia and
antheridia (without antherozoids) known in some cases; zoospores
common: Chytridiaceae. Ancylistaceae.
CLASS II.--Zygomycetes. Mycelium well developed; sexual reproduction
by zygospores; asexual reproduction by sporangia and conidia.
1. Mucorineae. Sexual reproduction as above, asexual by sporangia or
conidia or both: Mucoraceae. Mortierellaceae, Chaetocladiaceae,
Piptocephalidaceae.
2. Entomophthorineae. Sexual reproduction typical but with sometimes
inequality of the fusing gametes (gametangia ?): Entomophthoraceae.
B. HIGHER FUNGI. Fungi with segmental thallus; sexual reproduction
sometimes with typical antheridia and oogonia (ascogonia) but usually
much reduced.
CLASS I.--Ustilaginales. Forms with septate thallus, and reproduction
by chlamydospores which on germination produce sporidia; sexuality
doubtful.
CLASS II.--Ascomycetes. Thallus septate; spores developed in special
type of sporangium, the ascus, the number of spores being usually
eight. Sexual reproduction sometimes typical, usually reduced.
Exoascineae, Saccharomycetineae, Perisporinea, Discomycetes,
Pyrenomycetes, Tuberineae, Laboulbeniineae.
CLASS III.--Basidiales. Thallus septate. Conidia (basidiospores) borne
in fours on a special conidiophore, the basidium. Sexual reproduction
always much reduced.
1. Uredineae. Life-history in some cases very complex and with
well-marked sexual process and alternation of generations, in others
much reduced; basidium (promycelium) derived usually from a
thick-walled spore (teleutospore).
2. Basidiomycetes. Life-history always very simple, no well-marked
alternation of generations; basidium borne directly on the mycelium.
(A) Protobasidiomycetes. Basidia septate. Auriculariaceae,
Pilacreaceae, Tremellinaceae.
(B) Autobasidiomycetes. Basidia non-septate. Hymenomycetes,
Gasteromycetes.
A. PHYCOMYCETES.--Most of the recent work of importance in this group
deals with the cytology of sexual reproduction and of spore-formation,
and the effect of external conditions on the production of reproductive
organs.
_Monoblepharidaceae_ consists of a very small group of aquatic forms
living on fallen twigs in ponds and ditches. Only one genus,
_Monoblepharis_, can certainly be placed here, though a somewhat
similar genus, _Myrioblepharis_, with a peculiar multiciliate zoospore
like that of Vaucheria, is provisionally placed in the same group.
_Monoblepharis_ was first described by Cornu in 1871, but from that
time until 1895 when Roland Thaxter described several species from
America the genus was completely lost sight of. _Monoblepharis_ has
oogonia with single oospheres and antheridia developing a few amoeboid
uniciliate antherozoids; these creep to the opening of the oogonium
and then swim in. The resemblance between this genus and _Oedogonium_
among the algae is very striking, as is also that of _Myrioblepharis_
and _Vaucheria_.
_Peronosporaceae_ are a group of endophytic parasites--about 100
species--of great importance as comprising the agents of "damping off"
disease (_Pythium_), vine-mildew (_Plasmopara_), potato disease
(Phytophthora), onion-mildew (_Peronospora_). _Pythium_ is a
semi-aquatic form attacking seedlings which are too plentifully
supplied with water; its hyphae penetrate the cell-walls and rapidly
destroy the watery tissues of the living plant; then the fungus lives
in the dead remains. When the free ends of the hyphae emerge again
into the air they swell up into spherical bodies which may either fall
off and behave as conidia, each putting out a germ-tube and infecting
the host; or the germ-tube itself swells up into a zoosporangium which
develops a number of zoospores. In the rotting tissues branches of the
older mycelium similarly swell up and form antheridia and oogonia
(fig. 4). The contents of the antheridium are not set free, but that
organ penetrates the oogonium by means of a narrow outgrowth, the
fertilizing tube, and a male nucleus then passes over into the single
oosphere, which at first multinucleate becomes uninucleate before
fertilization. _Pythium_ is of interest as illustrating the dependence
of zoospore-formation on conditions and the indeterminate nature of
conidia. The other genera are more purely parasitic; the mycelium
usually sends haustoria into the cells of the host and puts out
branched, aerial conidiophores through the stomata, the branches of
which abstrict numerous "conidia"; these either germinate directly or
their contents break up into zoospores (fig. 5). The development of
the "conidia" as true conidial spores or as zoosporangia may occur in
one and the same species (_Cystopus candidus_, _Phytophthora
infestans_) as in _Pythium_ described above; in other cases the direct
conidial germination is characteristic of genera--e.g. _Peronospora_;
while others emit zoospores--e.g. _Plasmopara_, &c. In _Cystopus_
(_Albugo_) the "conidia" are abstricted in basipetal chain-like series
from the ends of hyphae which come to the surface in tufts and break
through the epidermis as white pustules. Each "conidium" contains
numerous nuclei and is really a zoosporangium, as after dispersal it
breaks up into a number of zoospores. The Peronosporaceae reproduce
themselves sexually by means of antheridia and oogonia as described in
_Pythium_. In _Cystopus Bliti_ the oosphere contains numerous nuclei,
and all the male nuclei from the antheridium pass into it, the male
and female nuclei then fusing in pairs. We thus have a process of
"multiple fertilization"; the oosphere really represents a large
number of undifferentiated gametes and has been termed a coenogamete.
Between _Cystopus Bliti_ on the one hand and _Pythium de Baryanum_ on
the other a number of cytologically intermediate forms are known. The
oospore on germination usually gives origin to a zoosporangium, but
may form directly a germ tube which infects the host.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 4.--Fertilization of the Peronosporeae. After Wager.
1, _Peronospora parasitica_. Young multinucleate oogonium (og) and
antheridium (an).
2, _Albugo candida_. Oogonium with the central uninucleate oosphere
and the fertilizing tube (a) of the antheridium which introduces the
male nucleus.
3, The same. Fertilized egg-cell (o) surrounded by the periplasm
(p).]
[Illustration: FIG. 5.--_Phytophthora infestans_. Fungus of Potato
Disease.
A, B, Section of Leaf of Potato with sporangiophores of
_Phytophthora infestans_ passing through the stomata D, on the under
surface of the leaf.
E, Sporangia.
F, G, H, J, Further development of the sporangia.
K, Germination of the zoospores formed in the sporangia.
L, M, N, Fertilization of the oogonium and development of the
oospore in _Peronospora_.]
_Saprolegniaceae_ are aquatic forms found growing usually on dead
insects lying in water but occasionally on living fish (e.g. the
salmon disease associated with _Saprolegnia ferax_). The chief genera
are _Saprolegnia, Achlya, Pythiopsis, Dictyuchus, Aplanes._ Motile
zoospores which escape from the zoosporangium are present except in
Aplanes. The sexual reproduction shows all transitions between forms
which are normally sexual, like the Peronosporaceae, to forms in which
no antheridium is developed and the oospheres develop
parthenogenetically. The oogonia, unlike the Peronosporaceae, contain
more than one oosphere. Klebs has shown that the development of
zoosporangia or of oogonia and pollinodia respectively in
_Saprolegnia_ is dependent on the external conditions; so long as a
continued stream of suitable food-material is ensured the mycelium
grows on without forming reproductive organs, but directly the
supplies of nitrogenous and carbonaceous food fall below a certain
degree of concentration sporangia are developed. Further reduction of
the supplies of food effects the formation of oogonia. This explains
the sequence of events in the case of a _Saprolegnia_-mycelium
radiating from a dead fly in water. Those parts nearest the fly and
best supplied develop barren hyphae only; in a zone at the periphery,
where the products of putrefaction dissolved in the water form a
dilute but easily accessible supply, the zoosporangia are developed in
abundance; oogonia, however, are only formed in the depths of this
radiating mycelium, where the supplies of available food materials are
least abundant.
_Chytridineae._--These parasitic and minute, chiefly aquatic, forms
may be looked upon as degenerate Oomycetes, since a sexual process and
feeble unicellular mycelium occur in some; or they may be regarded as
series of primitive forms leading up to higher members. There is no
means of deciding the question. They are usually included in
Oomycetes, but their simple structure, minute size, usually uniciliate
zoospores, and their negative characters would justify their retention
as a separate group. It contains less than 200 species, chiefly
parasitic on or in algae and other water-plants or animals, of various
kinds, or in other fungi, seedlings, pollen and higher plants. They
are often devoid of hyphae, or put forth fine protoplasmic filaments
into the cells of their hosts. After absorbing the cell-contents of
the latter, which it does in a few hours or days, the fungus puts out
a sporangium, the contents of which break up into numerous minute
swarm-spores, usually one-ciliate, rarely two-ciliate. Any one of
these soon comes to rest on a host-cell, and either pierces it and
empties its contents into its cavity, where the further development
occurs (_Olpidium_), or merely sends in delicate protoplasmic
filaments (_Rhizophydium_) or a short hyphal tube of, at most, two or
three cells, which acts as a haustorium, the further development
taking place outside the cell-wall of the host (_Chytridium_). In some
cases resting spores are formed inside the host (_Chytridium_), and
give rise to zoosporangia on germination. In a few species a sexual
process is described, consisting in the conjugation of similar cells
(_Zygochytrium_) or the union of two dissimilar ones (_Polyphagus_).
In the development of distinct antheridial and oogonial cells the
allied Ancylistineae show close alliances to _Pythium_ and the
Oomycetes. On the other hand, the uniciliate zoospores of _Polyphagus_
have slightly amoeboid movements, and in this and the
pseudopodium-like nature of the protoplasmic processes, such forms
suggest resemblances to the Myxomycetes. Opinions differ as to whether
the Chytridineae are degraded or primitive forms, and the group still
needs critical revision. Many new forms will doubtless be discovered,
as they are rarely collected on account of their minuteness. Some
forms cause damping off of seedlings--e.g. _Olpidium Brassicae_;
others discoloured spots and even tumour-like swellings--e.g.
_Synchytium Scabiosae_, _S. Succisae_, _Urophlyctis_, &c., on higher
plants. Analogies have been pointed out between Chytridiaceae and
unicellular algae, such as Chlorosphaeraceae, Protococcaceae,
"Palmellaceae," &c., some of which are parasitic, and suggestions may
be entertained as to possible origin from such algae.
The _Zygomycetes_, of which about 200 species are described, are
especially important from a theoretical standpoint, since they
furnished the series whence Brefeld derived the vast majority of the
fungi. They are characterized especially by the zygospores, but the
asexual organs (sporangia) exhibit interesting series of changes,
beginning with the typical sporangium of _Mucor_ containing numerous
endospores, passing to cases where, as in _Thamnidium_, these are
accompanied with more numerous small sporangia (sporangioles)
containing few spores, and thence to _Chaetocladium_ and
_Piptocephalis_, where the sporangioles form but one spore and fall
and germinate as a whole; that is to say, the monosporous sporangium
has become a conidium, and Brefeld regarded these and similar series
of changes as explaining the relation of ascus to conidium in higher
fungi. According to his view, the ascus is in effect the sporangium
with several spores, the conidium the sporangiole with but one spore,
and that not loose but fused with the sporangiole wall. On this basis,
with other interesting morphological comparisons, Brefeld erected his
hypothesis, now untenable, that the Ascomycetes and Basidiomycetes
diverge from the Zygomycetes, the former having particularly
specialized the ascus (sporangial) mode of reproduction, the latter
having specialized the conidial (indehiscent one-spored sporangiole)
mode. In addition to sporangia and the conidial spores referred to,
some Mucorini show a peculiar mode of vegetative reproduction by means
of gemmae or chlamydospores--i.e. short segments of the hyphae become
stored with fatty reserves and act as spores. The gemmae formed on
submerged Mucors may bud like a yeast, and even bring about alcoholic
fermentation in a saccharine solution.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 6.--_Mucor Mucedo._ Different stages in the formation and
germination of the zygospore. (After Brefeld, 1-4. 5 from v. Tavel,
_Pilze_.)
1, Two conjugating branches in contact.
2, Septation of the conjugating cells (a) from the suspensors (b).
3, More advanced stage, the conjugating cells (a) are still distinct
from one another; the warty thickenings of their walls have
commenced to form.
4, Ripe zygospore (b) between the suspensors (a).
5, Germinating zygospore with a germ-tube bearing a sporangium.]
The segments of the hyphae in this group usually contain several
nuclei. At the time of sporangial formation the protoplasm with
numerous nuclei streams into the swollen end of the sporangiophore and
there becomes cut off by a cell-wall to form the sporangium. The
protoplasm then becomes cut up by a series of clefts into a number of
smaller and smaller pieces which are unicellular in _Pilobolus_,
multicellular in _Sporodinia_. These then become surrounded by a
cell-wall and form the spores. This mode of spore-formation is totally
different from that in the ascus; hence one of the difficulties of the
acceptance of Brefeld's view of the homology of ascus and sporangium.
The cytology of zygospore-formation is not known in detail; the
so-called gametes which fuse are multinucleate and are no doubt of the
nature of gametangia. The fate of these nuclei is doubtful, probably
they fuse in pairs (fig. 6).
Blakeslee has lately made some very important observations of the
Zygomycetes. It is well known that while in some forms, e.g.
_Spordinia_, zygospores are easily obtained, in others, e.g. most
species of _Mucor_, they are very erratic in their appearance. This
has now been explained by Blakeslee, who finds that the Mucorinae can
be divided into two groups, termed homothallic and heterothallic
respectively. In the first group zygospores can arise by the union of
branches from the _same_ mycelium and so can be produced by the growth
from a single spore; this group includes _Spordinia grandis_,
_Spinellus fusiger_, some species of _Mucor_, &c. The majority of
forms, however, fall into the heterothallic group, in which the
association of branches from two mycelia _different in nature_ is
necessary for the formation of zygospores. These structures cannot
then be produced from the product of a single spore nor even from the
thalli derived from _any_ two spores. The two kinds of thalli
Blakeslee considers to have a differentiation of the nature of sex and
he distinguishes them as ( + ) and (-) forms; the former being usually
distinguished by a somewhat greater luxuriance of growth.
The classification of the Mucorini depends on the prevalence and
characters of the conidia, and of the sporangia and zygospores--e.g.
the presence or absence of a columella in the former, the formation of
an investment round the latter. Most genera are saprophytes, but
some--_Chaetocladium_, _Piptocephalis_--are parasites on other
Mucorini, and one or two are associated casually with the rotting of
tomatoes and other fruits, bulbs, &c., the fleshy parts of which are
rapidly destroyed if once the hyphae gain entrance. Even more
important is the question of mycosis in man and other animals,
referred to species of _Mucor_, and investigated by Lucet and
Costantin. Klebs has concluded that transpiration is the important
factor in determining the formation of sporangia, while
zygote-development depends on totally different conditions; these
results have been called in question by Falck.
The _Entomophthoraceae_ contain three genera, _Empusa_,
_Entomophthora_ and _Basidiobolus_. The two first genera consist of
forms which are parasitic on insects. _Empusa Muscae_ causes the
well-known epidemic in house-flies during the autumn; the dead,
affected flies are often found attached to the window surrounded by a
white halo of conidia. _B. ranarum_ is found in the alimentary canal
of the frog and growing on its excrement. In these three genera the
conidia are cast off with a jerk somewhat in the same way as the
sporangium of _Pilobolus_.
B. HIGHER FUNGI.--Now that Brefeld's view of the origin of these forms
from the Zygomycetes has been overthrown, the relationship of the higher
and lower forms of fungi is left in obscurity. The term _Eumycetes_ is
sometimes applied to this group to distinguish them from the
Phycomycetes, but as the same name is also applied to the fungi as a
whole to differentiate them from the Mycetozoa and Bacteria, the term
had best be dropped. The Higher Fungi fall into three groups: the
_Ustilaginales_, of doubtful position, and the two very sharply marked
groups _Basidiales_ and _Ascomycetes_.
[Illustration: From Vine's _Students' Text Book of Botany_, by
permission of Swan Sonnenschein & Co.
FIG. 7.--Germinating resting-gonidia. A, of _Ustilago receptaculorum_;
B, of _Tilletia Caries_.
sp, The gonidium.
pm, The promycelium.
d, The sporidia: in B the sporidia have coalesced in pairs at v.]
I. _Ustilaginales._--This includes two families Ustilaginaceae (smuts)
and Tilletiaceae (bunts). The bunts and smuts which damage our grain
and fodder plants comprise about 400 species of internal parasites,
found in all countries on herbaceous plants, and especially on
Monocotyledons. They are remarkable for their dark spores developed in
gall-like excrescences on the leaves, stems, &c., or in the fruits of
the host. The discovery of the yeast-conidia of these fungi, and their
thorough investigation by Brefeld, have thrown new lights on the
group, as also have the results elucidating the nature of the ordinary
dark spores--smuts, bunt, &c.--which by their mode of origin and
development are chlamydospores. When the latter germinate a slender
"promycelium" is put out; in _Ustilago_ and its allies this is
transversely septate, and bears lateral conidia (sporidia); in
_Tilletia_ and its allies non-septate, and bears a terminal tuft of
conidia (sporidia) (fig. 7). Brefeld regarded the promycelium as a
kind of _basidium_, bearing lateral or terminal conidia (comparable to
_basidiospores_), but since the number of basidiospores is not fixed,
and the basidium has not yet assumed very definite morphological
characters, Brefeld termed the group _Hemibasidii_, and regarded them
as a half-way stage in the evolution of the true Basidiomycetes from
Phycomycetes, the _Tilletia_ type leading to the true basidium
(Autobasidium), the _Ustilago_ type to the protobasidium, with lateral
spores; but this view is based on very poor evidence, so that it is
best to place these forms as a separate group, the _Ustilaginales_.
The yeast-conidia, which bud off from the conidia or their resulting
mycelium when sown in nutrient solutions, are developed in successive
crops by budding exactly as in the yeast plant, but they cannot
ferment sugar solutions. It is the rapid spread of these yeast-conidia
in manure and soil waters which makes it so difficult to get rid of
smuts, &c., in the fields, and they, like the ordinary conidia,
readily infect the seedling wheat, oats, barley or other cereals.
Infection in these cases occurs in the seedling at the place where
root and shoot meet, and the infecting hypha having entered the plant
goes on living in it and growing up with it as if it had no parasitic
action at all. When the flowers form, however, the mycelium sends
hyphae into the young ovaries and rapidly replaces the stores of sugar
and starch, &c., which would have gone to make the grain, by the
soot-like mass of spores so well known as smut, &c. These spores
adhere to the grain, and unless destroyed, by "steeping" or other
treatment, are sown with it, and again produce sporidia and
yeast-conidia which infect the seedlings. In other species the
infection occurs through the style of the flower, but the fungus after
reaching the ovule develops no further during that year but remains
dormant in the embryo of the seed. On germination, however, the fungus
behaves in the same way as one which has entered in the seedling
stage. The cytology of these forms is very little known; Dangeard
states that there is a fusion of two nuclei in the chlamydospore, but
this requires confirmation. Apart from this observation there is no
other trace of sexuality in the group.
II. _Ascomycetes._--This, except in the case of a few of the simpler
forms, is a very sharply marked group characterized by a special type
of sporangium, the ascus. In the development of the ascus we find two
nuclei at the base which fuse together to form the single nucleus of
the young _ascus_. The single nucleus divides by three successive
divisions to form eight nuclei lying free in the protoplasm of the
ascus. Then by a special method, described first by Harper, a mass of
protoplasm is cut out round each nucleus; thus eight uninucleate
ascospores are formed by free-cell formation. The protoplasm remaining
over is termed _epiplasm_ and often contains glycogen (fig. 8). In
some cases nuclear division is carried further before spore-formation
occurs, and the number of spores is then 16, 32 and 64, &c.; in a few
cases the number of spores is less than eight by abortion of some of
the eight nuclei. The ascus is thus one of the most sharply
characterized structures among the fungi.
In some forms we find definite male and female sexual organs
(_Sphaerotheca_, _Pyronema_, &c.), in others the antheridium is
abortive or absent, but the ascogonium (oogonium) is still present and
the female nuclei fuse in pairs (_Lachnea stercorea_, _Humaria
granulata_, _Ascobolus furfuraceus_); while in other forms ascogonium
and antheridium are both absent and fusion occurs between vegetative
nuclei (_Humaria rutilans_, and probably the majority of other forms).
In other cases the sexual fusion is apparently absent altogether, as
in _Exoascus_. In the first case (fig. 9) we have a true sexual
process, while in the second and third cases we have a _reduced_
sexual process in which the fusion of other nuclei has replaced the
fusion of the normal male and female nuclei. It is to be noted that
all the forms exhibit the fusion of nuclei in the ascus, so that those
with the normal or reduced sexual process described above have two
nuclear fusions in their life-history. The advantage or significance
of the second (ascus) fusion is not clearly understood.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 8.--Development of the Ascus.
A-C, _Pyronema confluens_. (After Harper.)
D, Young ascus of _Boudiera_ with eight spores. (After Claussen.)]
The group of the Hemiasci was founded by Brefeld to include forms
which were supposed to be a connecting link between Phycomycetes and
Ascomycetes. As mentioned before, the connexion between these two
groups is very doubtful, and the derivation of the ascus from an
ordinary sporangium of the Zygomycetes cannot be accepted. The
majority of the forms which were formerly included in this group have
been shown to be either true Phycomycetes (like _Ascoidea_) or true
Ascomycetes (like _Thelebolus_). _Eremascus_ and _Dipodascus_, which
are often placed among the Hemiasci, possibly do not belong to the
Ascomycetes series at all.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
Fig. 9.--_Sphaerotheca Castagnei_. Fertilization and Development of
the Perithecium. (After Harper.)
1, Oogonium (og) with the antheridial branch (az) applied to its
surface
2, Separation of antheridium (an).
3, Passage of the antheridial nucleus towards that of the oogonium.
4, Union of the nuclei.
5, Fertilized oogonium surrounded by two layers of hyphae derived
from the stalk-cell (st).
6, The multicellular ascogonium derived by division from the
oogonium; the terminal cell with the two nuclei (as) gives rise to
the ascus.]
_Exoascaceae_ are a small group of doubtful extent here used to
include _Exoascus_, _Taphrina_, _Ascorticium_ and _Endomyces_. The
mycelium is very much reduced in extent. The asci are borne directly
on the mycelium and are therefore fully exposed, being devoid from the
beginning of any investment. The _Taphrineae_, which include
_Exoascus_ and _Taphrina_, are important parasites--e.g. pocket-plums
and witches' brooms on birches, &c., are due to their action (fig.
10). _Exoascus_ and _Ascorticium_ present interesting parallels to
_Exobasidium_ and _Corticium_ among the Basidiomycetes.
_Saccharomycetaceae_ include the well-known yeasts which belong mainly
to the genus _Saccharomyces_. They are characterized by their
unicellular nature, their power of rapid budding, their capacity for
fermenting various sugars, and their power of forming endogenous
spores. The sporangium with its endogenous spores has been compared
with an ascus, and on these grounds the group is placed among the
Ascomycetes--a very doubtful association. The group has attained an
importance of late even beyond that to which it was brought by
Pasteur's researches on alcoholic fermentation, chiefly owing to the
exact results of the investigations of Hansen, who first applied the
methods of pure cultures to the study of these organisms, and showed
that many of the inconsistencies hitherto existing in the literature
were due to the coexistence in the cultures of several species or
races of yeasts morphologically almost indistinguishable, but
physiologically very different. About fifty species of _Saccharomyces_
are described more or less completely, but since many of these cannot
be distinguished by the microscope, and some have been found to
develop physiological races or varieties under special conditions of
growth, the limits are still far too ill-defined for complete
botanical treatment of the genus. A typical yeast is able to develop
new cells by budding when submerged in a saccharine solution, and to
ferment the sugar--i.e. so to break up its molecules that, apart from
small quantities used for its own substance, masses of it out of all
proportion to the mass of yeast used become resolved into other
bodies, such as carbon dioxide and alcohol, the process requiring
little or no oxygen. Brefeld regards the budding process as the
formation of conidia. Under other conditions, of which the temperature
is an important one, the nucleus in the yeast-cell divides, and each
daughter-nucleus again, and four spores are formed in the mother cell,
a process obviously comparable to the typical development of
ascospores in an ascus. Under yet other conditions the quiescent
yeast-cells floating on the surface of the fermented liquor grow out
into elongated sausage-shaped or cylindrical cells and branching
cell-series, which mat together into mycelium-like veils. At the
bottom of the fermented liquor the cells often obtain fatty contents
and thick walls, and behave as resting cells (chlamydospores). The
characters employed by experts for determining a species of yeast are
the sum of its peculiarities as regards form and size: the shapes,
colours, consistency, &c., of the colonies grown on certain definite
media; the optimum temperature for spore-formation, and for the
development of the "veils"; and the behaviour as regards the various
sugars.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 10.--_Taphrina Pruni._ Transverse section through the epidermis
of an infected plum. Four ripe asci, a1, a2, with eight spores, a3,
a4, with yeast-like conidia abstricted from the spores. After
Sadebeck.
st, Stalk-cells of the asci.
m, Filaments of the mycelium cut transversely.
cut, Cuticle.
sp, Epidermis.]
The following summary of some of the principal characteristics of
half-a-dozen species will serve to show how such peculiarities can be
utilized for systematic purposes:
+---------------------+---------------------------------+----------------------------------+----------------------------------------+
| | Optimum | Characters of | |
| Species. | Temperature for +----------+-----------+-----------+ Sugars Fermented and |
| +----------------+----------------+ Fermenta-| Cells. | Spores. | Products, &c. |
| | Spores. | Veils. | tion | | | |
+---------------------+----------------+----------------+----------+-----------+-----------+----------------------------------------+
|_S. cereviseae I_. | 30 deg. | 20 deg.-28 deg.| High | Rounded | Globoid | / Inverts maltose and saccharose and |
|_S. Pastorianus I_ | 27 deg.-5 deg. | 26 deg.-28 deg.| Low | Rounded | Globoid |< form alcohol 4-6 vol. %. |
|_S. ellipsoideus_ | 25 deg. | 33 deg.-34 deg.| Low | Rounded | Globoid | \ |
| | | | | | | |
|_S. anomalus_ | 28 deg.-31 deg.| ? | High | Elliptical| Hat-shaped| Ditto, and evolves a fragrant ether. |
| | | | | | |
|_S. Ludwigii_ | 30 deg.-31 deg.| ? | ? | Elongated | Globoid | Will not invert maltose. |
| | | | | | |
|_S. membranaefaciens_| 30 deg. | ? | High | Elongated | Globoid | Inverts neither maltose nor saccharose.|
+---------------------+----------------+----------------+----------+-----------+-----------+----------------------------------------+
Two questions of great theoretical importance have been raised over
and over again in connexion with yeasts, namely, (1) the morphological
one as to whether yeasts are merely degraded forms of higher fungi, as
would seem implied by their tendency to form elongated, hypha-like
cells in the veils, and their development of "ascospores" as well as
by the wide occurrence of yeast-like "sprouting forms" in other fungi
(e.g. _Mucor_, Exoasci, Ustilagineae, higher Ascomycetes and
Basidiomycetes); and (2) the question as to the physiological nature
and meaning of fermentation. With regard to the first question no
satisfactory proof has as yet been given that Saccharomycetes are
derivable by culture from any higher form, the recent statements to
that effect not having been confirmed. At the same time there are
strong grounds for insisting on the resemblances between _Endomyces_,
a hyphal fungus bearing yeast-like asci, and such a form as
_Saccharomyces anomalus_. Concerning the second question, the recent
investigations of Buchner and others have shown that a ferment
(zymase) can be extracted from yeast-cells which causes sugar to break
up into carbon dioxide and alcohol. It has since been shown by Buchner
and Albert that yeast-cells which have been killed by alcohol and
ether, or with acetone, still retain the enzyme. Such material is far
more active than the zymase obtained originally by Buchner from the
expressed juice of yeast-cells. Thus alcoholic fermentation is brought
into line with the other fermentations.
_Schizosaccharomyces_ includes a few species in which the cells do not
"bud" but become elongated and then divide transversely. In the
formation of sporangia two cells fuse together by means of outgrowths,
in a manner very similar to that of _Spirogyra_; sometimes, however,
the wall between two cells merely breaks down. The fused cell becomes
a sporangium, and in it eight spores are developed. In certain cases
single cells develop parthenogenetically, without fusion, each cell
producing, however, only four spores. In _Zygosaccharomyces_ described
by Barker (1901) we have a form of the usual sprouting type, but here
again there is a fusion of two cells to form a sporangium.
_Cytology._--The study of the nucleus of yeast-cells is rendered
difficult by the presence of other deeply staining granules termed by
Guillermond _metachromatic granules_. These have often been mistaken
for nuclei and have to be carefully distinguished by differential
stains. In the process of budding the nucleus divides apparently by a
process of direct division. In the formation of spores the nucleus of
the cell divides, the protoplasm collects round the nuclei to form the
spores by free-cell formation; the protoplasm (epiplasm) not used in
this process becomes disorganized. A fusion of nuclei was originally
described by Jansens and Leblanc, but it was observed neither by Wager
nor Guillermond and is probably absent. In _Schizosaccharomyces_ and
_Zygosaccharomyces_, however, we have a fusion of nuclei in connexion
with the conjugation of cells which precedes sporangium-formation. The
theory may be put forward that the ordinary forms have been derived
from sexual forms like _Schizosaccharomyces_ and _Zygosaccharomyces_
by a loss of sexuality, the sporangium being formed
parthenogenetically without any nuclear fusion. This suggests a
possible relationship to _Eremascus_, which can only doubtfully be
placed in the Ascomycetes (_vide supra_).
_Carpoascomycetes._--The other divisions of the Ascomycetes may be
distinguished as Carpoascomycetes because they do not bear the asci
free on the mycelium but enclosed in definite fruit bodies or
ascocarps. The ascocarps can be distinguished into two portions, a
mass of sterile or vegetative hyphae forming the main mass of the
fruit body, and surrounding the fertile ascogenous hyphae which bear
at their ends the asci. When the ascogonium (female organ) is present
the ascogenous hyphae arise from it, with or without its previous
fusion with an antheridium. In other cases the ascogenous hyphae arise
directly from the vegetative hyphae. In connexion with this condition
of reduction a fusion of nuclei has been observed in _Humaria
rutilans_ and is probably of frequent occurrence. The asci may be
derived from the terminal cell of the branches of the ascogenous
hyphae, but usually they are derived from the penultimate cell, the
tip curving over to form the so-called crozier. By this means the
ascus cell is brought uppermost, and after the fusion of the two
nuclei it develops enormously and produces the ascospores. The
ascospores escape from the asci in various ways, sometimes by a
special ejaculation-mechanism. The Ascomycetes, at least the
Carpoascomycetes, exhibit a well-marked alternation of sexual and
asexual generations. The ordinary mycelium is the gametophyte since it
bears the ascogonia and antheridia when present; the ascogenous hyphae
with their asci represent the sporophyte since they are derived from
the fertilized ascogonium. The matter is complicated by the apogamous
transition from gametophyte to sporophyte in the absence of the
ascogonium; also by the fact that there are normally two fusions in
the life-history as mentioned earlier. If there are two fusions one
would expect two reductions, and Harper has suggested that the
division of the nuclei into eight in the ascus, instead of into four
spores as in most reduction processes, is associated with a _double_
reduction process in the ascus. Miss Fraser in _Humaria rutilans_
finds two reductions: a normal synaptic reduction in the first nuclear
division of the ascus, and a peculiar reduction division termed
_brachymeiosis_ in the third ascus division.
Various types of ascocarp are characteristic of the different
divisions of the Carpoascomycetes: the cleistothecium, apothecium and
perithecium.
_Perisporineae._--This includes two chief families, Erysiphaceae and
Perisporiaceae. They are characterized by an ascocarp without any
opening to the exterior, the ascospores being set free by the decay or
rupture of the ascocarp wall; such a fruit-body is termed a
_cleistothecium_ (cleistocarp). The Erysiphaceae are a sharply marked
group of forms which live as parasites. They form a superficial
mycelium on the surface of the plant, the hyphae not usually
penetrating the tissues but merely sending haustoria into the
epidermal cells. Only in rare cases is the mycelium intercellular.
Owing to their appearance they go by the popular name of mildews.
_Sphaerotheca Humuli_ is the well known hop-mildew, _Sphaerotheca
Mors-Uvae_ is the gooseberry mildew, the recent advent of which has
led to special legislation in Great Britain to prevent its spreading,
as when rampant it makes the culture of gooseberries impossible.
_Erysiphe_, _Uncinula_ and _Phyllactinia_ are other well-known genera.
The form of the fruit body, the difference and the nature of special
outgrowths upon it--the appendages--are characteristic of the various
genera. Besides peritheca the members of the Erysiphaceae possess
conidia borne in simple chains. De Bary brought forward very strong
evidence for the origin of the ascocarp in _Sphaerotheca_ and
_Erysiphe_ by a sexual process, but Harper in 1895 was the first to
prove conclusively, by the observation of the nuclear fusion, that
there was a definite fertilization in _Sphaerotheca Humuli_ by the
fusion of a male (antheridial) nucleus with a female, ascogonial
(oogonial) nucleus. Since then Harper has shown that the same process
occurs in _Erysiphe_ and _Phyllactinia_.
[Illustration: FIG. 11.--Development of _Eurotium repens_. (After De
Bary.)
A, Small portion of mycelium with conidiophore (c), and archicarp
(as).
B, The spiral archicarp (as), with the antheridium (p).
D, The same, beginning to be surrounded by the hyphae forming the
perithecium wall.
D, The perithecium.
E, F, Sections of young perithecia.
w, Parietal cells.
f, Pseudo-parenchyma.
as, Ascogonium.
G, An ascus.
H, An ascospore.]
The Perisporiaceae are saprophytic forms, the two chief genera being
_Aspergillus_ and _Penicillium_. The blue-green mould _P. crustaceum_
and the green mould _A. herbariorium_ ( = _Eurotium herbariorum_) are
extraordinarily widely distributed, moulds being found on almost any
food-material which is exposed to the air. They have characteristic
conidiophores bearing numerous conidia, and also cleistothecia which
are spherical in form and yellowish in colour. The latter arise from
the crown of a spirally coiled archicarp (bearing an ascogonium at its
end) and a straight antheridium. Vegetative hyphae then grow up and
surround these and enclose them in a continuous sheath of plectenchyma
(fig. 11). It has lately been shown by Fraser and Chambers that in
_Eurotium_ both ascogonium and antheridium contain a number of nuclei
(i.e. are coenogametes), but that the antheridium disorganizes without
passing its contents into the ascogonium. There is apparently a
reduced sexual process by the fusion of the ascogonial (female) nuclei
in pairs. _Aspergillus Oryzae_ plays an important part in
saccharifying the starch of rice, maize, &c., by means of the abundant
diastase it secretes, and, in symbiosis with a yeast which ferments
the sugar formed, has long been used by the Japanese for the
preparation of the alcoholic liquor sake. The process has now been
successfully introduced into European commerce.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 12.--_Peziza aurantiaca._ (After Krombholz, nat. size.)]
[Illustration: FIG. 13.--_Ascobolus furfuraceus._ Diagrammatic section
of the fructification. (After Janczewski.)
m, Mycelium.
c, Archicarp.
l, Pollinodium.
s, Ascogenous filaments.
a, Asri.
r, p, The sterile tissue from which the paraphyses h spring.]
_Discomycetes._--Used in its widest sense this includes the
Hysteriaceae, Phacidiaceae, Helvellaceae, &c. The group is
characterized in general by the possession of an ascocarp which,
though usually a completely closed structure during the earlier stages
of development, at maturity opens out to form a bowl or saucer-shaped
organ, thus completely exposing the layer of asci which forms the
hymenium. Such an ascocarp goes by the name of _apothecium_. Owing to
the shape of the fruit-body many of these forms are known as
cup-fungi, the cup or apothecium often attaining a large size,
sometimes several inches across (fig. 12). Functional male and female
organs have been shown to exist in _Pyronema_ and _Boudiera_; in
_Lachnea stercorea_ both ascogonia and antheridia are present, but the
antheridium is non-functional, the ascogonial (female) nuclei fusing
in pairs; this is also the case in _Humaria granulata_ and _Ascobolus
furfuraceus_, where the antheridium is entirely absent. In _H.
rutilans_, however, both sexual organs are absent and the ascogenous
hyphae arise apogamously from the ordinary hyphae of the mycelim. In
all these cases the ascogonium and antheridium contain numerous
nuclei; they are to be looked upon as gametangia in which there is no
differentiation of gametes, and since they act as single gametes they
are termed coenogametes. In some forms as in _Ascobolus_ the
ascogonium is multicellular, the various cells communicating by pores
in the transverse walls (fig. 13).
In the Helvellaceae there is no apothecium but a large irregular fruit
body which at maturity bears the asci on its surface. The development
is only slightly known, but there is some evidence for believing that
the fruit-body is closed in its very early stages.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 14.--Perithecium of Podospora fimiseda in longitudinal section
After v. Tavel.
s, Asci.
a, Paraphyses.
e, Periphyses.
m, Mycelial hyphae.]
The genus _Peziza_ (in its widest sense) may be taken as the type of
the group. Most of them grow on living plants or on dead vegetable
remains, very often on fallen wood; a number, however, are found
growing on earth which is rich in humus. The genus _Sclerotinia_ may
be mentioned here; a number of forms have been investigated by
Woronin. The conidia are fragrant and are carried by bees to the
stigma of the bilberry; here they germinate with the pollen and the
hyphae pass with the pollen tubes down the style; the former infect
the ovules and produce sclerotia, therein reducing the fruits to a
mummified condition. From the sclerotia later the apothecium develops.
One species, _S. heteroica_, is _heteroecious_; the ascospores
infecting the leaves of _Vaccinium uliginosum_, while the conidia
which then arise infect only _Ledum palustre_. This is the only case
of heteroecism known in the vegetable kingdom outside the Uredineae.
_Pyrenomycetes._--This is an extraordinarily large and varied group of
forms which mostly live parasitically or saprophytically on vegetable
tissue, but a few are parasitic on insect-larvae. The group is
characterized by a special type of ascocarp, the _perithecium_. This
is typically of a flask-shaped form opening with a small pore at the
top. The asci live at the bottom often mixed with paraphyses, while
the upper "neck" of the flask is lined with special hyphae, the
periphyses, which aid in the ejection of the spores (fig. 14). The
simpler forms bear the perithecia directly on the mycelium, but the
more highly developed forms often bear them on a special mycelial
development--the stroma, which is often of large size and special
shape and colour, and of dense consistence. The cytological details of
development of the perithecia are not well known; most of them appear
to develop their ascogenous hyphae in an apogamous way without any
connexion with an ascogonium. Besides the special ascocarps, accessory
reproductive organs are known in the majority of cases in the form of
conidia.
_Tuberineae._--These are a small group of fungi including the
well-known truffles. They are found living saprophytically (in part
parasitically) underground in forests. The asci are developed in the
large dense fruit bodies (cleistothecia) and the spores escape by the
decay of the wall. The fruit-body is of complicated structure, but its
early stages of development are not known. Many of the fruit-bodies
have a pleasant flavour and are eaten under the name of truffles
(_Tuber brumale_ and other species). The exact life-history of the
truffle is not known.
_Laboulbeniineae_ are a group of about 150 species of fungi found on
insects, especially beetles, and principally known from the researches
of Thaxter in America. The plant is a small, dark brown, erect
structure (receptacle) of a few cells, and 1-10 mm. high, attached to
the insect by the lowermost end (foot), and easily mistaken for a hair
or similar appendage of the insect. The receptacle ends above in
appendages, each consisting of one or a few cells, some of which are
the male organs, others the female organs, and others again may be
barren hairs. The male organ (antheridium) consists of a few cells,
the terminal one of which either abstricts from its end, or emits from
its interior the non-motile spermatia, reminding us of those of the
Florideae. The female organ is essentially a flask-shaped structure;
the neck of the flask growing out as the trichogyne, and the belly
composed of an axial carpogenic cell surrounded by investing cells,
and with one cell (trichophoric) between it and the trichogyne. These
three elements--trichogyne, trichophoric cell, and carpogenic
cell--are regarded as the procarp. The spermatia have been shown by
Thaxter to fuse with the trichogyne, after which the axial cell below
(carpogenic cell) undergoes divisions, and ultimately forms asci
containing ascospores, while cells investing this form a perithecium,
the whole structure reminding us essentially of the fructification of
a Pyrenomycete. Many modifications in details occur, and the plants
may be dioecious. No injury is done to the infested insects. It has
lately been shown that there is a fusion of nuclei in connexion with
ascus formation, so that there can be no doubt of the position of this
extraordinary group of plants among the Ascomycetes. The various cells
of these organisms are connected by large pits which are traversed by
thick protoplasmic threads connecting one cell with the next. In this
point and in their method of fertilization the Laboulbeniineae suggest
a possible relationship of Ascomycetes and the Red Algae.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 15.--_Armillaria mellea._ (After Ruhland.)
A, Young basidium with the two primary nuclei.
B, After fusion of the two nuclei. _Hypholoma appendiculatum_.
C, A basidium before the four nuclei derived from the secondary
nucleus of the basidium have passed into the four basidiospores.
D, Passage of a nucleus through the sterigma into the basidiospore.]
_Basidiales._--This very large group of plants is characterized by the
possession of a special type of conidiophore--the basidium, which
gives its name to the group. The basidium is a unicellular or
multicellular structure from which four basidiospores arise as
outgrowths; it starts as a binucleate structure, but soon, like the
ascus, becomes uninucleate by the fusion of the two nuclei. Then two
successive nuclear divisions occur resulting in the formation of four
nuclei which later migrate respectively into the four basidiospores
(fig. 15). The Basidiales are further characterized by the complete
loss of normal sexuality, but at some time or other in the
life-history there takes place an association of two nuclei in a cell;
the two nuclei are derived from separate cells or possibly in some
cases are sister nuclei of the same cell. The two nuclei when once
associated are termed "conjugate" nuclei, and they always divide at
the same time, a half of each passing into each cell. This conjugate
condition is finally brought to a close by the nuclear fusion in the
basidium. Between the nuclear association and the nuclear fusion in
the basidium many thousands of cell generations may be intercalated.
This nuclear association of equivalent nuclei apparently represents a
reduced sexual process (like the fusion of female nuclei in _Humaria
granulata_ and of vegetative nuclei in _H. rutilans_, among the
Ascomycetes) in which, however, the actual fusion (normally, in a
sexual process, occurring immediately after association) is delayed
until the formation of the basidium. During the tetrad division in the
basidium nuclear reduction occurs. There is thus in all the Basidiales
an alternation of generations, obscured, however, by the apogamous
transition from the gametophyte to sporophyte. The sporophyte may be
considered to begin at the stage of nuclear association and end with
the nuclear reduction in the basidium.
[Illustration: FIG. 16.--_Puccinia graminis._
A, Mass of teleutospores (t) on a leaf of couch-grass.
e, Epidermis ruptured.
b, Sub-epidermal fibres. (After De Bary.)
B, Part of vertical section through leaf of Berberis vulgaris, with
a, aecidium fruits, p, peridium, and sp, spermogonia. (After Sachs.)
C, Mass of uredospores (ur), with one teleutospore (t).
sh, Sub-hymenial hyphae. (After De Bary.)]
_Uredineae._--This is a large group of about 2000 forms. They are all
intercellular parasites living mostly on the leaves of higher plants.
Owing to the presence of oily globules of an orange-yellow or
rusty-red colour in their hyphae and spores they are termed
Rust-Fungi. They are distinguished from the other fungi and the rest
of the Basidiales by the great variety of the spores and the great
elaboration of the life-history to be found in many cases. Five
different kinds of spores may be present--teleutospores, sporidia ( =
basidiospores), aecidiospores, spermatia and uredospores (fig. 16).
The teleutospore, with the sporidia which arise from it, is always
present, and the division into genera is based chiefly on its
characters. The teleutospore puts forth on germination a four-celled
structure, the promycelium or basidium, and this bears later four
sporidia or basidiospores, one on each cell. When the sporidia infect
a plant the mycelium so produced gives origin to aecidiospores and
spermatia; the aecidiospores on infection produce a mycelium which
bears uredospores and later teleutospores. This is the life-history of
the most complicated forms, of the so-called _eu_ forms. In the
_opsis_ forms the uredospores are absent, the mycelium from the
aecidiospores producing directly the teleutospores. In _brachy_ and
_hemi_ the aecidiospores are absent, the mycelium from the sporidia
giving origin directly to the uredospores; the former possess
spermatia, in the latter they are absent. In _lepto_ and _micro_ forms
both aecidiospores and uredospores are absent, the sporidia producing
a mycelium which gives rise directly to teleutospores; in the _lepto_
forms the teleutospores can germinate directly, in the _micro_ forms
only after a period of rest. We have thus a series showing a
progressive reduction in the complexity of the life-history, the
_lepto_ and _micro_ forms having a life-history like that of the
Basidiomycetes. The _eu_ and _opsis_ forms may exhibit the remarkable
phenomenon of heteroecism, i.e. the dependence of the fungus on two
distinct host-plants for the completion of the life-history.
Heteroecism is very common in this group and is now known in over one
hundred and fifty species. In all cases of heteroecism the sporidia
infect one host leading to the production of aecidiospores and
spermatia (if present), while the aecidiospores are only able to
infect another host on which the uredospores (if present) and the
teleutospores are developed. A few examples are appended:
+-----------------------------+------------------+------------------+
| Species. | Teleutospores on | Aecidiospores on |
+-----------------------------+------------------+------------------+
| _Coleosporium Senecionis_ | _Pinus_ | _Senecio_ |
| _Melampsora Rostrupi_ | _Populus_ | _Mecurialis_ |
| _Pucciniastrum Goeppertiana_| _Vaccinium_ | _Abies_ |
| _Gymnosporangium Sabinae_ | _Juniperus_ | _Pyrus_ |
| _Uromyces Pisi_ | _Pisum, &c._ | _Euphorbia_ |
| _Puccinia graminis_ | _Triticum, &c._ | _Berberis_ |
| _P. dispersa_ | _Secale, &c._ | _Anchusa_ |
| _P. coronata_ | _Agrostis_ | _Rhamnus_ |
| _P. Ari-Phalaridis_ | _Phalaris_ | _Arum_ |
| _P. Caricis_ | _Carex_ | _Urtica_ |
| _Cronartium Ribicola_ | _Ribes_ | _Pinus_ |
| _Chrysomyxa Rhododendri_ | _Rhododendron_ | _Picea_ |
+-----------------------------+------------------+------------------+
Some of the Uredineae also exhibit the peculiarity of the development
of biologic forms within a single morphological species, sometimes
termed specialization of parasitism; this will be dealt with later
under the section Physiology.
[Illustration: From Strasburger's _Lehrbuch der Botanik_, by
permission of Gustav Fischer.
FIG. 17.--_Phragmidium Violaceum._ (After Blackman.)
A, Portion of a young aecidium.
st, Sterile cell.
a, Fertile cells; at a2 the passage of a nucleus from the adjoining
cell is seen.
B, Formation of the first spore-mother-cell (sm), from the basal
cell (a) of one of the rows of spores.
C, A further stage in which from sm1 the first aecidiospore (a) and
the intercalary cell (z) have arisen.
sm2, The second spore-mother-cell.
D, Ripe aecidiospore.]
_Cytology of Uredineae._--The study of the nuclear behaviour of the
cells of the Uredineae has thrown great light on the question of
sexuality. This group like the rest of the Basidiales exhibits an
association of nuclei at some point in its life-history, but unlike
the case of the Basidiomycetes the point of association in the
Uredineae is very well defined in all those forms which possess
aecidiospores. We find thus that in the _eu_ and _opsis_ forms the
association of nuclei takes place at the base of the aecidium which
produces the aecidiospores. There we find an association of nuclei
either by the fusion of two similar cells as described by Christmann
or by the migration of the nucleus of a vegetative cell into a special
cell of the aecidium. After this association the nuclei continue in
the conjugate condition so that the aecidiospores, the
uredospore-bearing mycelium, the uredospores and the young
teleutospores all contain two paired nuclei in their cells (fig. 17).
Before the teleutospore reaches maturity the nuclei fuse, and the
uninucleate condition then continues again until aecidium formation.
In the _hemi_, _brachy_, _micro_ and _lepto_ forms, which possess no
aecidium, we find that the association takes place at various points
in the ordinary mycelium but always before the formation of the
uredospores in the _hemi_ and _brachy_ forms, and before the formation
of teleutospores in _micro_ and _lepto_ form. Whether the association
of nuclei in the ordinary mycelium takes place by the migration of a
nucleus from one cell to another or whether two daughter nuclei become
conjugate in one cell, is not yet clear. The most reasonable
interpretation of the spermatia is that they are abortive male cells.
They have never been found to cause infection, and they have not the
characters of conidia; the large size of their nuclei, the reduction
of their cytoplasm and the absence of reserve material and their thin
cell wall all point to their being male gametes. Although in the forms
without aecidia the two generations are not sharply marked off from
one another, we may look up the generation with single nuclei in the
cells as the gametophyte and that with conjugate nuclei as the
sporophyte. The subjoined diagram will indicate the relationship of
the forms.
_Basidiomycetes._--This group is characterized by its greatly reduced
life-history as compared with that of the _eu_ forms among the
Uredineae. All the forms have the same life-history as the _lepto_
forms of that group, so that there is no longer any trace of sexual
organs. There is also a further reduction in that the basidium is not
derived from a teleutospore but is borne directly on the mycelium.
Formerly, before the relationship of promycelium and basidium were
understood, the Uredineae were considered as quite independent of the
Basidiomycetes. Later, however, these Uredineae were placed as a mere
subdivision of the Basidiomycetes. Although the Uredineae clearly lead
on to the Basidiomycetes, yet owing to their retaining in many cases
definite traces of sexual organs they are clearly a more primitive
group. Their marked parasitic habit also separates them off, so that
they are best included with the Basidiomycetes in a larger cohort
which may be called Basidiales. Most of Basidiomycetes are
characterized by the large sporophore on which the basidia with its
basidiospores are borne.
[Illustration: From _Annals of Botany_, by permission of the Clarendon
Press.
FIG. 18.]
It must be clearly borne in mind that though the Basidiomycetes show
no traces of differentiated sexual organs yet, like the _micro_ and
_lepto_ forms of the Uredineae, they still show (in the association of
nuclei and later fusion of nuclei in the basidium), a reduced
fertilization which denotes their derivation, through the Uredineae,
from more typically sexual forms. No one has yet made out in any form
the exact way in which the association of nuclei takes place in the
group. The mycelium is always found to contain conjugate nuclei before
the formation of basidia, but the point at which the conjugate
condition arises seems very variable. Miss Nichols finds that it
occurs very soon after the germination of the spore in _Coprinus_, but
no fusion of cells or migration of nuclei was to be observed.
_Protobasidiomycetes._--This, by far the smaller division of
Basidiomycetes, includes those forms which have a septate basidium.
There are three families--Auriculariaceae, Pilacreaceae and
Tremellinaceae. The first named contains a small number of forms with
the basidium divided like the promycelium of the Uredineae. They are
characterized by their gelatinous consistence and large size of their
sporophore. _Hirneola_ (_Auricularia_) _Auricula-Judae_ is the
well-known Jew's Ear, so named from the resemblance of the sporophore
to a human ear.
[Illustration: FIG. 19.--_Amanita muscaria_.
A, The young plant.
B, The mature plant.
C, Longitudinal section of mature plant.
p, The _pileus_.
g, The gills.
a, The _annulus_, or remnant of _velum partiale_,
v, Remains of _volva_ or _velum universale_.
s, The stalk.]
The Pilacreaceae are a family found by Brefeld to contain the genus
_Pilacre_. _P. Petersii_ has a transversely divided basidium as in
_Auriculariaceae_, but the basidia are surrounded with a peridium-like
sheath. The _Tremellinaceae_ are characterized by the possession of
basidia which are divided by two _vertical_ walls at right angles to
one another. From each of the four segments in the case of _Tremella_
a long outgrowth arises which reaches to the surface of the hymenium
and bears the basidiospores. In _Dacryomyces_ only two outgrowths and
two spores are produced.
_Autobasidiomycetes._--In this by far the larger division of the
Basidiomycetes the basidia are undivided and the four basidiospores
are borne on short sterigmata nearly always at the apex of the
basidium. The group may be divided into two main divisions,
_Hymenomycetes_ and _Gasteromycetes_.
_Hymenomycetes_ are a very large group containing over 11,000 species,
most of which live in soil rich in humus or on fallen wood or stems, a
few only being parasites. In the simplest forms (e.g. _Exobasidium_)
the basidia are borne directly on the ordinary mycelium, but in the
majority of cases the basidia are found developed in layers (hymenium)
on special sporophores of characteristic form in the various groups.
In these sporophores (such as the well-known toadstools and mushrooms
where the ordinary vegetative mycelium is underground) we have
structures specially developed for bearing the basidiospores and
protecting them from rain, &c., and for the distribution of the
spores--see earlier part of article on distribution of spores (figs.
19 and 20). The underground mycelium in many cases spreads wider and
wider each year, often in a circular manner, and the sporophores
springing from it appear in the form of a ring--the so-called fairy
rings. _Armillaria melleus_ and _Polyporus annosus_ are examples of
parasitic forms which attack and destroy living trees, while _Merulius
lacrymans_ is the well-known "dry rot" fungus.
[Illustration: FIG. 20.--_Agaricus mucidus_. Portion of hymenium. s,
Sporidia; st, sterigmata; g, sterile cells; c, cystidium, with
operculum o.]
_Gasteromycetes_ are characterized by having closed sporophores or
fruit-bodies which only open after the spores are ripe and then often
merely by a small pore. The fruit-bodies are of very various shapes,
showing a differentiation into an outer _peridium_ and an inner
spore-bearing mass, the _gleba_. The gleba is usually differentiated
into a number of chambers which are lined directly by the hymenium
(basidial layer), or else the chambers contain an interwoven mass of
hyphae, the branches of which bear the basidia. By the breaking down
of the inner tissues the spores often come to lie as a loose powdery
mass in the interior of the hollow fruit-body, mixed sometimes with a
capillitium. The best-known genera are _Bovista, Lycoperdon_
(puff-ball) _Scleroderma, Geaster_ (earth-star, q.v.). In the
last-named genus the peridium is double and the outer layer becomes
ruptured and spreads out in the form of star-shaped pieces; the inner
layer, however, merely opens at the apex by a small pore.
The most complex members of the Gasteromycetes belong to the
_Phalloideae_, which is sometimes placed as a distinct division of the
Autobasidiomycetes. _Phallus impudicus_, the stink-horn, is
occasionally found growing in woods in Britain. The fruit-body before
it ruptures may reach the size of a hen's egg and is white in colour;
from this there grows out a hollow cylindrical structure which can be
distinguished at the distance of several yards by its disgusting
odour. It is highly poisonous.
_Physiology._--The physiology of the fungi comes under the head of that
of plants generally, and the works of Pfeffer, Sachs, Vines, Darwin and
Klebs may be consulted for details. But we may refer generally here to
certain phenomena peculiar to these plants, the life-actions of which
are restricted and specialized by their peculiar dependence on organic
supplies of carbon and nitrogen, so that most fungi resemble the
colourless cells of higher plants in their nutrition. Like these they
require water, small but indispensable quantities of salts of potassium,
magnesium, sulphur and phosphorus, and supplies of carbonaceous and
nitrogenous materials in different stages of complexity in the different
cases. Like these, also, they respire oxygen, and are independent of
light; and their various powers of growth, secretion, and general
metabolism, irritability, and response to external factors show similar
specific variations in both cases. It is quite a mistake to suppose
that, apart from the chlorophyll function, the physiology of the
fungus-cell is fundamentally different from that of ordinary
plant-cells. Nevertheless, certain biological phenomena in fungi are
especially pronounced, and of these the following require particular
notice.
_Parasitism._--Some fungi, though able to live as saprophytes,
occasionally enter the body of living plants, and are thus termed
facultative parasites. The occasion may be a wound (e.g. _Nectria_,
_Dasyscypha_, &c.), or the enfeeblement of the tissues of the host, or
invigoration of the fungus, the mycelium of which then becomes strong
enough to overcome the host's resistance (_Botrytis_). Many fungi,
however, cannot complete their life-history apart from the host-plant.
Such _obligate_ parasites may be epiphytic (_Erysipheae_), the
mycelium remaining on the outside and at most merely sending haustoria
into the epidermal cells, or endophytic (_Uredineae_, _Ustilagineae_,
&c.), when the mycelium is entirely inside the organs of the host. An
epiphytic fungus is not necessarily a parasite, however, as many
saprophytes (moulds, &c.) germinate and develop a loose mycelium on
living leaves, but only enter and destroy the tissues after the leaf
has fallen; in some cases, however, these saprophytic epiphytes can do
harm by intercepting light and air from the leaf (_Fumago_, &c.), and
such cases make it difficult to draw the line between saprophytism and
parasitism. Endophytic parasites may be intracellular, when the fungus
or its mycelium plunges into the cells and destroys their contents
directly (_Olpidium_, _Lagenidium_, _Sclerotinia_, &c.), but they are
far more frequently intercellular, at any rate while young, the
mycelium growing in the lacunae between the cells (_Peronospora_,
_Uredineae_) into which it may send short (_Cystopus_), or long and
branched (_Peronospora Calotheca_) haustoria, or it extends in the
middle lamella (_Ustilago_), or even in the solid substance of the
cell-wall (_Botrytis_). No sharp lines can be drawn, however, since
many mycelia are intercellular at first and subsequently become
intracellular (_Ustilagineae_), and the various stages doubtless
depend on the degrees of resistance which the host tissues are able to
offer. Similar gradations are observed in the direct effect of the
parasite on the host, which may be local (_Hemileia_) when the
mycelium never extends far from the point of infection, or general
(_Phytophthora_) when it runs throughout the plant. Destructive
parasites rapidly ruin the whole plant-body (_Pythium_), whereas
restrained parasites only tax the host slightly, and ill effects may
not be visible for a long time, or only when the fungus is epidemic
(_Rhytisma_). A parasite may be restricted during a long
incubation-period, however, and rampant and destructive later
(_Ustilago_). The latter fact, as well as the extraordinary
fastidiousness, so to speak, of parasites in their choice of hosts or
of organs for attack, point to reactions on the part of the
host-plant, as well as capacities on that of the parasite, which may
be partly explained in the light of what we now know regarding enzymes
and chemotropism. Some parasites attack many hosts and almost any
tissue or organ (_Botrytis cinerea_), others are restricted to one
family (_Cystopus Candidus_) or genus (_Phytophthora infestans_) or
even species (_Pucciniastrum Padi_), and it is customary to speak of
root-parasites, leaf-parasites, &c., in expression of the fact that a
given parasite occurs only on such organs--e.g. _Dematophora necatrix_
on roots, _Calyptospora Goeppertiana_ on stems, _Ustilago Scabiosae_
in anthers, _Claviceps purpurea_ in ovaries, &c. Associated with these
relations are the specializations which parasites show in regard to
the age of the host. Many parasites can enter a seedling, but are
unable to attack the same host when older--e.g. _Pythium_,
_Phytophthora omnivora_.
_Chemotropism._--Taken in conjunction with Pfeffer's beautiful
discovery that certain chemicals exert a distinct attractive influence
on fungus hyphae (_chemotropism_), and the results of Miyoshi's
experimental application of it, the phenomena of enzyme-secretion
throw considerable light on the processes of infection and parasitism
of fungi. Pfeffer showed that certain substances in definite
concentrations cause the tips of hyphae to turn towards them; other
substances, though not innutritious, repel them, as also do nutritious
bodies if too highly concentrated. Marshall Ward showed that the
hyphae of _Botrytis_ pierce the cell-walls of a lily by secreting a
cytase and dissolving a hole through the membrane. Miyoshi then
demonstrated that if _Botrytis_ is sown in a lamella of gelatine, and
this lamella is superposed on another similar one to which a
chemotropic substance is added, the tips of the hyphae at once turn
from the former and enter the latter. If a thin cellulose membrane is
interposed between the lamellae, the hyphae nevertheless turn
chemotropically from the one lamella to the other and pierce the
cellulose membrane in the process. The hyphae will also dissolve their
way through a lamella of collodion, paraffin, parchment paper,
elder-pith, or even cork or the wing of a fly, to do which it must
excrete very different enzymes. If the membrane is of some impermeable
substance, like gold leaf, the hyphae cannot dissolve its way through,
but the tip finds the most minute pore and traverses the barrier by
means of it, as it does a stoma on a leaf We may hence conclude that a
parasitic hyphae pierces some plants or their stomata and refuses to
enter others, because in the former case there are chemotropically
attractive substances present which are absent from the latter, or are
there replaced by repellent poisonous or protective substances such as
enzymes or antitoxins.
_Specialization of Parasitism._--The careful investigations of recent
years have shown that in several groups of fungi we cannot be content
to distinguish as units morphologically different species, but we are
compelled to go deeper and analyse further the species. It has been
shown especially in the _Uredineae_ and _Erysiphaceae_ that many forms
which can hardly be distinguished morphologically, or which cannot be
differentiated at all by structural characters, are not really
homogeneous but consist of a number of forms which are sharply
distinguishable by their infecting power. Eriksson found, for example,
that the well-known species _Puccinia graminis_ could be split up into
a number of forms which though morphologically similar were
physiologically distinct. He found that the species really consisted
of six distinct races, each having a more or less narrow range of
grasses on which it can live. The six races he named _P. graminis
Secalis_, _Tritici_, _Avenae_, _Airae_, _Agrostis_, _Poae_. The first
named will grow on rye and barley but not on wheat or oat. The form
_Tritici_ is the least sharply marked and will grow on wheat, barley,
rye and oat but not on the other grasses. The form _Avenae_ will grow
on oat and many grasses but not on the other three cereals mentioned.
The last three forms grow only on the genera _Aira_, _Agrostis_ and
_Poa_ respectively. All these forms have of course their
aecidium-stage on the barberry. The terms biologic forms, biological
species, physiological species, physiological races, specialized forms
have all been applied to these; perhaps the term biologic forms is the
most satisfactory. A similar specialization has been observed by
Marshall Ward in the _Puccinia_ parasitic on species of _Bromus_, and
by Neger, Marchal and especially Salmon in the Erysiphaceae. In the
last-named family the single morphological species _Erysiphe graminis_
is found growing on the cereals, barley, oat, wheat, rye and a number
of wild grasses (such as _Poa_, _Bromus_, _Dactylis_). On each of
these host-plants the fungus has become specialized so that the form
on barley cannot infect the other three cereals or the wild grasses
and so on. Just as the uredospores and aecidiospores both show these
specialized characters in the case of _Puccinia graminis_ so we find
that both the conidia and ascospores of _E. graminis_ show this
phenomenon. Salmon has further shown in investigating the relation of
_E. graminis_ to various species of the genus, _Bromus_, that certain
species may act as "bridging species," enabling the transfer of a
biologic form to a host-plant which it cannot normally infect. Thus
the biologic form on _B. racemosus_ cannot infect _B. commutatus_. If,
however, conidia from _B. racemosus_ are sown on _B. hordaceus_, the
conidia which develop on that plant are now able to infect _B.
commutatus_; thus _B. hordaceus_ acts as a bridging species. Salmon
also found that injury of a leaf by mechanical means, by heat, by
anaesthetics, &c., would affect the immunity of the plant and allow
infection by conidia which was not able to enter a normal leaf. The
effect of the abnormal conditions is probably to stop the production
of, or weaken or destroy the protective enzymes or antitoxins, the
presence of which normally confers immunity on the leaf.
_Symbiosis._--The remarkable case of life in common first observed in
lichens, where a fungus and an alga unite to form a compound
organism--the lichen--totally different from either, has now been
proved to be universal in these plants, and lichens are in all cases
merely algae enmeshed in the interwoven hyphae of fungi (see LICHENS).
This dualism, where the one constituent (alga) furnishes
carbohydrates, and the other (fungus) ensures a supply of mineral
matters, shade and moisture, has been termed _symbiosis_. Since then
numerous other cases of symbiosis have been demonstrated. Many trees
are found to have their smaller roots invaded by fungi and deformed by
their action, but so far from these being injurious, experiments go to
show that this mycorhiza (fungus-root) is necessary for the well-being
of the tree. This is also the case with numerous other plants of moors
and woodlands--e.g. Ericaceae, Pyrolaceae, Gentianaceae, Orchidaceae,
ferns, &c. Recent experiments have shown that the difficulties of
getting orchid seeds to germinate are due to the absence of the
necessary fungus, which must be in readiness to infect the young
seedling immediately after it emerges from the seed. The well-known
failures with rhododendrons, heaths, &c., in ordinary garden soils are
also explained by the need of the fungus-infected peat for their
roots. The role of the fungus appears to be to supply materials from
the leaf-mould around, in forms which ordinary root-hairs are
incapable of providing for the plant; in return the latter supports
the fungus at slight expense from its abundant stores of reserve
materials. Numerous other cases of symbiosis have been discovered
among the fungi of fermentation, of which those between _Aspergillus_
and yeast in sake manufacture, and between yeasts and bacteria in
kephir and in the ginger-beer plant are best worked out. For cases of
symbiosis see BACTERIOLOGY.
AUTHORITIES.--_General_: Engler and Prantl, _Die naturlichen
Pflanzenfamilien_, i. Teil (1892 onwards); Zopf, _Die Pilze_ (Breslau,
1890); De Bary, _Comparative Morphology of Fungi_, &c. (Oxford, 1887);
von Tafel, _Vergleichende Morphologie der Pilze_ (Jena, 1892);
Brefeld, _Unters. aus dem Gesamtgebiete der Mykologie_, Heft i. 13
(1872-1905); Lotsy, _Vortrage uber botanische Stammesgeschichte_
(Jena, 1907). _Distribution_, &c.: Cooke, _Introduction to the Study
of Fungi_ (London, 1895); Felix in _Zeitschr. d. deutsch. geologisch.
Gesellsch._ (1894-1896); Staub, _Sitzungsber. d. bot. Sec. d. Kgl.
ungarischen naturwiss. Gesellsch. zu Budapest_ (1897). _Anatomy_, &c.:
Bommer, "Sclerotes et cordons myceliens," _Mem. de l'Acad. Roy. de
Belg._ (1894); Mangin, "Observ. sur la membrane des mucorinees,"
_Journ. de Bot._ (1899); Zimmermann, _Die Morph. und Physiologie des
Pflanzenzellkernes_ (Jena, 1896); Wisselingh, "Microchem. Unters. uber
die Zellwande d. Fungi," _Pringsh. Jahrb._ B. 31, p. 619 (1898);
Istvanffvi, "Unters. uber die phys. Anat. der Pilze," _Prings. Jahrb._
(1896). _Spore Distribution_: Fulton, "Dispersal of the Spores of
Fungi by Insects," _Ann. Bot._ (1889); Falck, "Die Sporenverbreitung
bei den Basidiomyceten," _Beitr. zur Biol. d. Pflanzen_, ix. (1904).
_Spores and Sporophores_: Zopf, _Die Pilze_; also the works of von
Tafel and Brefeld. _Classification_: van Tieghem, _Journ. de bot._ p.
77 (1893), and the works of Brefeld, Engler and Prantl, von Tafel,
Saccardo and Lotsy already cited, _Oomycetes_: Wager, "On the
Fertilization of _Peronospora parasitica_," _Ann. Bot._ vol. xiv.
(1900); Stevens, "The Compound Oosphere of _Albugo Bliti_," _Bot.
Gaz._ vol. 28 (1899); "Gametogenesis and Fertilization in _Albugo_,"
ibid. vol. 32 (1901); Miyake, "The Fertilization of _Pythium de
Baryanum_," _Ann. of Bot._ vol. xv. (1901); Trow, "On Fertilization in
the Saprolegnieae," _Ann. of Bot._ vol. xviii. (1904); Thaxter, "New
and Peculiar Aquatic Fungi," _Bot. Gaz._ vol. 20 (1895); Lagerheim,
"Unters. uber die Monoblepharideae," _Bih. Svenska Vet. Acad.
Handlingar_, 25. Afd. iii. (1900); Woronin, "Beitrag zur Kenntnis der
Monoblepharideen," _Mem. de l'Acad. Imp. d. Sc. de St-Petersbourg_, 8
ser. vol. 16 (1902). _Zygomycetes_: Harper, "Cell-division in
Sporangia and Asci," _Ann. Bot._ vol. xiii. (1899); Klebs, _Die
Bedingungen der Fortpflanzung_, &c. (Jena, 1896), and "Zur Physiologie
der Fortpflanzung" _Prings. Jahr._ (1898 and 1899), "Uber _Sporodinia
grandis_," _Bot. Zeit._ (1902); Falck, "Die Bedingungen der
Zygotenbildung bei Sporodinia grandis," Cohn's Beitr. z. Biol. d.
Pflanzen, Bd. 8 (1902); Gruber "Verhalten der Zellkerne in den
Zygosporen von _Sporodinia grandis_," _Ber. d. deutschen bot. Ges._
Bd. 19 (1901); Blakeslee, "Sexual Reproduction in the Mucorineae,"
_Proc. Am. Acad._ (1904); "Zygospore germination in the Mucorineae,"
_Annales mycologici_ (1906). _Ustilagineae_: Plowright, _British
Uredineae and Ustilagineae_ (London, 1889); Massee, _British Fungi_
(Phycomycetes and Ustilagineae) (London, 1891); Brefeld, _Unters. aus
dem Gesamtgeb. der Mykol._ Hefte xi. and xii.; and Falck, "Die
Bluteninfektion bei den Brandpilzen," ibid. Heft xiii. 1905; Dangeard,
"La Reproduction sexuelle des Ustilaginees," C.R., Oct. 9, 1893;
Maire, "Recherches cytologiques et taxonomiques sur les
Basidiomyceten," _Annexe au Bull. de la Soc. Mycol. de France_ (1902).
_Saccharomycetaceae_: Jorgensen, _The Micro-organisms of Fermentation_
(1899); Barker, _Ann. of Bot._ vol. xiv. (1901); "On Spore-formation
among the Saccharomycetes," _Journ. of the Fed. Institute of Brewing_,
vol. 8 (1902); Guillermond, _Recherches cytologiques sur les levures_
(Paris, 1902); Hansen, _Centralbl. f. Bakt. u. Parasitenp._ Abt. ii.
Bd. 12 (1904). _Exoascaceae_: Giesenhagen, "_Taphrina, Exoascus,
Magnusiella_" (complete literature given), _Bot. Zeit._ Bd. 7 (1901).
_Erysiphaceae_: Harper, "Die Entwicklung des Perithecium bei
_Sphaerotheca castagnei_," _Ber. d. deut bot Ges._ (1896); "Sexual
Reproduction and the Organization of the Nucleus in certain Mildews,"
_Publ. Carnegie Institution_ (Washington, 1906); Blackman & Fraser,
"Fertilization in _Sphaerotheca_," _Ann. of Bot._ (1905).
_Perisporiaceae_: Brefeld, _Untersuchungen aus dem Gesamtgeb. der
Mykol._ Heft 10 (1891); Fraser and Chamber, _Annales mycologici_
(1907). _Discomycetes_: Harper, "Uber das Verhalten der Kerne bei
Ascomyceten," _Jahr. f. wiss. Bot._ Bd. 29 (1890); "Sexual
Reproduction in _Pyronema confluens_," _Ann. of Bot._ 14 (1900);
Claussen, "Zur Entw. der Ascomyceten," Boudiera, Bot. Zeit. Bd. 63
(1905); Dangeard, "Sur le _Pyronema confluens_," _Le Botaniste_, 9
serie (1903) (and numerous papers in same journal earlier and later);
Ramlow, "Zur Entwick. von _Thelebolus stercoren_," _Bot. Zeit._
(1906); Woronin, "Uber die Sclerotienkrankheit der Vaccineen Beeren,"
_Mem. de l'Acad. Imp. des Sciences de St-Petersbourg_, 7 serie, 36
(1888); Dittrich, "Zur Entwickelungsgeschichte der Helvellineen,"
Cohn's _Beitr. z. Biol. d. Pflanzen_ (1892). _Pyrenomycetes_: Fisch,
"Beitr. z. Entwickelungsgeschichte einiger Ascomyceten," _Bot. Zeit._
(1882); Frank, "Uber einige neue u. weniger bekannte Pflanzkrankh.,"
_Landw. Jahrb._ Bd. 12 (1883); Ward, "_Onygena equina_, a
horn-destroying fungus," _Phil. Trans._, vol. 191 (1899); Dawson, "On
the Biology of Poroniapunctata," Ann. of Bot. 14 (1900). _Tuberineae_:
Buchholtz, "Zur Morphologie u. Systematik der Fungi hypogaei," _Ann.
Mycol._ Bd. 1 (1903); Fischer in Engler and Prantl, _Die naturlichen
Pflanzenfamilien_ (1896). _Laboulbeniineae_: Thaxter, "Monograph of
the Laboulbeniaceae," _Mem. Amer. Acad. of Arts and Sciences_, vol. 12
(1895). _Uredineae_: Eriksson and Henning, _Die Getreideroste_
(Stockholm, 1896); Eriksson, _Botan. Gaz._ vol. 25 (1896); "On the
Vegetative Life of some Uredineae," Ann. of Bot. (1905); Klebahn, _Die
wirtwechselnden Rostpilze_ (Berlin, 1904); Sapin-Trouffy, "Recherches
histologiques sur la famille des Uredinees," _Le Botaniste_
(1896-1897); Blackman, "On the Fertilization, Alternation of
Generations and General Cytology of the Uredineae," _Ann. of Bot._
vol. 18 (1904); Blackman and Fraser, "Further Studies on the Sexuality
of Uredineae," _Ann. of Bot._ vol. 20 (1906); Christman, "Sexual
Reproduction of Rusts," _Ann. of Bot._ vol. 20 (1906); Ward, "The
Brooms and their Rust Fungus," _Ann. of Bot._ vol. 15 (1901).
_Basidiomycetes_: Dangeard, "La Reprod. sexuelle des Basidiomycetes,"
_Le Botaniste_ (1894 and 1900); Maire, "Recherches cytologiques et
taxonomiques sur les Basidiomycetes," _Annexe du Bull. de la Soc.
Mycol. de France_ (1902); Moller, "Protobasidiomyceten," _Schimper's
Mitt. aus den Tropen_, Heft 8 (Jena, 1895); Nichols, "The Nature and
Origin of the Binucleated Cells in certain Basidiomycetes," _Trans.
Wisconsin Acad. of Sciences_, vol. 15 (1905); Wager, "The Sexuality of
the Fungi," _Ann. of Bot._ 13 (1899); Woronin, "_Exobasidium
Vaccinii_," _Verh. Naturf. Ges. zu Freiburg_, Bd. 4 (1867).
_Fermentation_: Buchner, "Gahrung ohne Hefezellen," _Bot. Zeit._ Bd.
18 (1898); Albert, _Cent. f. Bakt._ Bd. 17 (1901); Green, _The
Soluble Ferments and Fermentation_ (Cambridge, 1899). _Parasitism_:
"On some Relations between Host and Parasite," _Proc. Roy. Soc_. vol.
47 (1890); "A Lily Disease," _Ann. of Botany_, vol. 2 (1888); Eriksson
& Hennings, _Die Getreideroste (vide supra_); Ward, "On the Question
of Predisposition and Immunity in Plants," _Proc. Cambridge Phil.
Soc_. vol. 11 (1902); also _Annals of Bot_. vol. 16 (1902) and vol. 19
(1905); Neger, "Beitr. z. Biol. d. Erysipheen" _Flora_, Bde. 88 and 90
(1901-1902); Salmon, "Cultural Experiments with 'Biologic Forms' of
the Erysiphaceae," _Phil. Trans_. (1904); "On Erysiphe graminis and
its adaptative parasitism within the genus, _Bromus_," _Ann. Mycol_.
vol. 11 (1904), also _Ann. of Bot_. vol. 19 (1905). _Symbiosis_: Ward,
"The Ginger-Beer Plant," _Phil. Trans. Roy. Soc_. (1892); "Symbiosis,"
_Ann. of Bot_. 13 (1899); Shalk, "Der Sinn der Mykorrhizenbildung,"
_Jahrb. f. wiss. Bot_. Bd. 34 (1900); Bernard, "On some Different
Cases of Germination," _Gardener's Chronicle_ (1900); Pierce, _Publ.
Univ. California_ (1900). (H. M. W.; V. H. B.)
FUNJ (FUNNIYEH, FUNG, FUNGHA), a very mixed negroid race, occupying
parts of Sennar and the hilly country to the south between the White and
Blue Niles. They traditionally come from west of the White Nile and are
affiliated by some to the Kordofan Nubas, by others, more justifiably,
to the negro Shilluks. These Funj, who became the dominant race in
Sennar in the 15th century, almost everywhere assimilated the speech,
religion and habits of the Arabs settled in that region. Until the 19th
century they were one of the most powerful of African peoples in the
eastern Sudan. About the end of the 15th century they overthrew the
kingdom of Aloa, between the two Niles, and conquered the neighbouring
peoples of the Sudan, Nubia and even Kordofan. The Funj had mixed much
with the Arabs before their conquests, and had been converted to Islam.
But they were still in many ways savages, for James Bruce (who traversed
the district in 1772) says that their most famous king, Malek-el-Gahman,
preferred human liver to any other food, and the Belgian traveller E.
Pruyssenaere (1826-1864) found them still performing pagan rites on
their sacred Mount Gula. Ernst Marno declared that as late as 1870 the
most southern branch of the race, the Boruns, a non-Arabic speaking
tribe, were cannibals. The Funj kings were content with levying tribute
on their neighbours, and in this loose way Shendi, Berber and Dongola
were once tributary. The Arab viziers gradually absorbed all power, the
Funj sovereignty becoming nominal; and in 1821 the Egyptians easily
destroyed the Funj domination. To-day the Funj are few, and represent no
real type. They are a bright, hospitable folk. Many of them are skilful
surgeons and go far afield in their work. The fellahin, indeed, call
surgeons "Senaari" (men of Sennar). See further SENNAR AND SUDAN
(Anglo-Egyptian).
FUNKIA, in botany, a genus of rather handsome, hardy, herbaceous plants
belonging to the natural order Liliaceae, and natives of China and
Japan. They are tuberous, with broadly ovate or heart-shaped leaves and
racemes of white or pale lilac, drooping, funnel-shaped flowers. They
are useful for the borders of a shrubbery, the lawn or rock-work, or may
be grown in pots for the greenhouse. The plants are propagated by
dividing the crowns in autumn or when growth begins in spring.
FUNNEL (through an O. Fr. _founil_, found in Breton, from Lat.
_infundibulum_, that through which anything is poured, from _fundere_,
to pour), a vessel shaped like a cone having a small tube at the apex
through which powder, liquid, &c., may be easily passed into another
vessel with a small opening. The term is used in metal-casting of the
hole through which the metal is poured into a mould, and in anatomy and
zoology of an _infundibulum_ or funnel-shaped organ. The word is thus
used generally of any shaft or passage to convey light, air or smoke, as
of the chimney of an engine or a steam-boat, or the flue of an ordinary
chimney. It is also used of a shaft or channel in rocks, and in the
decoying of wild-fowl is applied to the cone-shaped passage leading from
a pond and covered with a net, a "funnel-net," into which the birds are
decoyed.
FUR (connected with O. Fr. _forre_, a sheath or case; so "an outer
covering"), the name specially given to the covering of the skin in
certain animals which are natives of the colder climates, lying
alongside of another and longer covering, called the overhair. The fur
differs from the overhair, in that it is soft, silky, curly, downy and
barbed lengthwise, while the overhair is straight, smooth and
comparatively rigid. These properties of fur constitute its essential
value for felting purposes, and mark its difference from wool and silk;
the first, after some slight preparation by the aid of hot water,
readily unites its fibres into a strong and compact mass; the others can
best be managed by spinning and weaving.
On the living animal the overhair keeps the fur filaments apart,
prevents their tendency to felt, and protects them from injury--thus
securing to the animal an immunity from cold and storm; while, as a
matter of fact, this very overhair, though of an humbler name, is most
generally the beauty and pride of the pelt, and marks its chief value
with the furrier. We arrive thus at two distinct and opposite uses and
values of fur. Regarded as useful for felt it is denominated staple fur,
while with respect to its use with and on the pelt it is called fancy
fur.
_History._--The manufacture of fur into a felt is of comparatively
modern origin, while the use of fur pelts as a covering for the body,
for the couch, or for the tent is coeval with the earliest history of
all northern tribes and nations. Their use was not simply a barbarous
expedient to defend man from the rigours of an arctic winter; woven wool
alone cannot, in its most perfect form, accomplish this. The pelt or
skin is requisite to keep out the piercing wind and driving storm, while
the fur and overhair ward off the cold; and "furs" are as much a
necessity to-day among more northern peoples as they ever were in the
days of barbarism. With them the providing of this necessary covering
became the first purpose of their toil; subsequently it grew into an
object of barter and traffic, at first among themselves, and afterwards
with their neighbours of more temperate climes; and with the latter it
naturally became an article of fashion, of ornament and of luxury. This,
in brief, has been the history of its use in China, Tatary, Russia,
Siberia and North America, and at present the employment of fancy furs
among civilized nations has grown to be more extensive than at any
former period.
The supply of this demand in earlier times led to such severe
competition as to terminate in tribal pillages and even national wars;
and in modern times it has led to commercial ventures on the part of
individuals and companies, the account of which, told in its plainest
form, reads like the pages of romance. Furs have constituted the price
of redemption for royal captives, the gifts of emperors and kings, and
the peculiar badge of state functionaries. At the present day they vie
with precious gems and gold as ornaments and garniture for wealth and
fashion; but by their abundance, and the cheapness of some varieties,
they have recently come within the reach of men of moderate incomes. The
history of furs can be read in Marco Polo, as he grows eloquent with the
description of the rich skins of the khan of Tatary; in the early
fathers of the church, who lament their introduction into Rome and
Byzantium as an evidence of barbaric and debasing luxury; in the
political history of Russia, stretching out a powerful arm over Siberia
to secure her rich treasures; in the story of the French occupation of
Canada, and the ascent of the St Lawrence to Lake Superior, and the
subsequent contest to retain possession against England; in the history
of early settlements of New England, New York and Virginia; in Irving's
_Astoria_; in the records of the Hudson's Bay Company; and in the annals
of the fairs held at Nizhniy Novgorod and Leipzig. Here it may suffice
to give some account of the present condition of the trade in fancy
furs. The collection of skins is now chiefly a matter of private
enterprise. Few, if any, monopolies exist.
_Natural Supplies._--We are dependent upon the Carnivora, Rodentia,
Ungulata and Marsupialia for our supplies of furs, the first two classes
being by far of the greatest importance. The Carnivora include bears,
wolverines, wolves, raccoons, foxes, sables, martens, skunks, kolinskis,
fitch, fishers, ermines, cats, sea otters, fur seals, hair seals, lions,
tigers, leopards, lynxes, jackals, &c. The Rodentia include beavers,
nutrias, musk-rats or musquash, marmots, hamsters, chinchillas, hares,
rabbits, squirrels, &c. The Ungulata include Persian, Astrachan,
Crimean, Chinese and Tibet lambs, mouflon, guanaco, goats, ponies, &c.
The Marsupialia include opossums, wallabies and kangaroos. These, of
course, could be subdivided, but for general purposes of the fur trade
the above is deemed sufficient.
The question frequently arises, not only for those interested in the
production of fur apparel, but for those who derive so much comfort and
pleasure from its use, whether the supply of fur-bearing animals is
likely to be exhausted. Although it is a fact that the demand is ever
increasing, and that some of the rarer animals are decreasing in
numbers, yet on the other hand some kinds of furs are occasionally
neglected through vagaries of fashion, which give nature an opportunity
to replenish their source. These respites are, however, becoming fewer
every day, and what were formerly the most neglected kinds of furs are
becoming more and more sought after. The supply of some of the most
valuable, such as sable, silver and natural black fox, sea otter and
ermine, which are all taken from animals of a more or less shy nature,
does very gradually decrease with persistent hunting and the
encroachment of man upon the districts where they live, but the climate
of these vast regions is so cold and inhospitable that the probabilities
of man ever permanently inhabiting them in numbers sufficient to scare
away or exterminate the fur-bearing wild animals is unlikely. Besides
these there are many useful, though commonplace, fur-bearing animals
like mink, musquash, skunk, raccoon, opossum, hamster, rabbit, hares and
moles, that thrive by depredations upon cultivated land. Some of these
are reared upon extensive wild farms. In addition there are domestic
fur-bearing animals, such as Persian, Astrachan and Chinese lambs, and
goats, easily bred and available.
With regard to the rearing of the Persian lamb, there is a prevalent
idea that the skins of the unborn lamb are frequently used; this,
however, is a mistake. A few such skins have been taken, but they are
too delicate to be of any service. The youngest, known as "broadtails,"
are killed when a few days old, but for the well-developed curly fur,
the lambs must be six or seven weeks old. During these weeks their
bodies are covered with leather so that the fur may develop in close,
light and clean curls. The experiment has been tried of rearing rare,
wild, fur-bearing animals in captivity, and although climatic conditions
and food have been precisely as in their natural environment, the fur
has been poor in quality and bad in colour, totally unlike that taken
from animals in the wild state. The sensation of fear or the restriction
of movement and the obtaining of food without exertion evidently prevent
the normal development of the creature.
In mountainous districts in the more temperate zones some good supplies
are found. Chinchillas and nutrias are obtained from South America,
whence come also civet cats, jaguars, ocelots and pumas. Opossums and
wallabies, good useful furs, come from Australia and New Zealand. The
martens, foxes and otters imported from southern Europe and southern
Asia, are very mixed in quality, and the majority are poor compared with
those of Canada and the north.
Certain characteristics In the skin reveal to the expert from what
section of territory they come, but in classifying them it is considered
sufficient to mention territories only.
Some of the poorer sorts of furs, such as hamster, marmot, Chinese goats
and lambs, Tatar ponies, weasels, kaluga, various monkeys, antelopes,
foxes, otters, jackals and others from the warmer zones, which until
recently were neglected on account of their inferior quality of colour,
by the better class of the trade, are now being deftly dressed or dyed
in Europe and America, and good effects are produced, although the lack
of quality when compared with the better furs from colder climates which
possess full top hair, close underwool and supple leathers, is readily
manifest. It is only the pressure of increasing demand that makes
marketable hard pelts with harsh brittle hair of nondescript hue, and
these would, naturally, be the last to attract the notice of dealers.
As it is impossible that we shall ever discover any new fur-bearing
animals other than those we know, it behoves responsible authorities to
enforce close seasons and restrictions, as to the sex and age, in the
killing for the purpose of equalizing the numbers of the catches. As
evidence of indiscriminate slaughter the case of the American buffaloes
may be cited. At one time thousands of buffalo skins were obtainable and
provided material for most useful coats and rugs for rough wear in cold
regions, but to-day only a herd or so of the animals remain, and in
captivity.
The majority of animals taken for their fur are trapped or snared, the
gun being avoided as much as possible in order that the coat may be
quite undamaged. Many weary hours are spent in setting baits, traps and
wires, and, frequently, when the hunter retraces his steps to collect
the quarry it is only to find it gone, devoured by some large animal
that has visited his traps before him. After the skins have been
carefully removed--the sooner after death the better for the subsequent
condition of the fur--they are lightly tacked out, pelt outwards, and,
without being exposed to the sun or close contact with a fire, allowed
to dry in a hut or shady place where there is some warmth or movement of
air. With the exception of sealskins, which are pickled in brine, all
raw skins come to the various trade markets simply dried like this.
_Quality and Colour._--The best fur is obtained by killing animals when
the winter is at its height and the colder the season the better its
quality and colour. Fur skins taken out of season are indifferent, and
the hair is liable to shed itself freely; a good furrier will, however,
reject such faulty specimens in the manufacturing. The finest furs are
obtained from the Arctic and northern regions, and the lower the
latitude the less full and silky the fur, till, at the torrid zone, fur
gives place to harsh hair without any underwool. The finest and closest
wools are possessed by the amphibious Carnivora and Rodentia, viz.
seals, otters, beavers, nutrias and musquash, the beauty of which is not
seen until after the stiff water or top hairs are pulled out or
otherwise removed. In this class of animal the underneath wool of the
belly is thicker than that of the back, while the opposite is true of
those found on the land. The sea otter, one of the richest and rarest of
furs, especially for men's wear, is an exception to this unhairing
process, which it does not require, the hair being of the same length as
the wool, silky and bright, quite the reverse of the case of other
aquatic animals.
Of sealskins there are two distinct classes, the fur seals and the hair
seals. The latter have no growth of fur under the stiff top hair and are
killed, with few exceptions (generally of the marbled seals), on account
of the oil and leather they yield. The best fur seals are found off the
Alaska coast and down as far south as San Francisco.
It is found that in densely wooded districts furs are darker in colour
than in exposed regions, and that the quality of wool and hair is softer
and more silky than those from bare tracts of country, where nature
exacts from its creatures greater efforts to secure food, thereby
developing stronger limbs and a consequently coarser body covering.
As regards density of colour the skunk or black marten has the blackest
fur, and some cats of the domestic kind, specially reared for their fur,
are nearly black. Black bears have occasionally very black coats, but
the majority have a brownish underwool. The natural black fox is a
member of the silver fox family and is very rare, the skins bringing a
high price. Most silver foxes have dark necks and in some the dark shade
runs a quarter, half-way, or three-quarters, or even the whole length of
the skin, but it is rather of a brownish hue. Some Russian sables are of
a very dense bluish brown almost a black, which is the origin
undoubtedly of the term "sables," while some, from one district in
particular, have a quantity of silver hairs, evenly interspersed in the
fur, a peculiarity which has nothing to do with age. The best sea otters
have very dark coats which are highly esteemed, a few with silver hairs
in parts; where these are equally and evenly spread the skins are very
valuable. Otters and beavers that run dark in the hair or wool are more
valuable than the paler ones, the wools of which are frequently touched
with a chemical to produce a golden shade. This is also done with
nutrias after unhairing. The darker sorts of mink, musquash, raccoon
and wolverine are more valuable than the paler skins.
_Collective Supplies and Sales._--There are ten large American and
Canadian companies with extensive systems for gathering the annual hauls
of skins from the far-scattered trappers. These are the Hudson's Bay
Co., Russian Fur Co., Alaska Commercial Co., North American Commercial
Co., Russian Sealskin Co., Harmony Fur Co., Royal Greenland Fur Co.,
American Fur Co., Missouri Co. and Pacific Co. Most of the raw skins are
forwarded to about half-a-dozen brokers in London, who roughly sort them
in convenient lots, issuing catalogues to the traders of the world, and
after due time for examination of the goods by intending purchasers, the
lots are sold by public auction. The principal sales of general furs are
held in London in January and March, smaller offerings being made in
June and October; while the bulk of fur sealskins is sold separately in
December. The Hudson's Bay Co.'s sales take place before the others,
and, as no reserves are placed on any lot, the results are taken as
exactly indicating current values. While many buyers from America and
Russia are personally in attendance at the sales, many more are
represented by London and Leipzig agents who buy for them upon
commission. In addition to the fur skins coming from North America vast
numbers from Russia, Siberia, China, Japan, Australia and South America
are offered during the same periods at public auction. Fairs are also
held in Siberia, Russia and Germany for the distribution of fur skins as
follows:--
January: Frankfort-on-the- Small collection of provincial produce,
Oder such as otter, fox, fitch and marten.
February: Irbit, Siberia General Russian furs.
Easter: Leipzig, Germany General furs.
August: Nizhniy Novgorod, Persian lamb and general furs.
Russia
August: Kiakhta, Siberia Chinese furs and ermine.
December: Ishim, Siberia Chiefly squirrels.
Of course there are many transactions, generally in the cheaper and
coarser kinds of furs, used only in central Europe, Russia and Asia
which in no way interest the London market, and there are many direct
consignments of skins from collectors in America and Russia to London,
New York and Leipzig merchants. But the bulk of the fine furs of the
world is sold at the large public trade auction sales in London. The
chief exceptions are the Persian and Astrachan lambs, which are bought
at the Russian fairs, and are dressed and dyed in Leipzig, and the
ermine and Russian squirrels, which are dressed and manufactured into
linings either in Russia or Germany before offered for sale to the
wholesale merchants or manufacturers.
The annual collection of fur skins varies considerably in quantity
according to the demand and to the good or bad climatic conditions of
the season; and it is impossible to give a complete record, as many
skins are used in the country of their origin or exported direct to
merchants. But a fairly exact statement of the numbers sold in the great
public trade auction sales in London during the year 1905-1906 is
herewith set out.
_Year ending 31st of March 1906._ Total Number
of Skins.
Badger 28,634
Badger, Japanese 6,026
Bear 18,576
Beaver 80,514
Cat, Civet 157,915
Cat, House 126,703
" Wild 32,253
Chinchilla (La Plata), known also as Bastard 43,578
" Peruvian finest 5,603
Deer, Chinese 124,355
Ermine 40,641
Fisher 5,949
Fitch 77,578
Fox, Blue 1,893
" Cross 10,276
" Grey 59,561
" Japanese 81,429
" Kit 4,023
" Red 158,961
" Silver 2,510
" White 27,463
Goats, Chinese 261,190
Hares 41,256
Kangaroo 7,115
Kid, Chinese linings and skins equal to 5,080,047
Kolinsky 114,251
Lamb, Mongolian linings and skins equal to 214,072
" Slink " " " 167,372
" Tibet " " " 794,130
Leopard 3,574
Lynx 88,822
Marmot, linings and skins equal to 1,600,600
Marten, Baum 4,573
" Japanese 16,461
" Stone 12,939
Mink, Canadian and American 299,254
" Japanese 360,373
Mouflon 23,594
Musk-rat or Musquash, Brown 5,126,339
" " Black 41,788
Nutria 82,474
Opossum, American 902,065
" Australian 4,161,685
Otter, River 21,235
" Sea 522
Raccoon 310,712
Sable, Canadian and American 97,282
" Japanese 556
" Russian 26,399
Seals, Fur 77,000
" Hair 31,943
Skunk 1,068,408
Squirrel 194,596
" Linings each averaging 126 skins 1,982,736
Tiger 392
Wallaby 60,956
Wolf 56,642
Wolverine 1,726
Wombat 193,625
A brief account of the different qualities of the pelts, with some
general remarks as to their customary uses, follows. The prices quoted
are subject to constant fluctuation and represent purely trade prices
for bulk, and it should be explained that the very great variations are
due to different sizes, qualities and colours, and moreover are only
_first cost_, before skins are dressed and prepared. These preparations
are in some cases expensive, and there is generally a considerable
percentage of waste. The prices cannot be taken as a guide to the
wholesale price of a single and finished skin, but simply as _relative_
value.
The fullest and darkest skins of each kind are the most valuable, and,
in cases of bluish grey or white, the fuller, clearer and brighter are
the more expensive. A few albinos are found in every species, but
whatever their value to a museum, they are of little commercial
importance. Some odd lots of skins arrive designated simply as
"sundries," so no classification is possible, and this will account for
the absence of a few names of skins of which the imports are
insignificant in quantity, or are received direct by the wholesale
merchants.
_Names, Qualities and Uses of Pelts._[1]
ASTRACHAN.--See _Lambs_, below.
BADGER.--Size 2 X 1 ft. American sorts have coarse thick underwool of
a pale fawn or stone colour with a growth of longer black and white
hairs, 3 or 4 in. long; a very durable but clumsy fur. The best skins
are exported to France, Spain and Italy, and used for carriage rugs
and military purposes. Asiatic, including Japanese, skins are more
woolly. Russian and Prussian kinds are coarser and darker, and used
mostly for brush trade. Value 6d. to 19s.
BEAR, AUSTRALIAN.--See _Wombat_, below.
BEAR, BLACK.--Size 6 X 3 ft. Fine dark brown underwool with bright
black and flowing top hair 4 in. long. Cubs are nearly as long in the
hair although only about half the size and not only softer and better,
but have the advantage of being very much lighter in pelt. Widely
distributed in North America, the best come from Canada, are costly
and are used for military caps, boas, muffs, trimmings, carriage rugs
and coachmen's capes, and the fur wears exceedingly well. Value 17s.
6d. to 86s. Those from East India and warm climates are harsh, poor
and only fit for floor rugs.
BEAR, BROWN.--Size 6 X 3 ft. Similar in quality to the black, but far
more limited in number; the colours range from light yellow to a rich
dark brown. The best come from Hudson Bay territory and are valuable.
Used for muffs, trimmings, boas, and carriage rugs. Inferior sorts,
almost grizzly in effect and some very pale, are found in Europe and
Asia and are mostly used locally. In India there is a species called
Isabelline bear, which was formerly imported to Great Britain, but
does not now arrive in any quantity worth mentioning. Value 10s. 6d.
to 60s., Isabelline sort 10s. 6d. to 78s.
BEAR, GRIZZLY.--Size 8 X 4 ft. Coarse hair, heavy pelt, mostly dark
yellowish and brown colours, only found in western parts of United
States, Russia and Siberia. Used as carriage rugs and floor rugs, most
durable for latter purpose and of fine effect. They are about half the
value of brown bear. Value 15s. to 54s.
BEAR, ISABELLINE.--See _Bear_, _Brown_, above.
BEAR, WHITE.--Size 10 X 5 ft. The largest of all bears. Short close
hair except on flanks, colour white to yellow. An inhabitant of the
Arctic circle, best from Greenland. Used for floor rugs, very durable;
and very white specimens are valuable. Value 20s. to 520s.
BEAVER. Size 3 X 2 ft. The largest of rodents, it possesses a close
underwool of bluish-brown hue, nearly an inch in depth, with coarse,
bright, black or reddish-brown top hair, 3 in. long. Found widely in
North America. After being unhaired the darkest wools are the most
valuable, although many people prefer the bright, lighter brown tones.
Used for collars, cuffs, boas, muffs, trimmings, coat linings and
carriage aprons, and is of a most durable nature, in addition to
having a rich and good appearance. Value 10s. to 39s. 6d.
BROADTAIL.--See _Lambs_, below.
CARACAL.--A small lynx from India, the fur very poor, seldom imported.
CARACUL.--See _Goats_ and _Lambs_, below.
CAT, CIVET.--Size 9 X 4-1/2 in., short, thick and dark underwool with
silky black top hair with irregular and unique white markings. It is
similar to skunk, but is much lighter in weight, softer and less full,
without any disagreeable odour. Used for coat linings it is very warm
and durable. A few come from China, but the fur is yellowish-grey,
slightly spotted and worth little. Value 1s. 1d. to 1s. 11d.
CAT, HOUSE, &C.--18 X 9 in., mostly black and dark brown, imported
from Holland, Bavaria, America and Russia, where they are reared for
their coats. The best, from Holland, are used for coat linings.
Although in colour, weight and warmth they are excellent, the fur is
apt to become loose and to fall off with friction of wear. The black
are known as genet, although the true genet is a spotted wild cat.
Wild sorts of the tabby order are coarser, and not so good and silky
in effect as when domestically reared. Value of the black sorts 2d. to
3s. Wild 9d. to 14s. Some small wild cats, very poor flat fur of a
pale fawn colour with yellow spots, are imported from Australia and
used for linings. Value 5-1/2d. to 1s. 1d.
CHEETAH.--Size of a small leopard and similar in colour, but has black
spots in lieu of rings. Only a few are now imported, which are used
for mats. Value 2s. 6d. to 18s.
CHINCHILLA, PERUVIAN and BOLIVIAN.--Size 12 X 7 in., fur 1 to 1-1/4
in. deep. Delicate blue-grey with black shadings, one of nature's most
beautiful productions, though not a durable one. Used for ladies'
coats, stoles, muffs, hats and trimmings. Yearly becoming scarcer and
most costly. Value 8s. 6d. to 56s. 8d.
CHINCHILLA, LA PLATA, incorrectly named and known in the trade as
"bastard chinchilla," size 9 X 4 in., in a similar species, but owing
to lower altitudes and warmer climatic conditions of habitation is
smaller, with shorter and less beautiful fur, the underwool colour
being darker and the top colour less pure. Used exactly as the better
kind, and the picked skins are most effective. As with the best sort
it is not serviceable for constant wear. Value 4s. 2d. to 27s. 6d.
CHINCHILLONE.--Size 13 X 8 in., obtained also from South America. Fur
is longer and weaker and poorer and yellower than chinchilla. Probably
a crossbred animal, very limited importation. Value 3s. 6d. to 16s.
8d.
DEER, CHINESE and EAST INDIAN.--Small, light, pelted skins, the
majority of which are used for mats. Reindeer and other varieties are
of little interest for use other than trophy mats. Thousands are taken
for the leather trade. Value of Chinese 1s. 2d. to 1s. 6d. each.
DOG.--The only dogs that are used in the fur trade in civilized
countries are those imported from China, which are heavy and coarse,
and only used in the cheaper trade, chiefly for rugs. Value 6d. to 1s.
DOG WOLF.--See _Wolf_, below.
ERMINE.--Size 12 X 2-1/2 in. Underwool short and even, with a shade
longer top hair. Pelt light and close in texture, and durable. In the
height of winter the colour is pure white with exception of the tip of
tail, which is quite black. Supplies are obtained from Siberia and
America. Best are from Ishim in Siberia. Used for cloak linings,
stoles, muffs and trimmings, also for embellishment of British state,
parliamentary and legal robes. When this fur is symmetrically spotted
with black lamb pieces it is styled miniver, in which form it is used
at the grand coronation functions of British sovereigns. Value 1s. 3d.
to 8s. 6d.
FISHER.--Size 30 X 12 in., tail 12 to 18 in. long, the largest of the
martens; has a dark shaded deep underwool with fine, glossy, dark and
strong top hair 2 in. or more long. Best obtained from British
America. The tails are almost black and make up most handsomely into
trimmings, muffs, &c. Tails worked separately in these forms are as
rich and fine and more durable than any other fur suitable for a like
purpose. The fur of the skin itself is something like a dark silky
raccoon, but is not as attractive as the tails. Value 12s. to 46s.
FITCH.--Size 12 X 3 in., of the marten species, also known as the pole
cat. Yellow underwool 1/3 in. deep, black top hair, 1-1/2 to 1-3/4 in.
long, very fine and open in growth, and not close as in martens.
Largest skins come from Denmark, Holland and Germany. The Russian are
smaller, but more silky and, as now dyed, make a cheap and fair
substitute for sable. They are excellent for linings of ladies' coats,
being of light weight and fairly strong in the pelt. English mayors'
and civic officials' robes are frequently trimmed with this fur in
lieu of sable. Value of the German variety 2s. to 5s. 6d. and of the
Russian 7d. to 1s. 4d.
FOX, BLUE.--Size 24 X 8 in. Underwool thick and long. Top hair fine
and not so plentiful as in other foxes. Found in Alaska, Hudson Bay
territory, Archangel and Greenland. Although called blue, the colour
is a slaty or drab tone. Those from Archangel are more silky and of a
smoky bluish colour and are the most valuable. These are scarce and
consequently dear. The white foxes that are dyed smoke and celestial
blue are brilliant and totally unlike the browner shades of this fox.
Value 34s. to 195s.
FOX, COMMON.--The variation of size and quality is considerable, and
the colour is anything from grey to red. In Great Britain the animal
is now only regarded for the sport it provides. On the European
continent, however, some hundreds of thousands of skins, principally
German, Russian and Norwegian, are sold annually, for home use, and
for dyeing and exportation, chiefly to the United States. The
qualities do not compare with those species found in North America and
the Arctic circle. The Asiatic, African and South American varieties
are, with the exception of those taken in the mountains, poorly furred
and usually brittle and therefore of no great service. No commercial
value can be quoted.
FOX, CROSS.--Size 20 X 7 in., are about as large as the silver and
generally have a pale yellowish or orange tone with some silvery
points and a darkish cross marking on the shoulders. Some are very
similar to the pale red fox from the North-West of America and a few
are exceptionally large. The darkest and best come from Labrador and
Hudson Bay, and the ordinary sorts from the north-west of the United
States and, as with silver and other kinds, the quality is inferior
when taken from warmer latitudes. Value 10s. 6d. to 60s.
FOX, GREY.--Size 27 X 10 in. Has a close dark drab underwool with
yellowish grizzly, grey, regular and coarse top hair. The majority
used for the trade come from Virginia and the southern and western
parts of the United States. Those from the west are larger than the
average, with more fur of a brighter tone. The fur is fairly
serviceable for carriage rugs, the leather being stout, but its
harshness of quality and nondescript colour does not contribute to
make it a favourite. Value 9d. to 4s. 9d.
FOX, JAPANESE.--See _Fox, Red_, and _Raccoon_, below.
FOX, KIT.--Size 20 X 6 in. The underwool is short and soft, as is also
the top hair, which is of very pale grey mixed with some
yellowish-white hair. It is the smallest of foxes, and is found in
Canada and the northern section of the United States. It is similar in
colour and quality to the prairie fox and to many kinds from the
warmer zones, such as from Turkey, eastern Asia and elsewhere. Value
1s. 3d. to 5s. 6d.
FOX, RED.--Size 24 X 8 in., though a few kinds are much larger. The
underwool is long and soft and the hair plentiful and strong. It is
found widely in the northern parts of America and in smaller numbers
south of the United States, also in China, Japan and Australia. The
colours vary from pale yellowish to a dark red, some being very
brilliant. Those of Kamschatka are rich and fine in quality. Farther
north, especially near the sea, the fur is coarse. Where the best
coloured skins are not used for carriage rugs they are extensively
dyed, and badger and other white hairs are inserted to resemble silver
fox. They are also dyed a sable colour. The skins, being the strongest
of foxes', both in the fur and pelt, are serviceable. The preparations
in imitation of the natural black and silver sorts are very good and
attractive. Value 1s. to 41s.
FOX, SILVER. Size 30 X 10 in. Underwool close and fine. Top hair black
to silvery, 3 in. long. The fur upon the necks usually runs dark,
almost black, and in some cases the fur is black half-way down the
length of the skin, in rarer cases three-quarters of the length and,
in the most exceptional instances, the whole length, and when this is
the case they are known as "Natural Black Foxes" and fetch enormous
prices. The even silvery sorts are highly esteemed, and the fur is one
of the most effective and precious. The finest are taken in Labrador.
The farther south they are found, the poorer and coarser the fur. The
brush has invariably a white tip. Value L1 to L320.
FOX, WHITE.--Size 20 X 7 in. Animals of this species are generally
small in size and inhabit the extreme northern sections of Hudson Bay,
Newfoundland, Greenland, Labrador and Siberia. The Canadian are silky
in nature and inclined to a creamy colour, while the Siberian are more
woolly and rather whiter. Those taken in central Asia near or in
Chinese territory are poorer and yellowish. The underwool in all sorts
is generally of a bluish-grey tone, but the top hair in the depth of
winter is usually full enough in quantity to hide any such variation.
Those skins in which the underwool is quite white are rare and much
more expensive. In summer specimens of this species, as with other
white furred animals, have slightly discoloured coats. The skins that
are not perfectly white are dyed jet black, dark or light smoke,
violet-blue, blue-grey, and also in imitation of the drab shades of
the natural blue. Value 18s. to 66s.
GENET.--Size 10 X 4 in. The genet proper is a small white spotted cat
found in Europe, but the quantity is too small to be of commercial
interest. The name has been adopted for the black cats used so much in
the trade. (See CATS, above.) Value 1s. to 6s. 6d.
GOATS.--Size varies greatly. The European, Arabian and East Indian
kinds are seldom used for rugs, the skins are chiefly dressed as
leather for books and furniture, and the kids for boots and gloves,
and the finer wool and hair are woven into various materials. Many
from Russia are dyed black for floor and carriage rugs; the hair is
brittle, with poor underwool and not very durable; the cost, however,
is small. The Chinese export thousands of similar skins in black, grey
and white, usually ready dressed and made into rugs of two skins each.
A great many are dyed black and brown, in imitation of bear, and are
used largely in the western parts of the United States and Canada for
sleigh and carriage rugs. Many are used for their leather. Thousands
of the kids are also dyed black and worked into cross-shaped pieces,
in which shape they are largely exported to Germany, France, Great
Britain and America, and sold by the retail as caracal, kid or
caracul. The grey ones are in good demand for motor coats. The word
caracul has been adopted from the Turkish and signifies black-eared.
See also LAMBS, CARACUL. Value of Chinese white 3s. 6d. to 6s. 6d.:
grey, 4s. to 6s. 9d.
The Angora from the heights of central Asia Minor has curly, fleecy,
silky, white wool, 4 to 7 in. long. The fur is not used in Great
Britain, as formerly, and the greater quantity, known as mohair, is
now imported for purposes of weaving. This species of goat was some
years since introduced into Cape Colony, but its wool is not so good
as the Asiatic breed. Good business, however, is done with the
product, but chiefly for leather. Value 4s. to 12s. 6d.
The Mongolian goat has a very soft silk underwool, and after the long
top hair is removed it is dressed and imported and erroneously named
mouflon. The colour is a light fawn, but it is so pale that it lends
itself to be dyed any colour. It was popular some years since in the
cheaper trade, but it is not now much seen in England. Value 2s. to
6s.
The Tibet goat is similar to the Angora in the fineness of its wool,
and many are used in the making of cashmere shawls. The Tibet lamb so
largely imported and used for children's wear is often miscalled Tibet
goat. Value 3s. to 7s. 6d.
GUANACO.--Size 30 X 15 in. Is a species of goat found in Patagonia and
other parts of South America. It has a very long neck and exceedingly
soft woolly fur of a light reddish-fawn colour with very white flanks.
It is usually imported in small quantities, native dressed, and ready
made into rugs. The dressing is hard and brittle. If the skins are
dressed in Europe they afford a very comfortable rug, though a very
marked one in effect. They have a similar wool to the vicuna, but
coarser and redder; both are largely used in South America. Value 1s.
to 4s. 6d.
HAMSTER.--Size 8 X 3-1/2 in. A destructive rodent, is found in great
numbers in Russia and Germany. The fur is very flat and poor, of a
yellowish pale brown with a little marking of black. Being of a light
weight it is used for linings. Value 3d. to 1s.
HARE.--Size 24 X 9 in. The common hare of Europe does not much
interest the furrier, the fur being chiefly used by makers of hatters'
felt. The white hares, however, of Russia, Siberia and other regions
in the Arctic circle are very largely used in the cheaper trade of
Europe, America and the British colonies. The fur is of the whitest
when killed in winter, and that upon the flanks of the animal is very
much longer than that upon its back. The flanks are usually cut off
and made into muffs and stoles. The hair is, however, brittle and is
not at all durable. This fur is dyed jet black and various shades of
brown and grey, and manufactured into articles for the small drapers
and for exportation. The North American hares are also dyed black and
brown and used in the same way. Value of white 2d. to 5d.
JACKAL.--Size 2 to 3 ft. long. Is found in India and north and south
Africa. Indian are light brown and reddish, those from the Cape are
dark grey and rather silvery. Few are imported. Fur generally poor and
harsh, only suitable for carriage rugs. Value 1s. to 3s. 6d.
JAGUAR.--Size 7 to 10 ft. long. Is found in Mexico and British
Honduras. The markings are an irregular ring formation with a spot in
the centre. Leopards have rings only and cheetahs solid spots.
Suitable only for hearth-rugs. Supply very limited. Value 5s. to 45s.
KALUGA.--See _Souslik_, below.
KANGAROO.--The sizes vary considerably, some being huge, others quite
small. The larger varieties, viz. the red and the great, do not
usually interest furriers, the fur being harsh and poor without
underwool. They are tanned for the leather trade. The sorts used for
carriage aprons, coat linings and the outside of motor coats include:
blue kangaroo, bush kangaroo, bridled kangaroo, wallaroo, yellow
kangaroo, rock wallaby, swamp wallaby and short-tailed wallaby. Many
of the swamp sort are dyed to imitate skunk and look well. Generally
the colours are yellowish or brown. Some are dark brown as in the
swamp, which being strong are suitable for motor coats. The rock
wallabies are soft and woolly and often of a pretty bluish tone, and
make moderately useful carriage rugs and perambulator aprons. The
redder and browner sorts are also good for rugs as they are thick in
the pelt. On the European continent many of these are dyed. The best
of the lighter weights are frequently insufficiently strong in the
hair to stand the friction of wear in a coat lining. Value, kangaroo
9d. to 3s., wallaby 1-1/2 d. to 5s. 3d., wallaroo 1s. to 5s. 6d.
KIDS.--See _Goats_, above.
KOLINSKY.--Size 12 X 2-1/2 in. Is one of the marten tribe. The
underwool is short and rather weak, but regular, as is also the top
hair; the colour is usually yellow. They have been successfully dyed
and used as a substitute for sable. They are found in Siberia, Amoor,
China and Japan, but the best are from Siberia. They are light in
weight and therefore suitable for linings of coats. The tails are used
for artists' "sable" brushes. The fur has often been designated as red
or Tatar sable. Value 1s. 6d. to 4s. 6d.
LAMBS.--The sorts that primarily interest the fur trade in Europe and
America are those from south Russia, Persia and Afghanistan, which are
included under the following wholesale or retail commercial terms:
Persian lamb, broadtail, astrachan, Shiraz, Bokharan and caracul lamb.
With the public the general term astrachan is an old one, embracing
all the above curly sorts; the flatter kinds, as broadtail and caracul
lamb, have always been named separately. The Persian lambs, size 18 X
9 in., are the finest and the best of them. When dressed and dyed they
should have regular, close and bright curl, varying from a small to a
very large one, and if of equal size, regularity, tightness and
brightness, the value is comparatively a matter of fancy. Those that
are dull and loose, or very coarse and flat in the curl, are of far
less market value.
All the above enumerated lambs are naturally a rusty black or brown,
and with very few exceptions are dyed a jet black. Lustre, however,
cannot be imparted unless the wool was originally of a silky nature.
Broadtails, size 10 X 5 in., are the very young of the Persian sheep,
and are killed before the wool has time to develop beyond the flat
wavy state which can be best compared to a piece of moire silk. They
are naturally exceedingly light in weight, and those that are of an
even pattern, possessing a lustrous sheen, are costly. There is,
notwithstanding, a great demand for these from the fashionable world,
as not only are they very effective, but being so flat in the wool the
figure of the wearer can be shown as perfectly as in a garment made of
silk. It cannot be regarded as an economical fur, as the pelt is too
delicate to resist hard wear.
Persian Lamb price 12s. 6d. to 25s.
Broadtail " 10s. " 35s.
Astrachan, Shiraz and Bokharan lambs, size 22 by 9 in., are of a
coarser, looser curl, and chiefly used for coat linings, while the
Persians are used for outside of garments, collars, cuffs, stoles,
muffs, hats and trimmings and gloves. The so-called caracul lambs,
size 12 X 6 in., are the very young of the astrachan sheep, and the
pick of them are almost as effective as broadtails, although less fine
in the texture. See also remarks as to caracul kid under Goats, above.
Astrachan price 1s. to 5s. 6d.
Caracul Lamb " 2s. 6d " 10s. 6d.
Shiraz " 4s. 6d " 10s.
Bokharan " 1s. 6d " 3s. 6d.
Grey lambs, size 24 X 10 in., are obtained from the Crimea and known
in the trade as "crimmers." They are of a similar nature to the
caracul lambs, but looser in curl, ranging from a very light to a dark
grey. The best are the pale bluish greys, and are chiefly used for
ladies' coats, stoles, muffs and hats. Price 2s. to 6s. Mongolian
lambs, size 24 X 15 in., are of a short wavy loose curl, creamy white
colour, and are usually exported from China dressed, the majority
being ready-made into cross-shaped coats or linings. They are used
principally for linings of good evening wraps for ladies. Price 1s. to
2s. 6d. Slink lambs come from South America and China. The former are
very small and generally those that are stillborn. They have a
particularly thin pelt with very close wool of minute curl. The China
sorts are much larger. The smallest are used for glove linings and the
others for opera cloak linings. Price 1s. to 6s. 6d.
LEOPARD.--Size 3 to 6 ft. long. There are several kinds, the chief
being the snow or ounce, Chinese, Bengal, Persian, East Indian and
African. The first variety inhabit the Himalayas and are beautifully
covered with a deep soft fur quite long compared to the flat harsh
hair of the Bengal sort. The colours are pale orange and white with
very dark markings, a strong contrast making a fine effect. Most
artists prize these skins above all others. The Chinese are of a
medium orange brown colour, but full in fur. The East Indian are less
full and not so dark. The Bengal are dark and medium in colour, short
and hard hair, but useful for floor rugs, as they do not hold the dust
like the fuller and softer hair of the kinds previously named. They
are also used for drummers' aprons and saddle cloths in the Indian
army. The African are small with pale lemon colour grounds very
closely marked with black spots on the skin, the strong contrast
making a pleasing effect. Occasionally, where something very marked is
wanted, skating jackets and carriage aprons are made from the softest
and flattest of skins, but usually they are made into settee covers,
floor rugs and foot muffs. Value 2s. to 40s.
LION.--Size 5 to 6 ft. long. These skins are found in Africa, Arabia
and part of India, and are every year becoming scarcer. They are only
used for floor rugs, and the males are more highly esteemed on account
of the set-off of the mane. Value, lions' L10 to L100; lionesses' L5
to L25.
LYNX.--Size 45 X 20 in. The underwool is thinner than fox, but the top
hair is fine, silky and flowing, 4 in. long, of a pale grey, slightly
mottled with fine streaks and dark spots. The fur upon the flanks is
longer and white with very pronounced markings of dark spots, and this
part of the skin is generally worked separately from the rest and is
very effective for gown trimmings. Where the colour is of a sandy and
reddish hue the value is far less than where it is of a bluish tone.
They inhabit North America as far south as California, also Norway and
Sweden. Those from the Hudson Bay district and Sweden are the best and
are very similar. Those taken in Central Asia are mostly used locally.
For attire the skins manufactured in Europe are generally dyed black
or brown, in which state it has a similar appearance to dyed fox, but
having less thick underwool and finer hair flows freely. The finest
skins when dyed black are used very largely in America in place of the
dyed black fox so fashionable for mourning wear in Great Britain and
France. The British Hussar busbies are made of the dark brown lynx,
and it is the free silky easy movement of the fur with the least
disturbance in the atmosphere that gives it such a pleasing effect. It
is used for rugs in its natural state and also in Turkey as trimmings
for garments. Value 13s. 6d. to 56s.
LYNX CAT or BAY LYNX.--Is about half the size and depth of fur of a
lynx proper, and inhabits the central United States. It is a flat and
reddish fur compared to the lynx and is suitable for cheap carriage
aprons. A few come from Canada and are of better quality. Value 5s. to
15s.
MARMOT.--Size 18 X 12 in. Is a rodent and is found in considerable
numbers in the south of Prussia. The fur is a yellowish brown and
rather harsh and brittle and has no underwool. Since, however, the
value of all good furs has advanced, dyers and manufacturers have made
very successful efforts with this fur. The Viennese have been
particularly successful, and their method has been to dye the skins a
good brown and then not put in the dark stripes, which exist in sable
and mink, until the garment or article is finished, thus obtaining as
perfectly symmetrical effects as if the articles were made of small
skins instead of large ones. Marmots are also found in North America,
Canada and China; the best, however, come from Russia. It should
always be a cheap fur, having so few good qualities to recommend it.
Value 9d. to 2s. 6d.
MARTEN, AMERICAN.--See _Sable_, below.
MARTEN, BAUM.--Size 16 X 5 in. Is sometimes called the pine marten,
and is found in quantity in the wooded and mountainous districts of
Russia, Norway, Germany and Switzerland. It possesses a thick
underwool with strong top hair, and ranges from a pale to a dark
bluish brown. The best, from Norway, are very durable and of good
appearance and an excellent substitute for American sable. The tails
when split into two or three, with small strips of narrow tape so as
to separate the otherwise dense fur, formerly made very handsome sets
of trimmings, ties and muffs, and the probabilities are, as with other
fashions, such use will have its period of revival. Value 6s. to 85s.
MARTEN, BLACK.--See _Skunk_, below.
MARTEN, JAPANESE.--Size 16 X 5 in. Is of a woolly nature with rather
coarse top hair and quite yellow in colour. It is dyed for the cheap
trade for boas and muffs, but it is not an attractive fur at the best
of times. It lacks a silky, bright and fresh appearance, and therefore
is unlikely to be in great demand, except where economy is an object.
Value 6s. 6d. to 18s. 6d.
MARTEN, STONE.--Size and quality similar to the baum; the colour,
however, of the underwool is a stony white and the top hair is very
dark, almost black. They live in rocky and stony districts. Skins of a
pale bluish tone are generally used in their natural state for stoles,
boas and muffs, but the less clear coloured skins are dyed in
beautiful shades similar in density to the dark and valuable sables
from Russia, and are the most effective skins that can be purchased at
a reasonable price. The tails have also been worked, in the manner
explained with regard to the baum marten, as sets of trimmings and in
other forms. Stone martens are found in Russia, Bosnia, Turkey,
Greece, Germany, the Alps and France. The Bosnian and the French are
the best in colour. The Asiatic sorts are less woolly, but being silky
are useful when dyed. There are many from Afghanistan and India which
are too poor to interest the European markets. Value 7s. 6d. to 26s.
MINK.--Size 16 X 5 in. Is of the amphibious class and is found
throughout North America and in Russia, China and Japan. The underwool
is short, close and even, as is also the top hair, which is very
strong. The best skins are very dark and are obtained from Nova
Scotia. In the central states of America the colour is a good brown,
but in the north-west and south-west the fur is coarse and generally
pale. It is very durable for linings, and is an economical substitute
for sable for coats, capes, boas and trimmings. Values have greatly
increased, and the fur possessing good qualities as to colour and
durability will doubtless always be in good request. The Russian
species is dark but flat and poor in quality, and the Chinese and
Japanese are so pale that they are invariably dyed. These, however,
are of very inferior nature. Value of American 3s. 3d. to 40s.,
Japanese 3d. to 2s. 3d.
MOLE.--Size 3-1/2 X 2-1/2 in. Moles are plentiful in the British Isles
and Europe, and owing to their lovely velvety coats of exquisite blue
shade and to the dearness of other furs are much in demand. Though the
fur is cheap in itself, the expense of dressing and working up these
little skins is considerable, and they possess the unique charm of an
exceptional colour with little weight of pelt; the quality of
resistance to friction is, however, so slight as to make them
expensive in wear. The best are the dark blue from the Fen district of
Cambridgeshire in England. Value 1/2d. to 2d.
MONGOLIAN LAMBS.--See _Lambs_, above.
MONKEY, BLACK.--Size 18 X 10 in. Among the species of monkeys only one
interests to any extent the fur trade, and that is the black monkey
taken on the west coast of Africa (_Colobus satanas_). The hair is
very long, very black and bright with no underwool, and the white pelt
of the base of the hair, by reason of the great contrast of colour, is
very noticeable. The skins were in 1850 very fashionable in England
for stoles, muffs and trimmings, and in America also as recently as
1890. They are now mostly bought for Germany and the continent. Value
6d. to 1s. 6d.
MOUFLON.--Size 30 X 15 in. Is a sheep found in Russia and Corsica and
now very little in demand, and but few are imported into Great
Britain. Many Mongolian goats with the long hairs pulled out are sold
as mouflon. Value 4s. to 10s. 6d.
MUSK-OX.--Size 6 X 3 ft. These animals have a dense coat of fine, long
brown wool, with very long dark brown hair on the head, flanks and
tail, and, in the centre, a peculiar pale oval marking. There is no
other fur that is so thick, and it is eminently suitable for sleighing
rugs, for which purpose it is highly prized in Canada. The musk-ox
inhabits the north part of Greenland and part of Canada, but in very
limited numbers. Value 10s. to 130s.
MUSQUASH or MUSK-RAT, BROWN and BLACK RUSSIAN.--Size 12 X 8 in. A very
prolific rodent of the amphibious class obtained from Canada and the
United States, similar in habit to the English vole, with a fairly
thick and even brown underwool and rather strong top dark hair of
medium density. It is a very useful fur for men's coat linings and
ladies' driving or motoring coats, being warm, durable and not too
heavy. If the colour were less motley and the joins between the skins
could be made less noticeable, it would be largely in demand for
stoles, ties and muffs. As it is, this fur is only used for these
smaller articles for the cheaper trade. It has, however, of later
years been "unhaired," the underwool clipped very even and then dyed
seal colour, in which way very useful and attractive garments are
supplied at less than half the cost of the cheaper sealskins. They do
not wear as well, however, as the pelt and the wool are not of a
strength comparable to those of sealskin. With care, however, such a
garment lasts sufficiently long to warrant the present outlay. Value
5-1/2d. to 1s. 9d.
There is a so-called black variety found in Delaware and New Jersey,
but the number is very small compared to the brown species. They are
excellent for men's coat linings and the outside of ladies' coats, for
stoles, muffs, collars and cuffs. Value 10d. to 3s. 7d.
The Russian musquash is very small, 7 X 4 in., and is limited in
numbers compared to the brown. Only a few thousands are imported to
London. It is of a very pretty silvery-blue shade of even wool with
very little silky top hair, having silvery-white sides and altogether
a very marked effect. The odour, however, even after dressing is
rather pungent of musk, which is generally an objection. Value 4s. to
6s. 6d.
NUTRIA.--Size 20 X 12 in. Is a rodent known in natural history as the
coypu, about half the size of a beaver, and when unhaired has not more
than half, generally less, the depth of fur, which is also not so
close. Formerly the fur was only used for hatters' felt, but with the
rise in prices of furs these skins have been more carefully removed
and--with improved dressing, unhairing and silvering processes--the
best provides a very effective and suitable fur for ladies' coats,
capes, stoles, muffs, hats and gloves, while the lower qualities make
very useful, light-weighted and inexpensive linings for men's or
women's driving coats. It is also dyed sealskin colour, but its woolly
nature renders it less effective than the more silky musquash. They
are obtained from the northern part of South America. Value is. 6d. to
6s. 6d.
OCELOT.--Size 36 X 13 in. Is of the nature of a leopard and prettily
marked with stripes and oblong spots. Only a few are now imported from
South America for carriage aprons or mats. The numbers are very
limited. Value 1s. to 2s. 6d.
OPOSSUM, AMERICAN.--Size 18 X 10 in. Is a marsupial, a class with this
exception not met with out of Australia. The underwool is of a very
close frizzy nature, and nearly white, with long bluish grey mixed
with some black top hair. It is only found in the central sections of
the United States. About 1870 in England it was dyed dark brown or
black and used for boas, muffs and trimmings, but until recently has
been neglected on the continent. With, however, recent experiments in
brown and skunk coloured dyes, it bids fair to become a popular fur.
Value 2-1/2d. to 5s. 6d.
OPOSSUM, AUSTRALIAN.--Size 16 X 8 in. Is a totally different nature of
fur to the American. Although it has wool and top hair, the latter is
so sparse and fine that the coat may be considered as one of close
even wool. The colour varies according to the district of origin, from
a blue grey to yellow with reddish tones. Those from the neighbourhood
of Sydney are light clear blue, while those from Victoria are dark
iron grey and stronger in the wool. These animals are most prolific
and evidently increasing in numbers. Their fur is pretty, warm and as
yet inexpensive, and is useful for rugs, coat linings, stoles, muffs,
trimmings and perambulator aprons. The worst coloured ones are
frequently dyed black and brown. The most pleasing natural grey come
from Adelaide. The reddest are the cheapest. Value 3-3/4d. to 3s. 6d.
OPOSSUM, RINGTAILED.--Size 7 X 4 in. Has a very short close and dark
grey wool, some being almost black. There are but a few thousands
imported, and being so flat they are only of use for coat linings, but
they are very warm and light in weight. Value 6d. to 10d.
OPOSSUM, TASMANIAN (grey and black).--Size 20 X 10 in. Is of a similar
description, but darker and stronger in the wool and larger. Besides
these there are some very rich brown skins which were formerly in such
request in Europe, especially Russia, that undue killing occurred
until 1899, when the government stopped for a time the taking of any
of this class. They are excellent for carriage aprons, being not only
very light in weight and warm, but handsome. Value 2s. 6d. to 8s. 6d.
OTTER, RIVER.--The size varies considerably, as does the underwool and
the top hair, according to the country of origin. There are few rivers
in the world where they do not live. But it is in the colder northern
regions that they are found in the greatest numbers and with the best
fur or underwool, the top hair, which, with the exception of the
scarce and very rich dark brown specimens they have in common with
most aquatic animals, is pulled out before the skins are manufactured.
Most of the best river otter comes from Canada and the United States
and averages 36 X 18 in. in size. Skins from Germany and China are
smaller, and shorter in the wool. The colours of the under wools of
river otters vary, some being very dark, others almost yellow. Both as
a fur and as a pelt it is extremely strong, but owing to its short and
close wool it is usually made up for the linings, collars and cuffs of
men's coats. A large number of skins, after unhairing, is dyed seal
colour and used in America. Those from hot climates are very poor in
quality. Value 28s. to 118s.
OTTER, SEA.--Size 50 X 25 in. Possesses one of the most beautiful of
coats. Unlike other aquatic animals the skin undergoes no process of
unhairing, the fur being of a rich dense silky wool with the softest
and shortest of water hairs. The colours vary from pale grey brown to
a rich black, and many have even or uneven sprinkling of white or
silvery-white hairs. The blacker the wool and the more regular the
silver points, the more valuable the skin. Sea otters are,
unfortunately, decreasing in numbers, while the demand is increasing.
The fur is most highly esteemed in Russia and China; in the latter
country it is used to trim mandarins' state robes. In Europe and
America it is much used for collar, long facings and cuffs of a
gentleman's coat; such a set may cost from L200 to L600, and in all
probability will soon cost more. Taking into consideration the size,
it is not so costly as the natural black fox, or the darkest Russian
sable, which is now the most expensive of all. The smaller and young
sea otters of a grey or brown colour are of small value compared to
the large dark and silvery ones. Value L10 to L220. A single skin has
been known to fetch L400.
OUNCE.--See _Leopard_, above.
PERSIAN LAMBS.--See _Lambs_, above.
PLATYPUS.--Size 12 X 8 in. One of the most singular of fur-bearing
animals, being the link between bird and beast. It has fur similar to
otter, is of aquatic habits, being web-footed with spurs of a cock and
the bill of a duck. The skins are not obtained in any numbers, but
being brought over by travellers as curiosities and used for muffs,
collars and cuffs, &c., they are included here for reference. Value
2s. to 3s. 6d.
_Pony_ or _Tatar Foal._--Size 36 X 20 in. These skins are of
comparatively recent importation to the civilized world. They are
obtained from the young of the numerous herds of wild horses that roam
over the plains of Turkestan. The coat is usually a shade of brown,
sometimes greyish, fairly bright and with a suggestion of waviness.
Useful for motor coats. Value 3s. to 10s. 6d.
PUMA.--Size 4-1/2 X 3 ft. Is a native of South America, similar to a
lion in habits and colour of coat. The hair and pelt is, however, of
less strength, and only a few are now used for floor rugs. Value 5s.
to 10s.
RACCOON.--Size 20 X 12 in. Is an animal varying considerably in size
and in quality and colour of fur, according to the part of North
America in which it is found. In common parlance, it may be described
as a species of wild dog with close affinity to the bear. The
underwool is 1 to 1-1/2 in. deep, pale brown, with long top hairs of a
dark and silvery-grey mixture of a grizzly type, the best having a
bluish tone and the cheapest a yellowish or reddish-brown. A limited
number of very dark and black sorts exist and are highly valued for
trimmings. The very finest skins are chiefly used for stoles and
muffs, and the general run for coachmen's capes and carriage rugs,
which are very handsome when the tails, which are marked with rings of
dark and light fur alternately, are left on. Raccoons are used in
enormous quantities in Canada for men's coats, the fur outside. The
poorer qualities are extensively bought and made up in a similar way
for Austria-Hungary and Germany. These make excellent linings for
coats or footsacks for open driving in very cold climates. The worst
coloured skins are dyed black or brown and are used for British
military busbies, or caps, stoles, boas, muffs and coachmen's capes.
The best skins come from the northern parts of the United States. A
smaller and poorer species inhabits South America, and a very few are
found in the north of India, but these do not interest the European
trade. From Japan a similar animal is obtained in smaller quantities
with very good but longer fur, of yellowish motley light-brown shades.
It is more often imported and sold as Japanese fox, but its
resemblance to the fur of the American raccoon is so marked as to
surely identify it. When dyed dark blue or skunk colour it is
good-looking and is sold widely in Europe. Raccoon skins are also
frequently unhaired, and if the underwool is of good quality the
effect is similar to beaver. It is the most useful fur for use in
America or Russia, having a full quantity of fur which will retain
heat. Value 10d. to 26s.
SABLE, AMERICAN and CANADIAN.--Size 17 X 5 in. The skins are sold in
the trade sale as martens, but as there are many that are of a very
dark colour and the majority are almost as silky as the Russian sable,
the retail trade has for generations back applied the term of sable to
this fur. The prevailing colour is a medium brown, and many are quite
yellow. The dyeing of these very pale skins has been for so long well
executed that it has been possible to make very good useful and
effective articles of them at a moderate price compared to Russian
sable. The finest skins are found in the East Main and the Esquimaux
Bay, in the Hudson's Bay Company's districts, and the poorest in
Alaska. They are not found very far south of the northern boundary of
the United States. The best skins are excellent in quality, colour and
effect, and wear well. Value 27s. 3d. to 290s.
SABLE, CHINESE and JAPANESE.--Size 14 X 4-1/2 in. These are similar to
the Amur skins previously referred to, but of much poorer quality and
generally only suitable for linings. The very palest skins are dyed
and made by the Chinese into mandarins' coats, in which form they are
found in the London trade sales, but being overdressed they are
inclined to be loose in the hair and the colour of the dye is not
good. The Japanese kind are imported raw, but are few in numbers, very
pale and require dyeing. Value 15s. to 150s.
SABLE, RUSSIAN.--Size 15 X 5 in. These skins belong to a species of
marten, very similar to the European and American, but much more silky
in the nature of their fur. They have long been known as "sables,"
doubtless owing to the density of colour to which many of them attain,
and they have always been held in the highest esteem by connoisseurs
as possessing a combination of rare qualities. The underwool is close,
fine and very soft, the top hair is regular, fine, silky and flowing,
varying from 1-1/2 to 2-1/2 in. in depth. In colour they range from a
pale stony or yellowish shade to a rich dark brown, almost black with
a bluish tone. The pelts are exceedingly fine and close in texture
and, although of little weight, are very durable, and articles made of
them produce a sensation of warmth immediately they are put upon the
body.
The Yakutsk, Okhotsk and Kamschatka sorts are good, the last being the
largest and fullest furred, but of less density of colour than the
others. Many from other districts are pale or yellowish brown, and
those from Saghalien are poor in quality. The most valuable are the
darkest from Yakutsk in Siberia, particularly those that have silvery
hairs evenly distributed over the skin. These however are exceedingly
scarce, and when a number are required to match for a large garment,
considerable time may be necessary to collect them. This class of skin
is the most expensive fur in the world, reckoning values by a square
foot unit.
The Amur skins are paler, but often of a pretty bluish stony tone with
many frequently interspersed silvery hairs. The quality too is lower,
that is, the fur is not so close or deep, but they are very effective,
particularly for close-fitting garments, as they possess the least
appearance of bulk. The paler skins from all districts in Siberia are
now cleverly coloured or "topped," that is, just the tips of the hair
are stained dark, and it is only an expert who can detect them from
perfectly natural shades. If this colouring process is properly
executed it remains fairly fast. Notwithstanding the reported rights
of the Russian imperial authorities over some regions with respect to
these and other valuable fur-bearing animals, there are in addition to
the numbers regularly sent to the trade auction sales in London many
good parcels of raw skins to be easily bought direct, provided price
is not the first consideration. Value 25s. to 980s.
SEAL, FUR.--Sizes range from 24 X 15 in. to 55 X 25 in., the width
being taken at the widest part of the skin after preparation. The
centre of the skin between the fins is very narrow and the skins taper
at each end, particularly at the tail. The very small pups are of a
beautiful quality, but too tiny to make into garments, and, as the aim
of a good furrier is to avoid all lateral or cross seams, skins are
selected that are the length of the garment that is to be made. The
most useful skins for coats are the large pups 42 in. long, and the
quality is very good and uniform. The largest skins, known in the
trade as "wigs," which range up to 8 ft. in length, are uneven and
weak in the fur, and hunters do not seek to obtain them. The supply of
the best sort is chiefly from the North Pacific, viz. Pribilof
Islands, Alaska, north-west coast of America, Copper Island of the
Aleutian group near to Kamschatka, Robben Island and Japan. Other
kinds are taken from the South Pacific and South Atlantic Oceans,
around Cape Horn, the Falkland Islands up to Lobos Islands at the
entrance of the La Plata river, off the Cape of Good Hope and Crozet
Isles. With, however, the exception of the pick of the Lobos Island
seals the fur of the southern sea seals is very poor and only suitable
for the cheapest market. Formerly many skins were obtained from New
Zealand and Australia, but the importation is now small and the
quality not good. The preparation of seal skin occupies a longer time
than any other fur skin, but its fine rich effect when finished and
its many properties of warmth and durability well repay it. Value 10s.
to 232s.
SEAL, HAIR.--There are several varieties of these seals in the seas
stretching north from Scotland, around Newfoundland, Greenland and the
north-west coast of America, and they are far more numerous than fur
seals. Generally they have coarse rigid hair and none possess any
underwool. They are taken principally for the oil and leather they
yield. Some of the better haired sorts are dyed black and brown and
used for men's motor coats when quite a waterproof garment is wanted,
and they are used also for this quality in China. The young of the
Greenland seals are called whitecoats on account of the early growth
being of a yellowish white colour; the hair is 3/4 to 1 in. long, and
at this early stage of their life is soft compared to that of the
older seals. These fur skins are dyed black or dark brown and are used
for military caps and hearth-rugs. Value 2s. to 15s. There are fewer
hair seals in the southern than in the northern seas.
SHEEP.--Vary much in size and in quality of wool. Many of the domestic
kind in central and northern Europe and Canada are used for drivers'
and peasants' coat linings, &c. In Great Britain many coats of the
home-reared sheep, having wools two and a half to five inches long,
are dyed various colours and used as floor rugs. Skins with very short
wool are dyed black and used for military saddle-cloths. The bulk,
however, is used in the wool trade. The Hungarian peasants are very
fond of their natural brown sheep coats, the leather side of which is
not lined, but embellished by a very close fancy embroidery, worked
upon the leather itself; these garments are reversible, the fur being
worn inside when the weather is cold. Chinese sheep are largely used
for cheap rugs. Value of English sheep from 3s. to 10s.
SKUNK or BLACK MARTEN.--Size 15 X 8 in. The underwool is full and
fairly close with glossy, flowing top hair about 2-1/2 in. long. The
majority have two stripes of white hair, extending the whole length of
the skin, but these are cut out by the manufacturing furrier and sold
to the dealers in pieces for exportation. The animals are found widely
spread throughout North and South America. The skins which are of the
greatest interest to the European trade are those from North America,
the South American species being small, coarse and generally brown.
The best skins come from Ohio and New York. If it were not for its
disagreeable odour, skunk would be worth much more than the usual
market value, as it is naturally the blackest fur, silky in appearance
and most durable. The improved dressing processes have to a large
extent removed the naturally pungent scent. The fur is excellent for
stoles, boas, collars, cuffs, muffs and trimmings. Value 1s. 6d. to
11s.
SOUSLIK.--Size 7 in. X 2-1/4. Is a small rodent found in the south of
Russia and also in parts of America. It has very short hair and is a
poor fur even for the cheapest linings, which is the only use to which
the skin could be put. It is known as kaluga when imported in
ready-made linings from Russia where the skins are dressed and worked
in an inferior way. Value 1d. to 3d.
SQUIRREL.--Size 10 X 5 in. This measurement refers to the Russian and
Siberian sorts, which are the only kind imported for the fur. The
numerous other species are too poor in their coats to attract notice
from fur dealers. The back of the Russian squirrel has an even close
fur varying from a clear bluish-grey to a reddish-brown, the bellies
in the former being of a flat quality and white, in the latter
yellowish. The backs are worked into linings separately, as are the
bellies or "locks." The pelts, although very light, are tough and
durable, hence their good reputation for linings for ladies' walking
or driving coats. The best skins also provide excellent material for
coats, capes, stoles, ties, collars, cuffs, gloves, muffs, hoods and
light-weight carriage aprons. The tails are dark and very small, and
when required for ends of boas three or four are made as one. Value
per skin from 2-1/2d. to 1s. 1d.
TIBET LAMB.--Size 27 X 13 in. These pretty animals have a long, very
fine, silky and curly fleece of a creamy white. The majority are
consigned to the trade auction sales in London ready dressed and
worked into cross-shaped coats, and the remainder, a fourth of the
total, come as dressed skins. They are excellent for trimmings of
evening mantles and for children's ties, muffs and perambulator
aprons. The fur is too long and bulky for linings. Value per skin from
4s. 6d. to 8s. 6d.
TIGER.--Size varies considerably, largest about 10 ft. from nose to
root of tail. Tigers are found throughout India, Turkestan, China,
Mongolia and the East Indies. The coats of the Bengal kind are short
and of a dark orange brown with black stripes, those from east or
further India are similar in colour, but longer in the hair, while
those from north of the Himalayas and the mountains of China are not
only huge in size, but have a very long soft hair of delicate orange
brown with very white flanks, and marked generally with the blackest
of stripes. The last are of a noble appearance and exceedingly scarce.
They all make handsome floor rugs.
Value of the Indian from L3 to L15.
" " Chinese " L10 to L65.
VICUNA is a species of long-necked sheep native to South America,
bearing some resemblance to the guanaco, but the fur is shorter,
closer and much finer. The colour is a pale golden-brown and the fur
is held in great repute in South America for carriage rugs. The supply
is evidently small as the prices are high. There is scarcely a
commercial quotation in London, few coming in except from private
sources. 2s. 6d. to 5s. 6d. may be considered as the average value.
WALLABY.--See _Kangaroo_, above.
WALLAROO.--See _Kangaroo_, above.
WOLF.--Size 50 X 25 in. Is closely allied to the dog tribe and, like
the jackals, is found through a wide range of the world,--North and
South America, Europe and Asia. Good supplies are available from North
America and Siberia and a very few from China. The best are the full
furred ones of a very pale bluish-grey with fine flowing black top
hair, which are obtained from the Hudson Bay district. Those from the
United States and Asia are harsher in quality and browner. A few black
American specimens come into the market, but usually the quality is
poor compared to the lighter furred animal. The Siberian is smaller
than the North American and the Russian still smaller. Besides the
wolf proper a large number of prairie or dog wolves from America and
Asia are used for cheaper rugs. In size they are less than half that
of a large wolf and are of a motley sandy colour. Numbers of the
Russian are retained for home use. The finest wolves are very light
weighted and most suitable for carriage aprons, in fact, ideal for the
purpose, though lacking the strength of some other furs.
Wolves value 2s. 6d. to 64s.
Dog wolves " 1s. to 2s. 6d.
WOLVERINE.--Size 16 X 18 in. Is native to America, Siberia, Russia and
Scandinavia and generally partakes of the nature of a bear. The
underwool is full and thick with strong and bright top hair about
2-1/2 in. long. The colour is of two or three shades of brown in one
skin, the centre being an oval dark saddle, edged as it were with
quite a pale tone and merging to a darker one towards the flanks. This
peculiar character alone stamps it as a distinguished fur, in addition
to which it has the excellent advantage of being the most durable fur
for carriage aprons, as well as the richest in colour. It is not
prolific, added to which it is very difficult to match a number of
skins in quality as well as colour. Hence it is an expensive fur, but
its excellent qualities make it valuable. The darkest of the least
coarse skins are worth the most. Prices from 6s. to 37s.
WOMBAT, KOALA or Australian Bear.--Size 20 X 12 in. Has light grey or
brown close thick wool half an inch deep without any top hair, with a
rather thick spongy pelt. It is quite inexpensive and only suitable
for cheap rough coats, carriage rugs, perambulator aprons and linings
for footbags. The coats are largely used in western America and
Canada. Value 3d. to 1s. 8-1/2d.
_Preparing and Dressing._--A furrier or skin merchant must possess a
good eye for colour to be successful, the difference in value on this
subtle matter solely (in the rarer precious sorts, especially sables,
natural black, silver and blue fox, sea otters, chinchillas, fine mink,
&c.) being so considerable that not only a practised but an intuitive
sense of colour is necessary to accurately determine the exact merits of
every skin. In addition to this a knowledge is required of what the
condition of a pelt should be; a good judge knows by experience whether
a skin will turn out soft and strong, after dressing, and whether the
hair is in the best condition of strength and beauty. The dressing of
the pelt or skin that is to be preserved for fur is totally different to
the making of leather; in the latter tannic acid is used, but never
should be with a fur skin, as is so often done by natives of districts
where a regular fur trade is not carried on. The results of applying
tannic acid are to harden the pelt and discolour and weaken the fur. The
best methods for dressing fur skins are those of a tawer or currier, the
aim being to retain all the natural oil in the pelt, in order to
preserve the natural colour of the fur, and to render the pelt as supple
as possible. Generally the skins are placed in an alkali bath, then by
hand with a blunt wooden instrument the moisture of the pelt is worked
out and it is drawn carefully to and fro over a straight, dull-edged
knife to remove any superfluous flesh and unevenness. Special grease is
then rubbed in and the skin placed in a machine which softly and
continuously beats in the softening mixture, after which it is put into
a slowly revolving drum, fitted with wooden paddles, partly filled with
various kinds of fine hard sawdust according to the nature of the furs
dealt with. This process with a moderate degree of heat thoroughly
cleans it of external greasy matter, and all that is necessary before
manufacturing is to gently tap the fur upon a leather cushion stuffed
with horsehair with smooth canes of a flexibility suited to the strength
of the fur. After dressing most skins alter in shape and decrease in
size.
With regard to the merits of European dressing, it may be fairly taken
that English, German and French dressers have specialities of
excellence. In England, for instance, the dressing of sables, martens,
foxes, otters, seals, bears, lions, tigers and leopards is first rate;
while with skunk, mink, musquash, chinchillas, beavers, lambs and
squirrels, the Germans show better results, particularly in the last.
The pelt after the German dressing is dry, soft and white, which is due
to a finishing process where meal is used, thus they compare favourably
with the moister and consequently heavier English finish. In France they
do well with cheaper skins, such as musquash, rabbit and hare, which
they dye in addition to dressing. Russian dressing is seldom reliable;
not only is there an unpleasant odour, but in damp weather the pelts
often become clammy, which is due to the saline matter in the dressing
mixture. Chinese dressing is white and supple, but contains much powder,
which is disagreeable and difficult to get rid of, and in many instances
the skin is rendered so thin that the roots of the fur are weakened,
which means that it is liable to shed itself freely, when subject to
ordinary friction in handling or wearing. American and Canadian dressing
is gradually improving, but hitherto their results have been inferior to
the older European methods.
In the case of seal and beaver skins the process is a much more
difficult one, as the water or hard top hairs have to be removed by hand
after the pelt has been carefully rendered moist and warm. With seal
skins the process is longer than with any other fur preparation and the
series of processes engage many specialists, each man being constantly
kept upon one section of the work. The skins arrive simply salted. After
being purchased at the auction sales they are washed, then stretched
upon a hoop, when all blubber and unnecessary flesh is removed, and the
pelt is reduced to an equal thickness, but not so thin as it is finally
rendered. Subsequently the hard top hairs are taken out as in the case
of otters and beavers and the whole thoroughly cleaned in the revolving
drums. The close underwool, which is of a slightly wavy nature and
mostly of a pale drab colour, is then dyed by repeated applications of a
rich dark brown colour, one coat after another, each being allowed to
thoroughly dry before the next is put on, till the effect is almost a
lustrous black on the top. The whole is again put through the cleaning
process and evenly reduced in thickness by revolving emery wheels, and
eventually finished off in the palest buff colour.
The English dye for seals is to-day undoubtedly the best; its
constituents are more or less of a trade secret, but the principal
ingredients comprise gall nuts, copper dust, camphor and antimony, and
it would appear after years of careful watching that the atmosphere and
particularly the water of London are partly responsible for good and
lasting results. The Paris dyers do excellent work in this direction,
but the colour is not so durable, probably owing to a less pure water.
In America of late, strides have been made in seal dyeing, but
preference is still given to London work. In Paris, too, they obtain
beautiful results in the "topping" or colouring Russian sables and the
Germans are particularly successful in dyeing Persian lambs black and
foxes in all blue, grey, black and smoke colours and in the insertion of
white hairs in imitation of the real silver fox. Small quantities of
good beaver are dyed in Russia occasionally, and white hairs put in so
well that an effect similar to sea otter is obtained.
The process of inserting white hairs is called in the trade "pointing,
"and is either done by stitching them in with a needle or by adhesive
caoutchouc.
The Viennese are successful in dyeing marmot well, and their cleverness
in colouring it with a series of stripes to represent the natural
markings of sable which has been done after the garments have been made,
so as to obtain symmetry of lines, has secured for them a large trade
among the dealers of cheap furs in England and the continent.
_Manufacturing Methods and Specialities._--In the olden times the
Skinners' Company of the city of London was an association of furriers
and skin dressers established under royal charter granted by Edward III.
At that period the chief concern of the body was to prevent buyers from
being imposed upon by sellers who were much given to offering old furs
as new; a century later the Skinners' Company received other charters
empowering them to inspect not only warehouses and open markets, but
workrooms. In 1667 they were given power to scrutinize the preparing of
rabbit or cony wool for the wool trade and the registration of the then
customary seven years' apprenticeship. To-day all these privileges and
powers are in abeyance, and the interest that they took in the fur trade
has been gradually transferred to the leather-dressing craft.
The work done by English furriers was generally good, but since about
1865 has considerably improved on account of the influx of German
workmen, who have long been celebrated for excellent fur work, being In
their own country obliged to satisfy officially appointed experts and to
obtain a certificate of capacity before they can be there employed. The
French influence upon the trade has been, and still is, primarily one of
style and combination of colour, bad judgment in which will mar the
beauty of the most valuable furs. It is a recognized law among
high-class furriers that furs should be simply arranged, that is, that
an article should consist of one fur or of two furs of a suitable
contrast, to which lace may be in some cases added with advantage. As
illustrative of this, it may be explained that any brown tone of fur
such as sable, marten, mink, black marten, beaver, nutria, &c., will go
well upon black or very dark-brown furs, while those of a white or grey
nature, such as ermine, white lamb, chinchilla, blue fox, silver fox,
opossum, grey squirrel, grey lamb, will set well upon seal or black
furs, as Persian lamb, broadtail, astrachan, caracul lamb, &c. White is
also permissible upon some light browns and greys, but brown motley
colours and greys should never be in contrast. One neutralizes the other
and the effect is bad. The qualities, too have to be considered--the
fulness of one, the flatness of the other, or the coarseness or fineness
of the furs. The introduction of a third fur in the same garment or
indiscriminate selection of colours of silk linings, braids, buttons,
&c., often spoils an otherwise good article.
With regard to the natural colours of furs, the browns that command the
highest prices are those that are of a bluish rather than a reddish
tendency. With greys it is those that are bluish, not yellow, and with
white those that are purest, and with black the most dense, that are
most esteemed and that are the rarest.
Perhaps for ingenuity and the latest methods of manipulating skins in
the manufacturing of furs the Americans lead the way, but as fur cutters
are more or less of a roving and cosmopolitan character the larger fur
businesses in London, Berlin, Vienna, St Petersburg, Paris and New York
are guided by the same thorough and comparatively advanced principles.
During the period just mentioned the tailors' methods of scientific
pattern cutting have been adopted by the leading furriers in place of
the old chance methods of fur cutters, so that to-day a fur garment may
be as accurately and gracefully fitted as plush or velvet, and with all
good houses a material pattern is fitted and approved before the skins
are cut.
Through the advent of German and American fur sewing-machines since
about 1890 fur work has been done better and cheaper. There are,
however, certain parts of a garment, such as the putting in of sleeves
and placing on of collars, &c., that can only be sewn by hand. For
straight seams the machines are excellent, making as neat a seam as is
found in glove work, unless, of course, the pelts are especially heavy,
such as bears and sheep rugs.
A very great feature of German and Russian work is the fur linings
called rotondes, sacques or plates, which are made for their home use
and exportation chiefly to Great Britain, America and France.
In Weissenfels, near Leipzig, the dressing of Russian grey squirrel and
the making it into linings is a gigantic industry, and is the principal
support of the place. After the dressing process the backs of the
squirrels are made up separately from the under and thinner white and
grey parts, the first being known as squirrel-back and the other as
squirrel-lock linings. A few linings are made from entire skins and
others are made from the quite white pieces, which in some instances are
spotted with the black ear tips of the animals to resemble ermine. The
smaller and uneven pieces of heads and legs are made up into linings, so
there is absolutely no waste. Similar work is done in Russia on almost
as extensive a scale, but neither the dressing nor the work is so good
as the German.
The majority of heads, gills or throats, sides or flanks, paws and
pieces of skins cut up in the fur workshops of Great Britain, America
and France, weighing many tons, are chiefly exported to Leipzig, and
made up in neighbouring countries and Greece, where labour can be
obtained at an alarmingly low rate. Although the sewing, which is
necessarily done by hand, the sections being of so unequal and tortuous
a character, is rather roughly executed, the matching of colours and
qualities is excellent. The enormous quantities of pieces admit of good
selection and where odd colours prevail in a lining it is dyed. Many
squirrel-lock linings are dyed blue and brown and used for the outside
of cheap garments. They are of little weight, warm and effective, but
not of great durability.
The principal linings are as follows: Sable sides, sable heads and paws,
sable gills, mink sides, heads and gills, marten sides, heads and gills,
Persian lamb pieces and paws, caracul lamb pieces or paws, musquash
sides and heads, nutria sides, genet pieces, raccoon sides or flanks,
fox sides, kolinski whole skins, and small rodents as kaluga and
hamster. The white stripes cut out of skunks are made into rugs.
Another great source of inexpensive furs is China, and for many years
past enormous quantities of dressed furs, many of which are made up in
the form of linings and Chinese loose-shaped garments, have been
imported by England, Germany and France for the lower class of business;
the garments are only regarded as so much fur and are reworked. With,
however, the exception of the best white Tibet lambs, the majority of
Chinese furs can only be regarded as inferior material. While the work
is often cleverly done as to matching and manipulation of the pelt which
is very soft, there are great objections in the odour and the
brittleness or weakness of the fur. One of the most remarkable results
of the European intervention in the Boxer rising in China (1900) was the
absurd price paid for so-called "loot" of furs, particularly in
mandarins' coats of dyed and natural fox skins and pieces, and natural
ermine, poor in quality and yellowish in colour; from three to ten times
their value was paid for them when at the same time huge parcels of
similar quality were warehoused in the London docks, because purchasers
could not be found for them.
With regard to Japanese furs, there is little to commend them. The best
are a species of raccoon usually sold as fox, and, being of close long
quality of fur, they are serviceable for boas, collars, muffs and
carriage aprons. The sables, martens, minks and otters are poor in
quality, and all of a very yellow colour and they are generally dyed for
the cheap trade. A small number of very pretty guanaco and vicuna
carriage rugs are imported into Europe, and many come through travellers
and private sources, but generally they are so badly dressed that they
are quite brittle upon the leather side. Similar remarks are applicable
to opossum rugs made in Australia. From South Africa a quantity of
jackal, hyena, fox, leopard and sheep karosses, i.e. a peculiarly shaped
rug or covering used by native chiefs, is privately brought over. The
skins are invariably tanned and beautifully sewn, the furs are generally
flat in quality and not very strong in the hair, and are retained' more
as curiosities than for use as a warm covering.
_Hatters' Furs and Cloths and Shawls._--The hat trade is largely
interested in the fur piece trade, the best felt hats being made from
beaver and musquash wool and the cheaper sorts from nutria, hare and
rabbit wools. For weaving, the most valuable pieces are mohair taken
from the angora and vicuna. They are limited in quantity and costly, and
the trade depends upon various sorts of other sheep and goat wools for
the bulk of its productions.
_Frauds and Imitations._--The opportunities for cheating in the fur
trade are very considerable, and most serious frauds have been
perpetrated in the selling of sables that have been coloured or
"topped"; that is, just the tips of the hairs stained dark to represent
more expensive skins. It is only by years of experience that some of
these colourings can be detected. Where the skins are heavily dyed it is
comparatively easy to see the difference between a natural and a dyed
colour, as the underwool and top hair become almost alike and the
leather is also dark, whereas in natural skins the base of the underwool
is much paler than the top, or of a different colour, and the leather Is
white unless finished in a pale reddish tone as is sometimes the case
when mahogany sawdust is used in the final cleaning. As has been
explained, sable is a term applied for centuries past to the darker
sorts of the Russian Siberian martens, and for years past the same term
has been bestowed by the retail trade upon the American and Canadian
martens. The baum and stone martens caught in France, the north of
Turkey and Norway are of the same family, but coarser in underwool and
the top hair is less in quantity and not so silky. The kolinski, or as
it is sometimes styled Tatar sable, is the animal, the tail of which
supplies hair for artists' brushes. This is also of the marten species
and has been frequently offered, when dyed dark, as have baum and stone
martens, as Russian sables. Hares, too, are dyed a sable colour and
advertised as sable. The fur, apart from a clumsy appearance, is so
brittle, however, as to be of scarcely any service whatever.
Among the principal imitations of other furs is musquash, out of which
the top hair has been pulled and the undergrowth of wool clipped and
dyed exactly the same colour as is used for seal, which is then offered
as seal or red river seal. Its durability, however, is far less than
that of seal. Rabbit is prepared and dyed and frequently offered as
"electric sealskin." Nutria also is prepared to represent sealskin, and
in its natural colour, after the long hairs are plucked out, it is sold
as otter or beaver. The wool is, however, poor compared to the otter and
beaver, and the pelt thin and in no way comparable to them in strength.
White hares are frequently sold as white fox, but the fur is weak,
brittle and exceedingly poor compared to fox and possesses no thick
underwool. Foxes, too, and badger are dyed a brownish black, and white
hairs inserted to imitate silver fox, but the white hairs are too coarse
and the colour too dense to mislead any one who knows the real article.
But if sold upon its own merits, pointed fox is a durable fur.
Garments made of sealskin pieces and Persian lamb pieces are frequently
sold as if they were made of solid skins, the term "pieces" being simply
suppressed. The London Chamber of Commerce have issued to the British
trade a notice that any misleading term in advertising and all attempts
at deception are illegal, and offenders are liable under the Merchandise
Marks Act 1887.
The most usual misnaming of manufactured furs is as follow:--
Musquash, pulled and dyed Sold as seal.
Nutria, pulled and dyed Sold as seal.
Nutria, pulled and natural Sold as beaver.
Rabbit, sheared and dyed Sold as seal or electric seal.
Otter, pulled and dyed Sold as seal.
Marmot, dyed Sold as mink or sable.
Fitch, dyed Sold as sable.
Rabbit, dyed Sold as sable or French sable.
Hare, dyed Sold as sable, or fox, or lynx.
Musquash, dyed Sold as mink or sable.
Wallaby, dyed Sold as skunk.
White Rabbit Sold as ermine.
White Rabbit, dyed Sold as chinchilla.
White Hare, dyed or natural Sold as fox, foxaline, and
other similar names.
Goat, dyed Sold as bear, leopard, &c.
Dyed manufactured articles of
all kinds Sold as "natural."
White hairs inserted in foxes
and sables Sold as real or natural furs.
Kids Sold as lamb or broadtails.
American sable Sold as real Russian sable.
Mink Sold as sable.
_The Preservation of Furs._--For many years raw sealskins have been
preserved in cold storage, but it is only within a recent period, owing
to the difficulty there was in obtaining the necessary perfectly dry
atmosphere, that dressed and made-up furs have been preserved by
freezing. Furs kept in such a condition are not only immune from the
ravages of the larvae of moth, but all the natural oils in the pelt and
fur are conserved, so that its colour and life are prolonged, and the
natural deterioration is arrested. Sunlight has a tendency to bleach
furs and to encourage the development of moth eggs, therefore continued
exposure is to be avoided. When furs are wetted by rain they should be
well shaken and allowed to dry in a current of air without exposure to
sun or open fire.
Where a freezing store for furs is not accessible, furs should be well
shaken and afterwards packed in linen and kept in a perfectly cool dry
place, and examined in the summer at periods of not less than five
weeks. Naphthalene and the usual malodorous powders are not only very
disagreeable, but quite useless. Any chemical that is strong enough to
destroy the life in a moth egg would also be sufficiently potent to
injure the fur itself. In England moth life is practically continuous
all the year round, that is, as regards those moths that attack furs,
though the destructive element exists to a far greater extent during
spring and summer.
_Comparative Durability of Various Furs and Weight of Unlined Skins
per Square Foot._
The following estimates of durability refer to the use of fur when
made up "hair outside" in garments or stoles, not as a lining. The
durability of fur used as linings, which is affected by other
conditions, is set forth separately. Otter, with its water hairs
removed, the strongest of furs for external use, is, in this table,
taken as the standard at 100 and other furs marked accordingly:--
_The Precious Furs._
+---------------------------------------+-------------+------------+
| | Points of | [2]Weight |
| | Durability. | in oz. per |
| | | sq. ft. |
+---------------------------------------+-------------+------------+
| Sable | 60 | 2-1/2 |
| Sea | 75 | 3 |
| Fox, Silver or Black | 40 | 3 |
| " White | 20 | 3 |
| Ermine | 25 | 1-1/4 |
| Chinchilla | 15 | 1-1/2 |
| Sea-otter (for stoles or collars) | 100 | 4-1/4 |
+---------------------------------------+-------------+------------+
_The Less Valuable Furs._
+----------------------------------------+-------------+------------+
| | Points of | Weight |
| | Durability. | in oz. per |
| | | sq. ft. |
+----------------------------------------+-------------+------------+
| Sable "topped," i.e. top hairs coloured| 55 | 2-1/2 |
| " tinted, i.e. fur all coloured. | 50 | 2-1/2 |
| Baum Marten, natural | 65 | 2-3/4 |
| " " tinted | 45 | 2-3/4 |
| Stone Marten | 40 | 2-3/4 |
| Nutria | 27 | 3-1/4 |
| Musquash, natural | 37 | 3-1/4 |
| " water hairs removed, sheared | | |
| and seal finished. | 33 | 3-1/4 |
| Skunk | 70 | 2-3/4 |
| Mink | 70 | 3-1/4 |
| Lynx, natural | 25 | 2-3/4 |
| " tinted black | 20 | 2-3/4 |
| Marmot, tinted | 10 | 3 |
| Fox, tinted black | 25 | 3 |
| " " blue | 20 | 3 |
| Opossum | 37 | 3 |
| Otter (with water hairs) | 100 | 4 |
| " (water hairs removed) | 95 | 3-15/16 |
| Beaver (water hairs cut level with fur)| 90 | 4 |
| " (water hairs removed). | 85 | 3-15/16 |
| Moleskin | 7 | 1-3/4 |
| Persian Lamb | 65 | 3-1/4 |
| Grey " | 30 | 3-1/4 |
| Broadtail | 15 | 2-1/4 |
| Caracul Kid | 10 | 3-1/4 |
| " Lamb | 15 | 3-1/4 |
| Squirrel | 25 | 1-3/4 |
| Hare | 5 | 1-3/4 |
| Rabbit | 5 | 2-1/4 |
+----------------------------------------+-------------+------------+
_Quantities of Fur needed, in Square Feet._
The "Paris Model" figure is the basis of these estimates for ladies'
garments, the standard measurements being height 5 ft. 6 in., waist 23
in., bust 38 in.
Sq. Ft.
(approximate).
Straight stole 1/2 length (just below the waist line) 2-3/4
Straight stole 3/4 length (just below the knee) 3-3/4
Stole, broad enough at the neck to cover the top of
arm 3/4 length 5
The same, full length (to hem of skirt) 6
Eton jacket, without collar 13
Plain cape, 15 in. long 6-1/2
Deep cape, 30 in. long 15
Full cape with broad stole front, 3/4 length 15
Inverness cape (to knee) 25
Double-breasted, straight, semi-fitting coat, covering
hips 16
Double-breasted sacque jacket, 36 in. long, full sleeves 20
Same, 30 in. long 18
Same, 22 in. long 15
Long, full, shawl cape with points at back and front,
well below knee 15
Shorter shawl cape 16
Motoring or driving coat, 3/4 length 22
Motoring or driving coat, full length 27
_Weight and Durability of Furs for Men's Coat Linings._
Otter with the water hairs removed, the strongest fur suited for
linings, is here taken as the standard.
+---------------------------------+-----------+-------------+
| | Points of | Weight |
| |Durability.| in oz. per |
| | | sq. ft. |
+---------------------------------+-----------+-------------+
| Otter (the water hairs removed) | 100 | 3-15/16 |
| Beaver " " | 90 | 3-15/16 |
| Mink | 90 | 3-1/4 |
| Sealskin | 75 | 3 |
| Raccoon | 75 | 4-1/2 |
| Persian lamb or astrachan | 70 | 3-1/4 |
| Sable | 65 | 2-1/2 |
| Musquash | 55 | 3-1/2 |
| Nutria | 40 | 3-1/4 |
| Grey Opossum | 40 | 3 |
| Wallaby | 30 | 3-3/4 |
| Squirrel | 30 | 1-3/4 |
| Hamster | 15 | 1-1/4 |
| Rabbit | 10 | 2-1/4 |
+---------------------------------+----------