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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:07:10 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:07:10 -0700 |
| commit | 744ba1f59f0e64e04167cfbb5fe992f583704bad (patch) | |
| tree | 343e462041e9437183c78a22c0c076c064e9265b /37064-h | |
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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 + "Frost" to "Fyzabad" + +Author: Various + +Release Date: August 13, 2011 [EBook #37064] + +Language: English + +Character set encoding: ISO-8859-1 + +*** 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 + + + + + + +</pre> + + + +<table border="0" cellpadding="10" style="background-color: #dcdcdc; color: #696969; " summary="Transcriber's note"> +<tr> +<td style="width:25%; vertical-align:top"> +Transcriber’s note: +</td> +<td class="norm"> +A few typographical errors have been corrected. They +appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the +explanation will appear when the mouse pointer is moved over the marked +passage. Sections in Greek will yield a transliteration +when the pointer is moved over them, and words using diacritic characters in the +Latin Extended Additional block, which may not display in some fonts or browsers, will +display an unaccented version. <br /><br /> +<a name="artlinks">Links to other EB articles:</a> Links to articles residing in other EB volumes will +be made available when the respective volumes are introduced online. +</td> +</tr> +</table> +<div style="padding-top: 3em; "> </div> + +<h2>THE ENCYCLOPÆDIA BRITANNICA</h2> + +<h2>A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION</h2> + +<h3>ELEVENTH EDITION</h3> +<div style="padding-top: 3em; "> </div> + +<hr class="full" /> +<h3>VOLUME XI SLICE III<br /><br /> +Frost to Fyzabad</h3> +<hr class="full" /> +<div style="padding-top: 3em; "> </div> + +<p class="center1" style="font-size: 150%; font-family: 'verdana';">Articles in This Slice</p> +<table class="reg" style="width: 90%; font-size: 90%; border: gray 2px solid;" cellspacing="8" summary="Contents"> + +<tr><td class="tcl"><a href="#ar1">FROST</a></td> <td class="tcl"><a href="#ar59">FULMAR</a></td></tr> +<tr><td class="tcl"><a href="#ar2">FROSTBITE</a></td> <td class="tcl"><a href="#ar60">FULMINIC ACID</a></td></tr> +<tr><td class="tcl"><a href="#ar3">FROSTBURG</a></td> <td class="tcl"><a href="#ar61">FULTON, ROBERT</a></td></tr> +<tr><td class="tcl"><a href="#ar4">FROTHINGHAM, OCTAVIUS BROOKS</a></td> <td class="tcl"><a href="#ar62">FULTON</a> (Missouri, U.S.A.)</td></tr> +<tr><td class="tcl"><a href="#ar5">FROUDE, JAMES ANTHONY</a></td> <td class="tcl"><a href="#ar63">FULTON</a> (New York, U.S.A.)</td></tr> +<tr><td class="tcl"><a href="#ar6">FRUCTOSE</a></td> <td class="tcl"><a href="#ar64">FUM</a></td></tr> +<tr><td class="tcl"><a href="#ar7">FRUGONI, CARLO INNOCENZIO MARIA</a></td> <td class="tcl"><a href="#ar65">FUMARIC AND MALEIC ACIDS</a></td></tr> +<tr><td class="tcl"><a href="#ar8">FRUIT</a></td> <td class="tcl"><a href="#ar66">FUMAROLE</a></td></tr> +<tr><td class="tcl"><a href="#ar9">FRUIT AND FLOWER FARMING</a></td> <td class="tcl"><a href="#ar67">FUMIGATION</a></td></tr> +<tr><td class="tcl"><a href="#ar10">FRUMENTIUS</a></td> <td class="tcl"><a href="#ar68">FUMITORY</a></td></tr> +<tr><td class="tcl"><a href="#ar11">FRUNDSBERG, GEORG VON</a></td> <td class="tcl"><a href="#ar69">FUNCHAL</a></td></tr> +<tr><td class="tcl"><a href="#ar12">FRUSTUM</a></td> <td class="tcl"><a href="#ar70">FUNCTION</a></td></tr> +<tr><td class="tcl"><a href="#ar13">FRUYTIERS, PHILIP</a></td> <td class="tcl"><a href="#ar71">FUNDY, BAY OF</a></td></tr> +<tr><td class="tcl"><a href="#ar14">FRY</a></td> <td class="tcl"><a href="#ar72">FUNERAL RITES</a></td></tr> +<tr><td class="tcl"><a href="#ar15">FRY, SIR EDWARD</a></td> <td class="tcl"><a href="#ar73">FUNGI</a></td></tr> +<tr><td class="tcl"><a href="#ar16">FRY, ELIZABETH</a></td> <td class="tcl"><a href="#ar74">FUNJ</a></td></tr> +<tr><td class="tcl"><a href="#ar17">FRYXELL, ANDERS</a></td> <td class="tcl"><a href="#ar75">FUNKIA</a></td></tr> +<tr><td class="tcl"><a href="#ar18">FUAD PASHA</a></td> <td class="tcl"><a href="#ar76">FUNNEL</a></td></tr> +<tr><td class="tcl"><a href="#ar19">FUCHOW</a></td> <td class="tcl"><a href="#ar77">FUR</a></td></tr> +<tr><td class="tcl"><a href="#ar20">FUCHS, JOHANN NEPOMUK VON</a></td> <td class="tcl"><a href="#ar78">FURAZANES</a></td></tr> +<tr><td class="tcl"><a href="#ar21">FUCHS, LEONHARD</a></td> <td class="tcl"><a href="#ar79">FURETIÈRE, ANTOINE</a></td></tr> +<tr><td class="tcl"><a href="#ar22">FUCHSIA</a></td> <td class="tcl"><a href="#ar80">FURFOOZ</a></td></tr> +<tr><td class="tcl"><a href="#ar23">FUCHSINE</a></td> <td class="tcl"><a href="#ar81">FURFURANE</a></td></tr> +<tr><td class="tcl"><a href="#ar24">FUCINO, LAGO DI</a></td> <td class="tcl"><a href="#ar82">FURIES</a></td></tr> +<tr><td class="tcl"><a href="#ar25">FUEL</a></td> <td class="tcl"><a href="#ar83"> FURLONG</a></td></tr> +<tr><td class="tcl"><a href="#ar26">FUENTE OVEJUNA</a></td> <td class="tcl"><a href="#ar84">FURNACE</a></td></tr> +<tr><td class="tcl"><a href="#ar27">FUENTERRABIA</a></td> <td class="tcl"><a href="#ar85"> FURNEAUX, TOBIAS</a></td></tr> +<tr><td class="tcl"><a href="#ar28">FUERO</a></td> <td class="tcl"><a href="#ar86">FURNES</a></td></tr> +<tr><td class="tcl"><a href="#ar29">FUERTEVENTURA</a></td> <td class="tcl"><a href="#ar87">FURNESS, HORACE HOWARD</a></td></tr> +<tr><td class="tcl"><a href="#ar30">FUGGER</a></td> <td class="tcl"><a href="#ar88">FURNESS</a></td></tr> +<tr><td class="tcl"><a href="#ar31">FUGITIVE SLAVE LAWS</a></td> <td class="tcl"><a href="#ar89">FURNISS, HARRY</a></td></tr> +<tr><td class="tcl"><a href="#ar32">FUGLEMAN</a></td> <td class="tcl"><a href="#ar90"> FURNITURE</a></td></tr> +<tr><td class="tcl"><a href="#ar33">FUGUE</a></td> <td class="tcl"><a href="#ar91"> FURNIVALL, FREDERICK JAMES</a></td></tr> +<tr><td class="tcl"><a href="#ar34">FÜHRICH, JOSEPH VON</a></td> <td class="tcl"><a href="#ar92">FURSE, CHARLES WELLINGTON</a></td></tr> +<tr><td class="tcl"><a href="#ar35">FUJI</a></td> <td class="tcl"><a href="#ar93"> FÜRST, JULIUS</a></td></tr> +<tr><td class="tcl"><a href="#ar36">FU-KIEN</a></td> <td class="tcl"><a href="#ar94">FÜRSTENBERG</a></td></tr> +<tr><td class="tcl"><a href="#ar37">FUKUI</a></td> <td class="tcl"><a href="#ar95">FÜRSTENWALDE</a></td></tr> +<tr><td class="tcl"><a href="#ar38">FUKUOKA</a></td> <td class="tcl"><a href="#ar96">FÜRTH</a></td></tr> +<tr><td class="tcl"><a href="#ar39">FULA</a></td> <td class="tcl"><a href="#ar97"> FURTWÄNGLER, ADOLF</a></td></tr> +<tr><td class="tcl"><a href="#ar40">FULCHER OF CHARTRES</a></td> <td class="tcl"><a href="#ar98">FURZE</a></td></tr> +<tr><td class="tcl"><a href="#ar41">FULDA</a></td> <td class="tcl"><a href="#ar99">FUSARO, LAGO</a></td></tr> +<tr><td class="tcl"><a href="#ar42">FULGENTIUS, FABIUS PLANCIADES</a></td> <td class="tcl"><a href="#ar100">FUSELI, HENRY</a></td></tr> +<tr><td class="tcl"><a href="#ar43">FULGINIAE</a></td> <td class="tcl"><a href="#ar101">FUSEL OIL</a></td></tr> +<tr><td class="tcl"><a href="#ar44">FULGURITE</a></td> <td class="tcl"><a href="#ar102">FUSIBLE METAL</a></td></tr> +<tr><td class="tcl"><a href="#ar45">FULHAM</a></td> <td class="tcl"><a href="#ar103">FUSILIER</a></td></tr> +<tr><td class="tcl"><a href="#ar46">FULK</a> (king of Jerusalem)</td> <td class="tcl"><a href="#ar104">FUSION</a></td></tr> +<tr><td class="tcl"><a href="#ar47">FULK</a> (archbishop of Reims)</td> <td class="tcl"><a href="#ar105">FÜSSEN</a></td></tr> +<tr><td class="tcl"><a href="#ar48">FULKE, WILLIAM</a></td> <td class="tcl"><a href="#ar106">FUST, JOHANN</a></td></tr> +<tr><td class="tcl"><a href="#ar49">FULK NERRA</a></td> <td class="tcl"><a href="#ar107">FUSTEL DE COULANGES, NUMA DENIS</a></td></tr> +<tr><td class="tcl"><a href="#ar50">FÜLLEBORN, GEORG GUSTAV</a></td> <td class="tcl"><a href="#ar108">FUSTIAN</a></td></tr> +<tr><td class="tcl"><a href="#ar51">FULLER, ANDREW</a></td> <td class="tcl"><a href="#ar109">FUSTIC</a></td></tr> +<tr><td class="tcl"><a href="#ar52">FULLER, GEORGE</a></td> <td class="tcl"><a href="#ar110">FUTURES</a></td></tr> +<tr><td class="tcl"><a href="#ar53">FULLER, MARGARET</a></td> <td class="tcl"><a href="#ar111">FUX, JOHANN JOSEPH</a></td></tr> +<tr><td class="tcl"><a href="#ar54">FULLER, MELVILLE WESTON</a></td> <td class="tcl"><a href="#ar112">FUZE</a></td></tr> +<tr><td class="tcl"><a href="#ar55">FULLER, THOMAS</a></td> <td class="tcl"><a href="#ar113">FYNE, LOCH</a></td></tr> +<tr><td class="tcl"><a href="#ar56">FULLER, WILLIAM</a></td> <td class="tcl"><a href="#ar114">FYRD</a></td></tr> +<tr><td class="tcl"><a href="#ar57">FULLER'S EARTH</a></td> <td class="tcl"><a href="#ar115">FYT, JOHANNES</a></td></tr> +<tr><td class="tcl"><a href="#ar58">FULLERTON, LADY GEORGIANA CHARLOTTE</a></td> <td class="tcl"><a href="#ar116">FYZABAD</a></td></tr> +</table> + +<hr class="art" /> +<p><span class="pagenum"><a name="page251" id="page251"></a>251</span></p> +<p><span class="bold">FROST<a name="ar1" id="ar1"></a></span> (a common Teutonic word, cf. Dutch, <i>vorst</i>, Ger. <i>Frost</i>, +from the common Teutonic verb meaning “to freeze,” Dutch, +<i>vriezen</i>, Ger. <i>frieren</i>; the Indo-European root is seen in Lat. +<i>pruina</i>, hoar-frost, cf. <i>prurire</i>, to itch, burn, <i>pruna</i>, burning coal, +Sansk. <i>plush</i>, 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° F., the freezing-point of water; if the temperature be +between 28° and 32° it is a “light frost,” if below 28° 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°, and this may occur even +when the temperature of the air a few feet above the ground is +12°-16° 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° 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.</p> + + +<hr class="art" /> +<p><span class="bold">FROSTBITE,<a name="ar2" id="ar2"></a></span> a form of <span class="sc">mortification</span> (<i>q.v.</i>), 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.</p> + + +<hr class="art" /> +<p><span class="bold">FROSTBURG,<a name="ar3" id="ar3"></a></span> 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.</p> + + +<hr class="art" /> +<p><span class="bold">FROTHINGHAM, OCTAVIUS BROOKS<a name="ar4" id="ar4"></a></span> (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 +<i>Anti-Supernaturalism in the Pulpit</i>. 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 <i>Tribune</i>. 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 +<span class="pagenum"><a name="page252" id="page252"></a>252</span> +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.</p> + +<div class="condensed"> +<p>His principal published works are: <i>Stories from the Life of the +Teacher</i> (1863), <i>A Child’s Book of Religion</i> (1866), and other works +of religious teaching for children; several volumes of sermons; +<i>Beliefs of Unbelievers</i> (1876), <i>The Cradle of the Christ: a Study in +Primitive Christianity</i> (1877), <i>The Spirit of New Faith</i> (1877), +<i>The Rising and the Setting Faith</i> (1878), and other expositions of +the “new faith” he preached; <i>Life of Theodore Parker</i> (1874), +<i>Transcendentalism in New England</i> (1876), which is largely biographical, +<i>Gerrit Smith, a Biography</i> (1878), <i>George Ripley</i> (1882), +in the “American Men of Letters” series, <i>Memoir of William +Henry Channing</i> (1886), <i>Boston Unitarianism, 1820-1850</i> (1890), +really a biography of his father; and <i>Recollections and Impressions, +1822-1890</i> (1891).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FROUDE, JAMES ANTHONY<a name="ar5" id="ar5"></a></span> (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 <i>Lives of the +English Saints</i>. 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 <i>Shadows of the Clouds</i>, 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 <i>Nemesis of Faith</i>, an heretical +and unpleasant book, of which the earlier part seems to be +autobiographical.</p> + +<p>On the demand of the college he resigned his fellowship at +Oxford, and mainly at least supported himself by writing, +contributing largely to <i>Fraser’s Magazine</i> and the <i>Westminster +Review</i>. The excellence of his style was soon generally recognized. +The first two volumes of his <i>History of England +from the Fall of Wolsey to the Defeat of the Spanish Armada</i> +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.</p> + +<p>Froude’s work is often marred by prejudice and incorrect +statements. He wrote with a purpose. The keynote of his +<i>History</i> 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 +<i>English in Ireland</i> (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 <i>Divorce of +Catherine of Aragon</i> (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 <i>History</i> 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 <i>Life of Caesar</i> (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, <i>The +English in the West Indies</i> (1888) shows that he made little effort +to master his subject, and <i>Oceana</i> (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 <i>Saturday Review</i>. 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 <i>Contemporary Review</i> (1878-1879) on Froude’s +<i>Short Study</i> of Thomas Becket.</p> + +<p>Notwithstanding its defects, Froude’s <i>History</i> 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 <i>Short Studies on Great Subjects</i> (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 +<span class="pagenum"><a name="page253" id="page253"></a>253</span> +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.</p> + +<p>The merits of his work met with full recognition. Each +instalment of his <i>History</i>, 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.</p> + +<p>His long life was full of literary work. Besides his labours as +an author, he was for fourteen years editor of <i>Fraser’s Magazine</i>. +He was one of Carlyle’s literary executors, and brought some +sharp criticism upon himself by publishing Carlyle’s <i>Reminiscences</i> +and the <i>Memorials of Jane Welsh Carlyle</i>, 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 <i>Life of Carlyle</i>. His earlier study of +Irish history afforded him suggestions for a historical novel +entitled <i>The Two Chiefs of Dunboy</i> (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 <span class="sc"><a href="#artlinks">South +Africa</a></span>: <i>History</i>). 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.</p> + +<div class="condensed"> +<p>Froude’s <i>Life</i>, by Herbert Paul, was published in 1905.</p> +</div> +<div class="author">(W. Hu.)</div> + + +<hr class="art" /> +<p><span class="bold">FRUCTOSE<a name="ar6" id="ar6"></a></span>, <span class="sc">Laevulose</span>, or <span class="sc">Fruit-Sugar</span>, a carbohydrate +of the formula C<span class="su">6</span>H<span class="su">12</span>O<span class="su">6</span>. It is closely related to ordinary <i>d</i>-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° 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.</p> + +<p>Chemically, fructose is an oxyketone or ketose, its structural +formula being CH<span class="su">2</span>OH·(CH·OH)<span class="su">3</span>·CO·CH<span class="su">2</span>OH; this result followed +from its conversion by H. Kiliani into methylbutylacetic +acid. The form described above is <i>laevo</i>-rotatory, but it is +termed <i>d</i>-fructose, since it is related to <i>d</i>-glucose. Solutions +exhibit mutarotation, fresh solutions having a specific rotation +of −104.0°, which gradually diminishes to −92°. It was +synthesized by Emil Fischer, who found the synthetic sugar +which he named α-acrose to be (<i>d</i> + <i>l</i>)-fructose, and by splitting +this mixture he obtained both the d and <i>l</i> 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 <span class="sc"><a href="#artlinks">Sugar</a></span>.</p> + + +<hr class="art" /> +<p><span class="bold">FRUGONI, CARLO INNOCENZIO MARIA<a name="ar7" id="ar7"></a></span> (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.</p> + +<div class="condensed"> +<p>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.</p> +</div> + +<p><span class="pagenum"><a name="page254" id="page254"></a>254</span></p> + + +<hr class="art" /> +<p><span class="bold">FRUIT<a name="ar8" id="ar8"></a></span> (through the French from the Lat. <i>fructus</i>; <i>frui</i>, 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.</p> + +<p>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 <i>pericarp</i>; 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 <i>pseudocarps</i>. 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.</p> + +<div class="condensed"> +<p>The fruit, like the ovary, may be formed of a single carpel or of +several. It may have one cell or cavity, being <i>unilocular</i>; or many, +<i>multilocular</i>, &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, <i>g</i>, +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 <i>Cathartocarpus Fistula</i> 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 (<i>Linum</i>) by the +folding inwards of the back of the carpels a five-celled ovary becomes +a ten-celled fruit. In <i>Astragalus</i> the folding inwards of the dorsal +suture converts a one-celled ovary into a two-celled fruit; and in +<i>Oxytropis</i> 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.</p> + +<div class="center pt2"><img style="width:473px; height:497px; vertical-align: middle;" src="images/img254.jpg" alt="" /></div> + +<p><span class="sc">Fig.</span> 1.—Samara or winged fruit of Ash (<i>Fraxinus</i>). 1, Entire, +with its wing <i>a</i>; 2, lower portion cut transversely, to show that it +consists of two cells; one of which, <i>l</i>, is abortive, and is reduced to +a very small cavity, while the other is much enlarged and filled +with a seed <i>g</i>.</p> + +<p><span class="sc">Fig.</span> 2.—Fruit of the Strawberry (<i>Fragaria vesca</i>), consisting of +an enlarged succulent receptacle, bearing on its surface the small +dry seed-like fruits (achenes). (After Duchartre.)</p> + +<p class="f80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p><span class="sc">Fig.</span> 3.—Fruit of the Rose cut vertically. <i>s’</i>, Fleshy hollowed +receptacle; <i>s</i>, persistent sepals; <i>fr</i>, ripe carpels; <i>e</i>, stamens, +withered.</p> + +<p><span class="sc">Fig.</span> 4.—Peduncle of Fig (<i>Ficus Carica</i>), ending in a hollow +receptacle enclosing numerous male and female flowers.</p> + +<p><span class="sc">Fig.</span> 5.—Fruit of Cherry (<i>Prunus Cerasus</i>) in longitudinal section. +<i>ep</i>, Epicarp; <i>m</i>, mesocarp; <i>en</i>, endocarp.</p> + +<p class="f80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p class="pt2">The pericarp consists usually of three layers, the external, or +<i>epicarp</i> (fig. 5, <i>ep</i>); the middle, or <i>mesocarp</i>, <i>m</i>; and the internal, +or <i>endocarp</i>, <i>en</i>. 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 <i>sarcocarp</i>; while the endocarp, +hardened by the production of woody cells, forms the <i>stone</i> or +<i>putamen</i> 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 (<i>Colutea arborescens</i>) 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 <i>apiculate</i>; +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.</p> + +<p><span class="pagenum"><a name="page255" id="page255"></a>255</span></p> + +<table class="flt" style="float: right; width: 250px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:199px; height:177px" src="images/img255a.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig.</span> 6.—Seed-vessel or capsule +of Campion, opening by ten +teeth at the apex. The calyx <i>c</i> +is seen surrounding the seed-vessel.</td></tr> + +<tr><td class="caption1"><span class="sc">Fig.</span> 7.—Capsule of Poppy, +opening by pores <i>p</i>, under the +radiating peltate stigma <i>s</i>.</td></tr></table> + +<p>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 <i>dehiscent</i>. But some fruits are <i>indehiscent</i>, falling to +<span class="sidenote">Dehiscence of fruits.</span> +the ground entire, and the seeds eventually reaching the +soil by their decay. By dehiscence the pericarp becomes divided +into different pieces, or <i>valves</i>, 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 <i>Campanula</i>. 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.</p> + +<div class="center pt2" style="clear: both;"><img style="width:515px; height:417px; vertical-align: middle;" src="images/img255b.jpg" alt="" /></div> + +<table class="flt" style="float: left; width: 200px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:149px; height:252px" src="images/img255c.jpg" alt="" /></td></tr></table> + +<p><span class="sc">Fig.</span> 8.—Dry dehiscent fruit. The pod +(legume) of the Pea; <i>r</i>, the dorsal suture; +<i>b</i>, the ventral; <i>c</i>, calyx; <i>s</i>, seeds.</p> + +<p class="f80">From Vines’ <i>Students’ Text-Book of Botany</i>, by permission +of Swan Sonnenschein & Co.</p> + +<p><span class="sc">Fig.</span> 9.—(1) Fruit or capsule of Meadow +Saffron (<i>Colchicum autumnale</i>), dehiscing along +the septa (septicidally); (2) same cut across, +showing the three chambers with the seeds +attached along the middle line (axile placentation).</p> + +<p><span class="sc">Fig.</span> 10.—Diagram to illustrate the septicidal +dehiscence in a pentalocular capsule. +The loculaments <i>l</i> correspond to the number of the carpels, which +separate by splitting through the septa, <i>s</i>.</p> + +<p><span class="sc">Fig.</span> 11.—The seed vessel (capsule) of the Flower-de-Luce (<i>Iris</i>), +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.</p> + +<p><span class="sc">Fig.</span> 12.—Diagram to illustrate loculicidal dehiscence. The loculaments +<i>l</i>, split at the back, and the valves separate, bearing the +septa <i>s</i> on their centres.</p> + +<p><span class="sc">Fig.</span> 13.—Diagram to illustrate septifragal dehiscence, in which +the dehiscence takes place through the back of the loculaments <i>l</i>, +and the valves separate from the septa <i>s</i>, which are left attached to +the placentas in the centre.</p> + +<p class="pt2">Dehiscent fruits, when composed +of single carpels, may open +by the ventral suture only, as in the paeony, hellebore, <i>Aquilegia</i> (fig. +28) and <i>Caltha</i>; by the dorsal suture only, as in magnolias and some +<i>Proteaceae</i>, or by both together, as in the pea (fig. 8) and bean; +in these cases the dehiscence is <i>sutural</i>. 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, <i>Colchicum</i>, &c.; this dehiscence, in consequence of +taking place through the septum, is called <i>septicidal</i> (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 <i>loculicidal</i> 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 <i>septifragal</i> (fig. 13), and where, as in <i>Linum +catharticum</i> and <i>Calluna</i>, the splitting takes place first of all in a +septicidal manner, the fruit is described as <i>septicidally septifragal</i>; +while in other cases, as in thorn apple (<i>Datura Stramonium</i>), where +the splitting is at first loculicidal, the dehiscence is <i>loculicidally +septifragal</i>. In all those forms the separation of the valves takes +place either from above downwards or from below upwards. In +<i>Saxifraga</i> 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 <i>replum</i>, 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 <i>circumscissile</i>, when the upper part of the +united carpels falls off in the form of a lid or operculum, as in <i>Anagallis</i> +and in henbane (<i>Hyoscyamus</i>) (fig. 16).</p> + +<div class="center pt2"><img style="width:513px; height:383px; vertical-align: middle;" src="images/img255d.jpg" alt="" /></div> + +<p><span class="sc">Fig. 14.</span>—Siliqua or seed-vessel of Wallflower (<i>Cheiranthus Cheiri</i>), +opening by two valves, which separate from the base upwards, +leaving the seeds attached to the dissepiment which is supported by +the replum.</p> + +<p class="f80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p><span class="sc">Fig. 15.</span>—Capsule of an Orchid (<i>Xylobium</i>). <i>v</i>, valve.</p> + +<p><span class="sc">Fig. 16.</span>—Seed-vessel of <i>Anagallisarvensis</i>, opening by circumscissile +dehiscence.</p> + +<p class="f80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p><span class="sc">Fig. 17.</span>—Lomentum of <i>Hedysarum</i> which, when ripe, separates +transversely into single-seeded portions or mericarps.</p> + +<p><span class="sc">Fig. 18.</span>—Fruit of <i>Geranium pratense</i>, after splitting.</p> + +<p class="pt2">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 <i>carpophore</i> or <i>podocarp</i>, which splits +into two (fig. 25) and suspends them; hence the fruit is termed a +<i>cremocarp</i>, which divides into two <i>mericarps</i>. The general term +<i>schizocarp</i> is applied to all dry fruits, which break up into two or +more one-seeded indehiscent mericarps, as in <i>Hedysarum</i> (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 <i>cocci</i>. They are well +<span class="pagenum"><a name="page256" id="page256"></a>256</span> +seen in the order Euphorbiaceae, where there are usually three such +carpels, and the fruit is termed tricoccus. In many of them, as +<i>Hura crepitans</i>, the cocci separate with great force and elasticity. +In many leguminous plants, such as <i>Ornithopus</i>, <i>Hedysarum</i> (fig. 17), +<i>Entada</i>, <i>Coronilla</i> and the gum-arabic plant (<i>Acacia arabica</i>), 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.</p> + +<p>Fruits are formed by one flower, or are the product of several +flowers combined. In the former case they are either <i>apocarpous</i>, +of one mature carpel or of several separate free carpels; or <i>syncarpous</i>, +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 <i>anthocarpous</i>. 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.</p> + +<div class="center pt2"><img style="width:464px; height:366px; vertical-align: middle;" src="images/img256.jpg" alt="" /></div> + +<p class="f80">From Vines’ <i>Students’ Text-Book of Botany</i>, by +permission of Swan Sonnenschein & Co.</p> + +<p><span class="sc">Fig. 19.</span>—Dry one-seeded fruit of dock (<i>Rumex</i>) cut vertically. +ov, Pericarp formed from ovary wall; <i>s</i>, seed; <i>e</i>, endosperm; <i>pl</i>, +embryo with radicle pointing upwards and cotyledons downwards—enlarged.</p> + +<p><span class="sc">Fig. 20.</span>—Achene of <i>Ranunculus arvensis</i> in longitudinal section; +<i>e</i>, endosperm; <i>pl</i>, embryo. (After Baillon, enlarged.)</p> + +<p class="f80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p><span class="sc">Fig. 21.</span>—Fruit of Common Sycamore (<i>Acer Pseudoplatanus</i>), +dividing into two mericarps <i>m</i>; <i>s</i>, pedicel; <i>fl</i>, wings (nat. size).</p> + +<p class="pt2">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 +<span class="sidenote">Dispersal of fruit or seed.</span> +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; +<i>e.g.</i> in gorse, by the spiral curving of the valves; in <i>Impatiens</i>, 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; <i>e.g.</i> 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 <i>Asclepias</i> (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 (<i>Anastatica hierocuntica</i>), 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], <i>Campanula</i>, +&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 (<i>Galium Aparine</i>), +a common hedge-row plant, <i>Ranunculus arvensis</i> (fig. 20), carrot, +<i>Geum</i>, &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.</p> + +<div class="center pt2"><img style="width:469px; height:373px; vertical-align: middle;" src="images/img256a.jpg" alt="" /></div> + +<p><span class="sc">Fig. 22.</span>—Vertical section of a grain of wheat, showing embryo +below at the base of the endosperm <i>e</i>; <i>s</i>, scutellum separating +embryo from endosperm; <i>f.l</i>, foliage leaf; <i>p.s</i>, sheath of plumule; +<i>p.r</i>, primary root; <i>s.p.r</i>, sheath of primary root.</p> + +<p><span class="sc">Fig. 23.</span>—Fruit of Comfrey (<i>Symphytum</i>) surrounded by persistent +calyx, <i>c</i>. The style s appears to arise from the base of the carpels, +enlarged.</p> + +<p><span class="sc">Fig. 24.</span>—Ovary of <i>Foeniculum officinale</i> with pendulous ovules, in +longitudinal section. (After Berg and Schmidt, magnified.)</p> + +<p class="f80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p><span class="sc">Fig. 25.</span>—Fruit of <i>Carum Carui</i>. A, Ovary of the flower; B, ripe +fruit. The two carpels have separated so as to form two mericarps +(<i>m</i>). Part of the septum constitutes the carpophore (<i>a</i>). <i>p</i>, Top of +flower-stalk; <i>d</i>, disk on top of ovary; <i>n</i>, stigma.</p> + +<p class="f80">From Vines’ <i>Students’ Text-Book of Botany</i>, by permission of Swan Sonnenschein +& Co.</p> + +<p class="pt2">Simple fruits have either a <i>dry</i> or <i>succulent</i> pericarp. The <i>achene</i> +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 +<span class="sidenote">Forms of fruit.</span> +cashew, where it is supported on a fleshy peduncle; or +<i>aggregate</i>, as in <i>Ranunculus</i> (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 <i>Dorstenia</i> 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 <i>Clematis</i>. In Compositae, the fruit +is an inferior achene (<i>cypsela</i>), to which the pappus (modified calyx) +remains adherent. Such is also the nature of the fruit in +Dipsacaceae (<i>e.g.</i> scabious). When the pericarp is thin, and +appears like a bladder surrounding the seed, the achene is termed +a <i>utricle</i>, as in Amarantaceae. When the pericarp is extended in +the form of a winged appendage, a <i>samara</i> or <i>samaroid achene</i> 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 <i>Fraxinus</i> (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 <i>caryopsis</i> is given. The one-seeded portions (mericarps) of +schizocarps often take the form of achenes, <i>e.g.</i> the mericarps of the +<span class="pagenum"><a name="page257" id="page257"></a>257</span> +mallows or of umbellifers (figs. 24, 25). In Labiatae and Boraginaceae +(<i>e.g.</i> 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.</p> + +<table class="flt" style="float: right; width: 330px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:274px; height:301px" src="images/img257a.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Strasburger’s <i>Lehrbuch der Botanik</i>, +by permission of Gustav Fischer.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 26.</span>—Cupule of <i>Quercus +Aegilops</i>. <i>cp</i>, cupule; <i>gl</i>, fruit. +(After Duchartre.)</td></tr></table> + +<p>The <i>nut</i> or <i>glans</i> 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 <i>husk</i>, and by the acorn, in +which the bracts and receptacle +form a <i>cupula</i> or <i>cup</i> (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.</p> + +<p>The <i>drupe</i> 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 <i>drupels</i> are aggregated so as to +constitute an <i>etaerio</i>.</p> + +<p>The <i>follicle</i> 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 <i>Magnolia</i> and <i>Banksia</i>. Occasionally, follicles dehisce by the +dorsal suture, as in <i>Magnolia grandiflora</i> and <i>Banksia</i>.</p> + +<div class="center pt2" style="clear: both;"><img style="width:431px; height:302px; vertical-align: middle;" src="images/img257b.jpg" alt="" /></div> + +<p><span class="sc">Fig. 27.</span>—Fruit of Columbine (<i>Aquilegia</i>), formed of five follicles.</p> + +<p><span class="sc">Fig. 28.</span>—Single follicle, showing dehiscence by the ventral suture.</p> + +<p><span class="sc">Fig. 29.</span>—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.</p> + +<p><span class="sc">Fig. 30.</span>—Section of the fruit of the Apple (<i>Pyrus Malus</i>), or pome, +consisting of a fleshy covering formed by the floral receptacle and +the true fruit or core with five cavities with seeds.</p> + +<p class="pt2">The <i>legume</i> or <i>pod</i> 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 <i>Arachis</i>, <i>Cathartocarpus Fistula</i> 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 +<i>lomentum</i> or <i>lomentaceous legume</i> of <i>Hedysarum</i> (fig. 17), <i>Coronilla</i>, +<i>Ornithopus</i>, <i>Entada</i> and of some Acacias. In <i>Medicago</i> the legume +is twisted like a snail, and in <i>Caesalpinia coriaria</i>, or Divi-divi, it is +vermiform or curved like a worm. Sometimes the number of seeds +is reduced, as in <i>Erythrina monosperma</i> and <i>Geoffroya superba</i>, +which are one-seeded, and in <i>Pterocarpus</i> and <i>Dalbergia</i>, which are +two-seeded.</p> + +<table class="flt" style="float: right; width: 270px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:223px; height:229px" src="images/img257c.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 31.</span>—Transverse section +of the fruit of the Melon +(<i>Cucumis Melo</i>), 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.</td></tr></table> + +<p>The <i>berry</i> (<i>bacca</i>) is a term applied generally to all fruits with +seeds immersed in pulp, and includes fruits of very various origin. +In <i>Actaea</i> (baneberry) or <i>Berberis</i> +(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.</p> + +<p>The <i>pepo</i>, 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.</p> + +<p>The <i>hesperidium</i> 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.</p> + +<p>The <i>pome</i> (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.</p> + +<p>The name <i>capsule</i> 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 +<i>Colchicum</i> (fig. 9), lily and iris (fig. 11). The <i>porose capsule</i> is seen +in the poppy (fig. 7), <i>Antirrhinum</i> and <i>Campanula</i>. In <i>Campanula</i> +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 <i>pyxidium</i>, as in pimpernel (<i>Anagallis +arvensis</i>) (fig. 16), henbane and monkey-pot (<i>Lecythis</i>). The capsule +assumes a screw-like form in <i>Helicteres</i>, and a star-like form in star-anise +(<i>Illicium anisatum</i>). 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 +(<i>Hura crepitans</i>) 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).</p> + +<p>The <i>siliqua</i> 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 <i>siliqua</i> (fig. 14); when broad +and short, <i>silicula</i> (fig. 33). It occurs in cruciferous plants, as wallflower, +cabbage and cress. In <i>Glaucium</i> and <i>Eschscholtzia</i> (Papaveraceae) +the dissepiment is of a spongy nature. It may become +transversely constricted (<i>lomentaceous</i>), as in radish (<i>Raphanus</i>) +and sea-kale, and it may be reduced, as in woad (<i>Isatis</i>), to a +one-seeded +condition.</p> + +<p>It sometimes happens that the ovaries of two flowers unite so as +to form a double fruit (<i>syncarp</i>). 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 <i>multiple</i> or <i>confluent</i>. +The term <i>anthocarpous</i> has also been applied as indicating that the +floral envelopes as well as the carpels are concerned in the formation +of the fruit.</p> + +<p>The <i>sorosis</i> is a succulent multiple fruit formed by the confluence +<span class="pagenum"><a name="page258" id="page258"></a>258</span> +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.</p> + +<p>The <i>syconus</i> 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 <i>Dorstenia</i> the axis is less deeply hollowed, and of a +harder texture, the fruit exhibiting often very anomalous forms.</p> + +<p>The <i>strobilus</i>, or <i>cone</i>, 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 <i>galbulus</i>. 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.</p> + +<div class="center pt2"><img style="width:522px; height:349px; vertical-align: middle;" src="images/img258a.jpg" alt="" /></div> + +<p><span class="sc">Fig. 32.</span>—Honesty (<i>Lunaria biennis</i>), showing the septum after +the carpels have fallen away.</p> + +<p class="f80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p><span class="sc">Fig. 33.</span>—Silicula or pouch of shepherd’s purse (<i>Capsella</i>), opening +by two folded valves, which separate from above downwards. The +partition is narrow, hence the silicula is angustiseptal.</p> + +<p>From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</p> + +<p><span class="sc">Fig. 34.</span>—Fruit of the pine-apple (<i>Ananassa sativa</i>), 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.</p> + +<p class="pt2">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 <i>Citrus</i>, 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.</p> + + +<p class="pt2 center"><i>Arrangement of Fruits.</i></p> + +<p>A. True fruits—developed from the ovary alone.</p> +<div class="list"> + <p>1. Pericarp not fleshy or fibrous.</p> +</div> +<div class="list1"> + <p>i. Indehiscent—not opening to allow the escape of the + seeds—generally one-seeded. Achene; caryopsis; + cypsela; nut; schizocarp.</p> + <p>ii. Dehiscent—the pericarp splits to allow the escape + of the seeds—generally many-seeded. Follicle; + legume; siliqua; capsule.</p> +</div> +<div class="list"> + <p>2. Pericarp generally differentiated into distinct layers, one + of which is succulent or fibrous. Drupe; berry.</p> +</div> +<p>B. Pseudocarps—the development extends beyond the ovary. +Pome; syconus; sorosis.</p> + +<p class="pt1"><i>The Seed.</i>—The <i>seed</i> 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 <i>angiospermous</i>; while in <i>gymnospermous</i> 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 <i>Cuphea</i>, 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 (<i>q.v.</i>) there is a development of cellular tissue, the +endosperm, more or less filling the embryo-sac. In Gymnosperms +(<i>q.v.</i>) 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 <i>tegmen</i> can be distinguished +from a tougher outer coat or <i>testa</i>; 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 <span class="sc"><a href="#artlinks">Angiosperms</a></span>. +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 <i>Abrus precatorius</i> and +<i>Adenanthera pavonina</i> 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 <i>Asclepias</i> (fig. 36) it has hairs attached to +it; and in mahogany, <i>Bignonia</i>, and the pines and firs it is expanded +in the form of wing-like appendages (fig. 37). In <i>Collomia</i>, <i>Acanthodium</i>, +<i>Cobaea scandens</i> and other seeds, it contains spiral cells, from +which, when moistened with water, the fibres uncoil in a beautiful +manner; and in flax (<i>Linum</i>) 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 <i>Fraxinus</i> (fig. 1) and <i>Acer</i> (fig. 21).</p> + +<div class="center pt2"><img style="width:408px; height:199px; vertical-align: middle;" src="images/img258b.jpg" alt="" /></div> + +<p><span class="sc">Fig. 35.</span>—Seed of Pea (<i>Pisum</i>) with one cotyledon removed. <i>c</i>, +Remaining cotyledon; <i>ch</i>, chalaza-point at which the nourishing +vessels enter; <i>e</i>, tegmen or inner coat; <i>f</i>, funicle or stalk; <i>g</i>, +plumule of embryo; <i>m</i>, micropyle; <i>pl</i>, placenta; <i>r</i>, radicle of +embryo; <i>t</i>, tigellum or stalk between root and plumule; <i>te</i>, testa.</p> + +<p><span class="sc">Fig. 36.</span>—Seed of <i>Asclepias</i>, with a cluster of hairs arising from +the edges of the micropyle.</p> + +<p class="pt2">Sometimes there is an additional covering to the seed, formed +after fertilization, to which the name <i>arillus</i> 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 <i>mace</i>. In such instances it has been called +an <i>arillode</i> (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 <i>strophioles</i>, or <i>caruncles</i>, has been given, the seeds being +strophiolate or carunculate. These tumours may occur near the +base of the seed, as in <i>Polygala</i>, or at the apex, as in Castor-oil +plant (<i>Ricinus</i>); or they may occur in the course of the raphe, as in +blood-root (<i>Sanguinaria</i>) and <i>Asarabacca</i>. 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 +<span class="pagenum"><a name="page259" id="page259"></a>259</span> +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, <i>m</i>) of the seed may be recognizable by +the naked eye, as in the pea and bean tribe, <i>Iris</i>, &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 <i>embryotega</i>. The chalaza (fig. +38, <i>ch</i>) 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).</p> + +<p>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 (<i>compressed</i>), or from above downwards (<i>depressed</i>). They +may be round, oval, triangular, polygonal, rolled up like a snail, as in +<i>Physostemon</i>, or coiled up like a snake, as in <i>Ophiocaryon paradoxum</i>.</p> + +<div class="center pt2"><img style="width:507px; height:206px; vertical-align: middle;" src="images/img259a.jpg" alt="" /></div> + +<p><span class="sc">Fig. 37.</span>—Seed of Pine (<i>Pinus</i>), with a membranous appendage +<i>w</i> to the testa, called a wing.</p> + +<p><span class="sc">Fig. 38.</span>—Young anatropal seed of the white Water-lily (<i>Nymphaea +alba</i>), cut vertically. It is attached to the placenta by the funicle f, +cellular prolongations from which form an aril <i>a a</i>. The vessels of +the cord are prolonged to the base of the nucellus n by means of +the raphe <i>r</i>. The base of the nucellus is indicated by the chalaza ch, +while the apex is at the micropyle <i>m</i>. The covering of the seed is +marked <i>i. n</i> is the nucellus or perisperm, enclosing the embryo-sac es, +<span class="correction" title="amended from is">in</span> which the endosperm is formed. The embryo <i>e</i>, with its suspensor, +is contained in the sac, the radicle pointing to the micropyle <i>m</i>.</p> + +<p><span class="sc">Fig. 39.</span>—Arillode <i>a</i>, or false aril, of the Spindle-tree (<i>Euonymus</i>), +arising from the micropyle <i>f</i>.</p> + +<p><span class="sc">Fig. 40.</span>—Anatropal seed of the Orange (<i>Citrus Aurantium</i>) +opened to show the chalaza <i>c</i>, which forms a brown spot at one end.</p> + +<p><span class="sc">Fig. 41.</span>—Entire anatropal seed of the Orange (<i>Citrus Aurantium</i>), +with its rugose or wrinkled testa, and the raphe <i>r</i> ramifying in the +thickness of the testa on one side.</p> + +<p class="pt2">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 <span class="correction" title="amended from occupied">occupies</span> the whole cavity of +the embryo-sac, or is formed only at certain portions of it, at the +apex, as in <i>Rhinanthus</i>, at the base, as in <i>Vaccinium</i>, or in the middle, +as in <i>Veronica</i>. 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 <i>perisperm</i>. 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 +<i>albumen</i> to the outer nutrient mass or perisperm, and designated +the endosperm as <i>vitellus</i>. 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 <i>exalbuminous</i>, +as in Compositae, Cruciferae and most Leguminosae (<i>e.g.</i> pea, fig. 35). +When either endosperm or perisperm or both are present the seed +is said to be <i>albuminous</i>.</p> + +<table class="flt" style="float: right; width: 275px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:223px; height:132px" src="images/img259b.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 42.</span>—The dicotyledonous +embryo of the Pea laid open. +<i>c</i>, <i>c</i>, The two fleshy cotyledons, +or seed-lobes, which remain under +ground when the plant sprouts; +<i>r</i>, the radicular extremity of the +axis whence the root arises; <i>t</i>, +the axis (hypocotyl) bearing the +young stalk and leaves <i>g</i> (plumule), +which lie in a depression of +the cotyledons <i>f</i>.</td></tr></table> + +<p>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 <i>Croton</i>, <i>Ricinus</i> 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 (<i>Phytelephas</i>). 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 <i>ruminated</i>. 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.</p> + +<p>The embryo consists of an axis bearing the <i>cotyledons</i> (fig. 42, <i>c</i>), +or the first leaves of the plant. To that part of this axis immediately +beneath the cotyledons the terms <i>hypocotyl</i>, <i>caulicle</i> or <i>tigellum</i> (<i>t</i>) +have been applied, and continuous backwards with it is the young +root or <i>radicle</i> (<i>r</i>), the descending axis, their point of union being +the collar or neck. The terminal growing bud of the axis is called +the <i>plumule</i> or <i>gemmule</i> (<i>g</i>), 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 <i>Cuscuta</i>, the embryo appears as an elongated +axis without divisions; and in <i>Caryocar</i> 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 <i>Orobanche</i>, <i>Monotropa</i>, &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 (<i>e.g.</i> +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.</p> + +<p>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 <i>antitropal</i>, +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 +<i>homotropal</i> (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 <i>amphitropal</i> (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.</p> + +<p><span class="pagenum"><a name="page260" id="page260"></a>260</span></p> + +<p>Plants in which there are two cotyledons produced in the embryo +are <i>dicotyledonous</i>. 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 <i>Carapa +guianensis</i> 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 <i>pseudomonocotyledonous</i>. +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 <i>Ricinus</i>, +<i>Jatropha</i>, <i>Euonymus</i>, &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.</p> + +<div class="center pt2"><img style="width:420px; height:297px; vertical-align: middle;" src="images/img260a.jpg" alt="" /></div> + +<p><span class="sc">Fig. 43.</span>—Seed of Pansy (<i>Viola tricolor</i>) cut vertically. The embryo +<i>pl</i> 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 <i>ch</i>, while the radicle, or the +other extremity of the embryo, points to the micropyle, close to the +hilum <i>h</i>. The hilum or base of the seed, and the chalaza or base of +the nucellus are united by means of the raphe <i>r</i>.</p> + +<p><span class="sc">Fig. 44.</span>—Seed of the Red Campion (<i>Lychnis</i>), 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 (<i>amphitropal</i>).</p> + +<p><span class="sc">Fig. 45.</span>—Mature dicotyledonous embryo of the Almond, with +one of the cotyledons removed. <i>r</i>, Radicle; <i>t</i>, young stem or +caulicle; <i>c</i>, one of the cotyledons left; <i>i</i>, line of insertion of the +cotyledon which has been removed; <i>g</i>, plumule.</p> + +<p><span class="sc">Fig. 46.</span>—Exalbuminous seed of Wallflower (Cheiranthus) cut +vertically. The radicle <i>r</i> is folded on the edges of the cotyledons <i>c</i> +which are accumbent.</p> + +<p><span class="sc">Fig. 47.</span>—Transverse section of the seed of the Wallflower (<i>Cheiranthus</i>), +showing the radicle <i>r</i> folded on the edges of the accumbent +cotyledons <i>c</i>.</p> + +<p><span class="sc">Fig. 48.</span>—Transverse section of the seed of the Dame’s Violet +(<i>Hesperis</i>). The radicle <i>r</i> is folded on the back of the cotyledons <i>c</i>, +which are said to be incumbent.</p> + +<p class="pt2">Cotyledons are usually entire and sessile. But they occasionally +become lobed, as in the walnut and the lime; or petiolate, as in +<i>Geranium molle</i>; 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 <i>Bunias</i>, 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 <i>accumbent</i>. In others, as <i>Hesperis</i>, +the cotyledons (fig. 48) are applied to each other by their faces, +and the radicle, <i>r</i>, is folded on their back, so as to be dorsal, and +the cotyledons are <i>incumbent</i>. Again, the cotyledons are <i>conduplicate</i> +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.</p> + +<p>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 <i>polycotyledonous</i>. In species of <i>Streptocarpus</i> the +cotyledons are permanent, and act the part of leaves. One of them +is frequently largely developed, while the other is small or abortive.</p> + +<div class="center pt2"><img style="width:484px; height:242px; vertical-align: middle;" src="images/img260b.jpg" alt="" /></div> + +<p><span class="sc">Fig. 49.</span>—Polycotylodonous embryo of the Pine (<i>Pinus</i>) beginning +to sprout. <i>t</i>, Hypocotyl; <i>r</i>, radicle. The cotyledons <i>c</i> are numerous. +Within the cotyledons the primordial leaves are seen, constituting +the plumule or first bud of the plant.</p> + +<p><span class="sc">Fig. 50.</span>—Embryo of a species of Arrow-grass (<i>Triglochin</i>), showing +a uniform conical mass, with a slit <i>s</i> near the lower part. The +cotyledon <i>c</i> envelops the young bud, which protrudes at the slit +during germination. The radicle is developed from the lower part +of the axis <i>r</i>.</p> + +<p><span class="sc">Fig. 51.</span>—Grain of wheat (<i>Triticum</i>) germinating, showing (<i>b</i>) +the cotyledon and (<i>c</i>) the rootlets surrounded by their sheaths +(<i>coleorrhizae</i>).</p> + +<p><span class="sc">Fig. 52.</span>—Embryo of <i>Caryocar</i>. <i>t</i>, Thick hypocotyl, forming nearly +the whole mass, becoming narrowed and curved at its extremity, +and applied to the groove <i>s</i>. In the figure this narrowed portion is +slightly separated from the groove; <i>c</i>, two rudimentary cotyledons.</p> + +<p class="pt2">In those plants in which there is only a single cotyledon in the +embryo, hence called <i>monocotyledonous</i>, 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 <i>scutellum</i>. 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 <span class="sc"><a href="#artlinks">Grasses</a></span>). +In many aquatic monocotyledons (<i>e.g.</i> <i>Potamogeton</i>, <i>Ruppia</i> 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 <i>macropodous</i> embryos. A similar +case is that of <i>Caryocar</i> 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 <i>epiblast</i>. 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.</p> +</div> +<div class="author">(A. B. R.)</div> + + +<hr class="art" /> +<p><span class="bold">FRUIT AND FLOWER FARMING.<a name="ar9" id="ar9"></a></span> 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 +<span class="sc"><a href="#artlinks">Horticulture</a></span>.</p> + +<p class="pt2 center sc">Great Britain</p> + +<p class="pt2 center"><span class="sc">Table</span> I.—<i>Extent of Orchards in Great Britain in each Year, +1887 to 1901.</i></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tcc rb lb tb bb">Year.</td> <td class="tcc rb2 lb tb bb">Acres.</td> <td class="tcc rb lb tb bb">Year.</td> <td class="tcc rb2 lb tb bb">Acres.</td> <td class="tcc rb lb tb bb">Year.</td> <td class="tcc rb lb tb bb">Acres.</td></tr> + +<tr><td class="tcc rb lb">1887</td> <td class="tcc rb2">202,234</td> <td class="tcc rb">1892</td> <td class="tcc rb2">208,950</td> <td class="tcc rb">1897</td> <td class="tcc rb">224,116</td></tr> +<tr><td class="tcc rb lb">1888</td> <td class="tcc rb2">199,178</td> <td class="tcc rb">1893</td> <td class="tcc rb2">211,664</td> <td class="tcc rb">1898</td> <td class="tcc rb">226,059</td></tr> +<tr><td class="tcc rb lb">1889</td> <td class="tcc rb2">199,897</td> <td class="tcc rb">1894</td> <td class="tcc rb2">214,187</td> <td class="tcc rb">1899</td> <td class="tcc rb">228,603</td></tr> +<tr><td class="tcc rb lb">1890</td> <td class="tcc rb2">202,305</td> <td class="tcc rb">1895</td> <td class="tcc rb2">218,428</td> <td class="tcc rb">1900</td> <td class="tcc rb">232,129</td></tr> +<tr><td class="tcc rb lb bb">1891</td> <td class="tcc rb2 bb">209,996</td> <td class="tcc rb bb">1896</td> <td class="tcc rb2 bb">221,254</td> <td class="tcc rb bb">1901</td> <td class="tcc rb bb">234,660</td></tr> + +</table> + +<p class="pt2 center"><span class="sc">Table</span> II.—<i>Areas under Orchards in England, Wales and +Scotland—Acres.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">Year.</td> <td class="tcc allb">England.</td> <td class="tcc allb">Wales.</td> <td class="tcc allb">Scotland.</td> <td class="tcc allb">Great Britain.</td></tr> + +<tr><td class="tcc lb rb">1896</td> <td class="tcc rb">215,642</td> <td class="tcc rb">3677</td> <td class="tcc rb">1935</td> <td class="tcc rb">221,254</td></tr> +<tr><td class="tcc lb rb">1897</td> <td class="tcc rb">218,261</td> <td class="tcc rb">3707</td> <td class="tcc rb">2148</td> <td class="tcc rb">224,116</td></tr> +<tr><td class="tcc lb rb">1898</td> <td class="tcc rb">220,220</td> <td class="tcc rb">3690</td> <td class="tcc rb">2149</td> <td class="tcc rb">226,059</td></tr> +<tr><td class="tcc lb rb">1899</td> <td class="tcc rb">222,712</td> <td class="tcc rb">3666</td> <td class="tcc rb">2225</td> <td class="tcc rb">228,603</td></tr> +<tr><td class="tcc lb rb">1900</td> <td class="tcc rb">226,164</td> <td class="tcc rb">3695</td> <td class="tcc rb">2270</td> <td class="tcc rb">232,129</td></tr> +<tr><td class="tcc lb rb">1901</td> <td class="tcc rb">228,580</td> <td class="tcc rb">3767</td> <td class="tcc rb">2313</td> <td class="tcc rb">234,660</td></tr> +<tr><td class="tcc lb rb bb">1908</td> <td class="tcc rb bb">244,430</td> <td class="tcc rb bb">3577</td> <td class="tcc rb bb">2290</td> <td class="tcc rb bb">250,297</td></tr> +</table> + +<p>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 +<span class="pagenum"><a name="page261" id="page261"></a>261</span> +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:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc rb lb tb bb">County.</td> <td class="tcc rb2 lb tb bb">Acres.</td> <td class="tcc rb lb tb bb">County.</td> <td class="tcc rb2 lb tb bb">Acres.</td> <td class="tcc rb lb tb bb">County.</td> <td class="tcc rb lb tb bb">Acres.</td></tr> + +<tr><td class="tcl lb rb">Kent</td> <td class="tcc rb2">32,751</td> <td class="tcl rb">Worcester</td> <td class="tcc rb2">23,653</td> <td class="tcl rb">Salop</td> <td class="tcc rb">4685</td></tr> +<tr><td class="tcl lb rb">Devon</td> <td class="tcc rb2">27,200</td> <td class="tcl rb">Gloucester</td> <td class="tcc rb2">20,424</td> <td class="tcl rb">Dorset</td> <td class="tcc rb">4464</td></tr> +<tr><td class="tcl lb rb">Hereford</td> <td class="tcc rb2">28,316</td> <td class="tcl rb">Cornwall</td> <td class="tcc rb2">5,415</td> <td class="tcl rb">Monmouth</td> <td class="tcc rb">3914</td></tr> +<tr><td class="tcl lb rb bb">Somerset</td> <td class="tcc rb2 bb">25,279</td> <td class="tcl rb bb">Middlesex</td> <td class="tcc rb2 bb">5,300</td> <td class="tcl rb bb">Wilts</td> <td class="tcc rb bb">3630</td></tr> +</table> + +<p class="noind">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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p class="pt2 center"><span class="sc">Table</span> III.—<i>Areas of Small Fruit in Great Britain</i>.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc rb lb tb bb">Year.</td> <td class="tcc rb2 lb tb bb">Acres.</td> <td class="tcc rb lb tb bb">Year.</td> <td class="tcc rb2 lb tb bb">Acres.</td> <td class="tcc rb lb tb bb">Year.</td> <td class="tcc rb lb tb bb">Acres.</td></tr> + +<tr><td class="tcc lb rb">1890</td> <td class="tcc rb2">46,234</td> <td class="tcc rb">1894</td> <td class="tcc rb2">68,415</td> <td class="tcc rb">1898</td> <td class="tcc rb">69,753</td></tr> +<tr><td class="tcc lb rb">1891</td> <td class="tcc rb2">58,704</td> <td class="tcc rb">1895</td> <td class="tcc rb2">74,547</td> <td class="tcc rb">1899</td> <td class="tcc rb">71,526</td></tr> +<tr><td class="tcc lb rb">1892</td> <td class="tcc rb2">62,148</td> <td class="tcc rb">1896</td> <td class="tcc rb2">76,245</td> <td class="tcc rb">1900</td> <td class="tcc rb">73,780</td></tr> +<tr><td class="tcc lb rb bb">1893</td> <td class="tcc rb2 bb">65,487</td> <td class="tcc rb bb">1897</td> <td class="tcc rb2 bb">69,792</td> <td class="tcc rb bb">1901</td> <td class="tcc rb bb">74,999</td></tr> +</table> + +<p class="pt2 center"><span class="sc">Table</span> IV.—<i>Areas under Small Fruit in England, Wales and +Scotland—Acres</i>.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">Year.</td> <td class="tcc allb">England.</td> <td class="tcc allb">Wales.</td> <td class="tcc allb">Scotland.</td> <td class="tcc allb">Great Britain.</td></tr> + +<tr><td class="tcc lb rb">1898</td> <td class="tcc rb">63,438</td> <td class="tcc rb">1044</td> <td class="tcc rb">5271</td> <td class="tcc rb">69,753</td></tr> +<tr><td class="tcc lb rb">1899</td> <td class="tcc rb">64,867</td> <td class="tcc rb">1106</td> <td class="tcc rb">5553</td> <td class="tcc rb">71,526</td></tr> +<tr><td class="tcc lb rb">1900</td> <td class="tcc rb">66,749</td> <td class="tcc rb">1109</td> <td class="tcc rb">5922</td> <td class="tcc rb">73,780</td></tr> +<tr><td class="tcc lb rb">1901</td> <td class="tcc rb">67,828</td> <td class="tcc rb">1092</td> <td class="tcc rb">6079</td> <td class="tcc rb">74,999</td></tr> +<tr><td class="tcc lb rb bb">1908</td> <td class="tcc rb bb">75,750</td> <td class="tcc rb bb">1200</td> <td class="tcc rb bb">7930</td> <td class="tcc rb bb">84,880</td></tr> +</table> + +<p>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 2½ 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.</p> + +<p>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, +<span class="pagenum"><a name="page262" id="page262"></a>262</span> +Cornwall and Middlesex each has more than 500 acres devoted +to strawberry cultivation.</p> + +<p>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:</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb"> </td> <td class="tccm allb">1907.</td> <td class="tccm allb">1908.</td> <td class="tccm allb">Increase or<br />Decrease.</td></tr> +<tr><td class="tcc lb rb"> </td> <td class="tcc rb">Acres.</td> <td class="tcc rb">Acres.</td> <td class="tcc rb">Acres.</td></tr> +<tr><td class="tcl lb rb">Small Fruit—</td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb">   Strawberries</td> <td class="tcr rb">27,827</td> <td class="tcr rb">28,815</td> <td class="tcr rb">+ 988</td></tr> +<tr><td class="tcl lb rb">   Raspberries</td> <td class="tcr rb">8,878</td> <td class="tcr rb">9,323</td> <td class="tcr rb">+ 445</td></tr> +<tr><td class="tcl lb rb">   Currants and Gooseberries</td> <td class="tcr rb">25,590</td> <td class="tcr rb">26,241</td> <td class="tcr rb">+ 651</td></tr> +<tr><td class="tcl lb rb">   Other kinds</td> <td class="tcr rb">19,880</td> <td class="tcr rb">20,501</td> <td class="tcr rb">+ 621</td></tr> +<tr><td class="tcl lb rb"> </td> <td class="tcr allb">82,175</td> <td class="tcr allb">84,880</td> <td class="tcr allb">+2705</td></tr> +<tr><td class="tcl lb rb">Orchards—</td> <td class="tcr rb"> </td> <td class="tcr rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb">   Apples</td> <td class="tcr rb">172,643</td> <td class="tcr rb">172,751</td> <td class="tcr rb">+ 108</td></tr> +<tr><td class="tcl lb rb">   Pears</td> <td class="tcr rb">8,911</td> <td class="tcr rb">9,604</td> <td class="tcr rb">+ 693</td></tr> +<tr><td class="tcl lb rb">   Cherries</td> <td class="tcr rb">12,027</td> <td class="tcr rb">11,868</td> <td class="tcr rb">− 159</td></tr> +<tr><td class="tcl lb rb">   Plums</td> <td class="tcr rb">14,901</td> <td class="tcr rb">15,683</td> <td class="tcr rb">+ 782</td></tr> +<tr><td class="tcl lb rb">   Other kinds</td> <td class="tcr rb">41,694</td> <td class="tcr rb">40,391</td> <td class="tcr rb">−1303</td></tr> +<tr><td class="tcl lb rb bb"> </td> <td class="tcr allb">250,176</td> <td class="tcr allb">250,297</td> <td class="tcr allb">+ 121</td></tr> + +</table> + +<p>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.</p> + +<p>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.</p> + +<p>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:</p> + +<div class="condensed"> +<p>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 £844,000 in 1893 and £2,079,000 in 1908. Grapes rank next +to apples in point of value, and over the seventeen years the amount +ranged between £394,000 in 1892 and £728,000 in 1908. On the +average, the annual outlay on imported pears is slightly in excess +of that on plums. The extremes shown are £167,000 in 1895 and +£515,000 in 1908. In the case of plums, the smallest outlay tabulated +is £166,000 in 1895, whilst the largest is £498,000 in 1897. The +amounts expended upon imported cherries varied between £96,000 +in 1895 and £308,000 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 £25,846; in 1901 +the quantity was 13,463 cwt. and the value £32,350. The latter +rose in 1908 to £60,000. 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 £87,170 (1908, £121,850); +of raw gooseberries 26,045 cwt., valued at £14,626 (1908, £25,520); +and of raw strawberries, 52,225 cwt., valued at £85,949. In 1907 +only 44,000 cwt. of strawberries were imported. In 1901 the +quantities and values were respectively—currants, 70,402 cwt., +£75,308; gooseberries, 21,735 cwt., £11,420; strawberries, 38,604 +cwt., £51,290. 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 £792,339, which is equivalent to a +fraction under 2½d. per ℔. For 1901 the quantity was 793,991 cwt., +and the value £734,051; for 1906, there were 1,124,700 cwt., valued +at £953,475; for 1907, 1,135,499 cwt., valued at £1,020,805; and +for 1908, 1,160,283 cwt., valued at £955,983.</p> + +<div class="list"> +<p class="pt2"><span class="sc">Table V</span>.—<i>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.</i></p> +</div> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb" rowspan="2">Year.</td> <td class="tccm allb" colspan="5">Quantities.</td></tr> +<tr><td class="tcc allb">Apples.</td> <td class="tcc allb">Pears.</td> <td class="tcc allb">Plums.</td> <td class="tcc allb">Cherries.</td> <td class="tcc allb">Grapes.</td></tr> + +<tr><td class="tcc lb rb">1892</td> <td class="tcc rb">4515</td> <td class="tcr rb">637</td> <td class="tcr rb">413</td> <td class="tcc rb">217</td> <td class="tcr rb">762</td></tr> +<tr><td class="tcc lb rb">1893</td> <td class="tcc rb">3460</td> <td class="tcr rb">915</td> <td class="tcr rb">777</td> <td class="tcc rb">346</td> <td class="tcr rb">979</td></tr> +<tr><td class="tcc lb rb">1894</td> <td class="tcc rb">4969</td> <td class="tcr rb">1310</td> <td class="tcr rb">777</td> <td class="tcc rb">311</td> <td class="tcr rb">833</td></tr> +<tr><td class="tcc lb rb">1895</td> <td class="tcc rb">3292</td> <td class="tcr rb">407</td> <td class="tcr rb">401</td> <td class="tcc rb">196</td> <td class="tcr rb">865</td></tr> +<tr><td class="tcc lb rb">1896</td> <td class="tcc rb">6177</td> <td class="tcr rb">483</td> <td class="tcr rb">560</td> <td class="tcc rb">219</td> <td class="tcr rb">883</td></tr> +<tr><td class="tcc lb rb">1897</td> <td class="tcc rb">4200</td> <td class="tcr rb">1052</td> <td class="tcr rb">1044</td> <td class="tcc rb">312</td> <td class="tcr rb">994</td></tr> +<tr><td class="tcc lb rb">1898</td> <td class="tcc rb">3459</td> <td class="tcr rb">492</td> <td class="tcr rb">922</td> <td class="tcc rb">402</td> <td class="tcr rb">1136</td></tr> +<tr><td class="tcc lb rb">1899</td> <td class="tcc rb">3861</td> <td class="tcr rb">572</td> <td class="tcr rb">558</td> <td class="tcc rb">281</td> <td class="tcr rb">1158</td></tr> +<tr><td class="tcc lb rb">1900</td> <td class="tcc rb">2129*</td> <td class="tcr rb">477*</td> <td class="tcr rb">423*</td> <td class="tcc rb">243*</td> <td class="tcr rb">593*</td></tr> +<tr><td class="tcc lb rb">1901</td> <td class="tcc rb">1830*</td> <td class="tcr rb">349*</td> <td class="tcr rb">264*</td> <td class="tcc rb">213*</td> <td class="tcr rb">680*</td></tr> + +<tr><td class="tcc allb" colspan="6">Values.</td></tr> + +<tr><td class="tcc lb rb">1892</td> <td class="tcc rb">1354</td> <td class="tcr rb">297</td> <td class="tcr rb">200</td> <td class="tcc rb">135</td> <td class="tcr rb">394</td></tr> +<tr><td class="tcc lb rb">1893</td> <td class="tcc rb"> 844</td> <td class="tcr rb">347</td> <td class="tcr rb">332</td> <td class="tcc rb">195</td> <td class="tcr rb">530</td></tr> +<tr><td class="tcc lb rb">1894</td> <td class="tcc rb">1389</td> <td class="tcr rb">411</td> <td class="tcr rb">302</td> <td class="tcc rb">167</td> <td class="tcr rb">470</td></tr> +<tr><td class="tcc lb rb">1895</td> <td class="tcc rb"> 960</td> <td class="tcr rb">167</td> <td class="tcr rb">166</td> <td class="tcc rb"> 96</td> <td class="tcr rb">487</td></tr> +<tr><td class="tcc lb rb">1896</td> <td class="tcc rb">1582</td> <td class="tcr rb">207</td> <td class="tcr rb">242</td> <td class="tcc rb">106</td> <td class="tcr rb">443</td></tr> +<tr><td class="tcc lb rb">1897</td> <td class="tcc rb">1187</td> <td class="tcr rb">378</td> <td class="tcr rb">498</td> <td class="tcc rb">178</td> <td class="tcr rb">495</td></tr> +<tr><td class="tcc lb rb">1898</td> <td class="tcc rb">1108</td> <td class="tcr rb">222</td> <td class="tcr rb">435</td> <td class="tcc rb">231</td> <td class="tcr rb">550</td></tr> +<tr><td class="tcc lb rb">1899</td> <td class="tcc rb">1186</td> <td class="tcr rb">266</td> <td class="tcr rb">294</td> <td class="tcc rb">154</td> <td class="tcr rb">588</td></tr> +<tr><td class="tcc lb rb">1900</td> <td class="tcc rb">1225</td> <td class="tcr rb">367</td> <td class="tcr rb">393</td> <td class="tcc rb">308</td> <td class="tcr rb">595</td></tr> +<tr><td class="tcc lb rb">1901</td> <td class="tcc rb">1183</td> <td class="tcr rb">296</td> <td class="tcr rb">244</td> <td class="tcc rb">214</td> <td class="tcr rb">695</td></tr> +<tr><td class="tcc lb rb bb">1908</td> <td class="tcc rb bb">2079</td> <td class="tcr rb bb">515</td> <td class="tcr rb bb">428</td> <td class="tcc rb bb">235</td> <td class="tcr rb bb">728</td></tr> + +<tr><td class="tcc" colspan="6">* Thousands of cwts.</td></tr> +</table> + +<p>In 1908 the outlay of the United Kingdom upon imported raw +fruits, such as can easily be produced at home, was £4,195,654, +made up as follows:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Apples</td> <td class="tcr rb">£2,079,703</td> <td class="tcl">Plums</td> <td class="tcr">£428,966</td></tr> +<tr><td class="tcl">Grapes</td> <td class="tcr rb">728,026</td> <td class="tcl">Currants</td> <td class="tcr">121,852</td></tr> +<tr><td class="tcl">Pears</td> <td class="tcr rb">515,914</td> <td class="tcl">Apricots and peaches</td> <td class="tcr">60,141</td></tr> +<tr><td class="tcl">Cherries</td> <td class="tcr rb">235,523</td> <td class="tcl">Gooseberries</td> <td class="tcr">25,529</td></tr> +</table> + +<p>In addition about £280,000 was spent upon “unenumerated” raw +fruit, and £560,000 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 £5,000,000, was made +up of £2,269,651 for oranges, £471,713 for lemons, £1,769,249 for +bananas, and £560,301 for almond-nuts; these cannot be grown on +an industrial scale in the British Isles.</p> + +<p>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.).</p> +</div> + +<p><i>Fruit-growing in Kent</i>.—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 +<span class="pagenum"><a name="page263" id="page263"></a>263</span> +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,<a name="fa1a" id="fa1a" href="#ft1a"><span class="sp">1</span></a> whose remarks, with various additions and +modifications, are here reproduced.</p> + +<div class="condensed"> +<p>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.</p> + +<p>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 Chrétien, Beurré +de Capiaumont, Fertility, Beurré Riche, Chissel, Beurré 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 £20 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 £80 per acre being +not uncommon. The crop on a large cherry orchard in Mid Kent +has been sold for more than £100 per acre.</p> + +<p>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 1½ ft. from plant to plant and 2½ ft. from row +to row. Or currant or gooseberry bushes are set between the half +standards, and strawberry plants between these.</p> + +<p>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.</p> + +<p>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.</p> + +<p>Kentish fruit-growers are becoming more particular in picking, +<span class="pagenum"><a name="page264" id="page264"></a>264</span> +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.</p> + +<p>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 ℔, which would make the gross return of the +100 acres amount to £4660. 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 5½ to 7 ft. +high upon stems from 1½ to 2 ft. high, and are trained so as to form +a cup of from 7 to 8 ft. in diameter.</p> + +<p>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.</p> + +<p><i>Fruit-growing in other Districts.</i>—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.</p> + +<p>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 £10 per ton, whereas it has since been down to £4. Huntingdonshire +fruit is sent chiefly to Yorkshire, Scotland and South Wales, +but railway freights are high.</p> + +<p>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 <span class="scs">A.M.</span>, and the first lot is made +into jam by 6 <span class="scs">A.M.</span></p> + +<p>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.</p> + +<p>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 ℔ 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 <i>Journal of the Royal Agricultural +Society</i> in 1898 and 1899.</p> + +<p>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.</p> + +<p><i>The Woburn Experimental Fruit Farm.</i>—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 +<span class="pagenum"><a name="page265" id="page265"></a>265</span> +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.</p> + +<p>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.</p> + +<p>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.</p> + +<p><i>Fruit Culture in Ireland.</i>—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.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">1. <i>Orchard Fruit</i>—</td> <td class="tcr">Statute Acres.</td></tr> +<tr><td class="tcl">    Apples</td> <td class="tcr">5829</td></tr> +<tr><td class="tcl">    Pears</td> <td class="tcr">224</td></tr> +<tr><td class="tcl">    Plums</td> <td class="tcr">223</td></tr> +<tr><td class="tcl">    Damsons</td> <td class="tcr">138</td></tr> +<tr><td class="tcl">    Other kinds</td> <td class="tcr">129</td></tr> +<tr><td class="tcl"> </td> <td class="tcr">——</td></tr> +<tr><td class="tcl"> </td> <td class="tcr">Total      6543</td></tr> + +<tr><td class="tcl">2. <i>Small Fruit</i>—</td> <td class="tcr"> </td></tr> +<tr><td class="tcl">    Currants, black</td> <td class="tcr">234</td></tr> +<tr><td class="tcl">    Currants, red and white</td> <td class="tcr">159</td></tr> +<tr><td class="tcl">    Gooseberries</td> <td class="tcr">675</td></tr> +<tr><td class="tcl">    Raspberries</td> <td class="tcr">374</td></tr> +<tr><td class="tcl">    Strawberries</td> <td class="tcr">994</td></tr> +<tr><td class="tcl">    Mixed fruit</td> <td class="tcr">2470</td></tr> +<tr><td class="tcl"> </td> <td class="tcr">——</td></tr> +<tr><td class="tcl"> </td> <td class="tcr">Total      4906</td></tr> +</table> + +<p>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.</p> +</div> + +<p><i>The Flower-growing Industry.</i>—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.</p> + +<p>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, +<span class="pagenum"><a name="page266" id="page266"></a>266</span> +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.</p> + +<p>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 £1; 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 3½ 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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>The imports of fresh flowers into the United Kingdom were not +separately shown prior to 1900. In that year, however, their value +amounted to £200,585, in 1901 to £225,011, in 1906 to £233,884, in +1907 to £233,641, and in 1908 to £229,802, 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.</p> + +<p class="pt2 center"><span class="sc">Table VI.</span>—<i>Values of Fresh Flowers imported into the United +Kingdom.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">Month.</td> <td class="tcc allb">1906.</td> <td class="tcc allb">1907.</td> <td class="tcc allb">1908.</td></tr> + +<tr><td class="tcl lb rb">January</td> <td class="tcr rb">£31,035</td> <td class="tcr rb">£18,545</td> <td class="tcr rb">£29,180</td></tr> +<tr><td class="tcl lb rb">February</td> <td class="tcr rb">34,647</td> <td class="tcr rb">25,541</td> <td class="tcr rb">30,541</td></tr> +<tr><td class="tcl lb rb">March</td> <td class="tcr rb">50,232</td> <td class="tcr rb">42,611</td> <td class="tcr rb">35,185</td></tr> +<tr><td class="tcl lb rb">April</td> <td class="tcr rb">30,809</td> <td class="tcr rb">50,418</td> <td class="tcr rb">42,681</td></tr> +<tr><td class="tcl lb rb">May</td> <td class="tcr rb">22,980</td> <td class="tcr rb">21,767</td> <td class="tcr rb">23,129</td></tr> +<tr><td class="tcl lb rb">June</td> <td class="tcr rb">17,641</td> <td class="tcr rb">18,358</td> <td class="tcr rb">16,904</td></tr> +<tr><td class="tcl lb rb">July</td> <td class="tcr rb">3,386</td> <td class="tcr rb">4,509</td> <td class="tcr rb">3,467</td></tr> +<tr><td class="tcl lb rb">August</td> <td class="tcr rb">1,646</td> <td class="tcr rb">1,539</td> <td class="tcr rb">1,081</td></tr> +<tr><td class="tcl lb rb">September</td> <td class="tcr rb">852</td> <td class="tcr rb">736</td> <td class="tcr rb">953</td></tr> +<tr><td class="tcl lb rb">October</td> <td class="tcr rb">4,481</td> <td class="tcr rb">3,180</td> <td class="tcr rb">4,504</td></tr> +<tr><td class="tcl lb rb">November</td> <td class="tcr rb">17,506</td> <td class="tcr rb">15,763</td> <td class="tcr rb">15,097</td></tr> +<tr><td class="tcl lb rb">December</td> <td class="tcr rb">18,669</td> <td class="tcr rb">30,674</td> <td class="tcr rb">27,080</td></tr> + +<tr><td class="tcl lb rb bb">    Total</td> <td class="tcr allb">£233,884</td> <td class="tcr allb">£233,641</td> <td class="tcr allb">£229,802</td></tr> +</table> + +</div> + +<p><i>Hothouse Culture of Fruit and Flowers.</i>—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, +<span class="pagenum"><a name="page267" id="page267"></a>267</span> +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 <i>sprengeri</i> +and <i>plumosus</i> 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 £792,339, and in 1901 to 793,991 cwt., valued at +£734,051. 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.</p> + +<p class="pt2 center"><span class="sc">Table</span> VII.—<i>Quantities of Tomatoes imported into the United +Kingdom.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">Month.</td> <td class="tcc allb">1906.</td> <td class="tcc allb">1907.</td> <td class="tcc allb">1908.</td></tr> +<tr><td class="tcl lb rb">January</td> <td class="tcr rb">61,940</td> <td class="tcr rb">56,022</td> <td class="tcr rb">73,409</td></tr> +<tr><td class="tcl lb rb">February</td> <td class="tcr rb">58,187</td> <td class="tcr rb">58,289</td> <td class="tcr rb">69,350</td></tr> +<tr><td class="tcl lb rb">March</td> <td class="tcr rb">106,458</td> <td class="tcr rb">98,028</td> <td class="tcr rb">86,928</td></tr> +<tr><td class="tcl lb rb">April</td> <td class="tcr rb">103,273</td> <td class="tcr rb">109,057</td> <td class="tcr rb">74,917</td></tr> +<tr><td class="tcl lb rb">May</td> <td class="tcr rb">67,933</td> <td class="tcr rb">114,041</td> <td class="tcr rb">88,901</td></tr> +<tr><td class="tcl lb rb">June</td> <td class="tcr rb">62,906</td> <td class="tcr rb">144,379</td> <td class="tcr rb">127,793</td></tr> +<tr><td class="tcl lb rb">July</td> <td class="tcr rb">238,362</td> <td class="tcr rb">150,907</td> <td class="tcr rb">171,978</td></tr> +<tr><td class="tcl lb rb">August</td> <td class="tcr rb">180,046</td> <td class="tcr rb">102,600</td> <td class="tcr rb">124,757</td></tr> +<tr><td class="tcl lb rb">September</td> <td class="tcr rb">114,860</td> <td class="tcr rb">101,198</td> <td class="tcr rb">119,224</td></tr> +<tr><td class="tcl lb rb">October</td> <td class="tcr rb">52,678</td> <td class="tcr rb">67,860</td> <td class="tcr rb">75,722</td></tr> +<tr><td class="tcl lb rb">November</td> <td class="tcr rb">41,513</td> <td class="tcr rb">66,522</td> <td class="tcr rb">74,292</td></tr> +<tr><td class="tcl lb rb">December</td> <td class="tcr rb">36,316</td> <td class="tcr rb">66,591</td> <td class="tcr rb">73,012</td></tr> + +<tr><td class="tcl lb">   Total</td> <td class="tcr allb">1,124,472</td> <td class="tcr allb">1,135,494</td> <td class="tcr allb">1,160,283</td></tr> + +<tr><td class="tcl lb bb">   Value</td> <td class="tcr allb">£953,475</td> <td class="tcr allb">£1,135,499</td> <td class="tcr allb">£1,160,283</td></tr> +</table> + +<p>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.</p> + +<p>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.</p> + +<p><i>Conditions of the Fruit and Flower growing Industries.</i>—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.</p> + +<p>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 +<span class="pagenum"><a name="page268" id="page268"></a>268</span> +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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>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.</p> +</div> +<div class="author">(W. Fr.; J. Ws.)</div> + +<p class="pt2 center sc">United States</p> + +<p>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° 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.</p> + +<p><i>Fruit Culture.</i>—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 <i>Prunus triflora</i>) 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.</p> + +<p>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 (<i>Pyrus baccata</i>) have produced +<span class="pagenum"><a name="page269" id="page269"></a>269</span> +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.</p> + +<p>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.</p> + +<p>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>i.e.</i> about 48 to the acre.</p> + +<p>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.</p> + +<p>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 +£211 per acre against £103 per acre from unsprayed orchards.</p> + +<p>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 <span class="scs">A.M.</span>, 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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p><i>Flower Culture.</i>—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 +<span class="pagenum"><a name="page270" id="page270"></a>270</span> +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.</p> + +<div class="condensed"> +<p><i>Vegetables.</i>—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.</p> +</div> +<div class="author">(J. Ws.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1a" id="ft1a" href="#fa1a"><span class="fn">1</span></a> <i>Jour. Roy. Agric. Soc.</i>, 1899.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FRUMENTIUS<a name="ar10" id="ar10"></a></span> (<i>c.</i> 300-<i>c.</i> 360), the founder of the Abyssinian +church, traditionally identified in Abyssinian literature with +Abba Salama or Father of Peace (but see <span class="sc"><a href="#artlinks">Ethiopia</a></span>), 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 +<i>Epistola ad Constantinum</i> 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.</p> + + +<hr class="art" /> +<p><span class="bold">FRUNDSBERG, GEORG VON<a name="ar11" id="ar11"></a></span> (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 <i>Landsknechte</i> (<i>q.v.</i>), 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 Württemberg 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.</p> + +<div class="condensed"> +<p>See Adam Reissner, <i>Historia Herrn Georgs und Herrn Kaspars +von Frundsberg</i> (Frankfort, 1568). A German translation of this +work was published at Frankfort in 1572. F.W. Barthold, <i>Georg +von Frundsberg</i> (Hamburg, 1833); J. Heilmann, <i>Kriegsgeschichte +von Bayern, Franken, Pfalz und Schwaben</i> (Munich, 1868).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FRUSTUM<a name="ar12" id="ar12"></a></span> (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.</p> + + +<hr class="art" /> +<p><span class="bold">FRUYTIERS, PHILIP<a name="ar13" id="ar13"></a></span> (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.</p> + + +<hr class="art" /> +<p><span class="bold">FRY,<a name="ar14" id="ar14"></a></span> the name of a well-known English Quaker family, +originally living in Wiltshire. About the middle of the 18th +century <span class="sc">Joseph Fry</span> (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, <span class="sc">Joseph Storrs +Fry</span> (1767-1835).</p> + +<p>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 <span class="sc">Francis Fry</span> (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.</p> + + +<hr class="art" /> +<p><span class="bold">FRY, SIR EDWARD<a name="ar15" id="ar15"></a></span> (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 +<span class="pagenum"><a name="page271" id="page271"></a>271</span> +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 <i>A Treatise on the Specific Performance of Public Contracts</i> +(London, 1858, and many subsequent editions).</p> + + +<hr class="art" /> +<p><span class="bold">FRY, ELIZABETH<a name="ar16" id="ar16"></a></span> (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.</p> + +<p>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, Zürich, 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.</p> + +<p>Two interesting volumes of <i>Memoirs, with Extracts from her +Journals and Letters</i>, edited by two of her daughters, were published +in 1847. See also <i>Elizabeth Fry</i>, by G. King Lewis (1910).</p> + + +<hr class="art" /> +<p><span class="bold">FRYXELL, ANDERS<a name="ar17" id="ar17"></a></span> (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 <i>Stories from Swedish +History</i>. 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 <i>Swedish Grammar</i>, 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 <i>Characteristics +of Sweden between 1592 and 1600</i> (1830), his <i>Origins of the Inaccuracy +with which the History of Sweden in Catholic Times has +been Treated</i> (1847), and his <i>Contributions to the Literary History +of Sweden</i>. 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 <i>History of My History</i>, +which was really a literary autobiography and displays the +persistency and tirelessness of his industry.</p> +<div class="author">(E. G.)</div> + + +<hr class="art" /> +<p><span class="bold">FUAD PASHA<a name="ar18" id="ar18"></a></span> (1815-1869), Turkish statesman, was the son +of the distinguished poet Kechéji-zadé 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUCHOW,<a name="ar19" id="ar19"></a></span> <span class="sc">Fu-Chau, Foochow</span>, 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° 5′ N. and 119º 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. +<span class="pagenum"><a name="page272" id="page272"></a>272</span> +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 £1,531,617, and exports to the value +of £3,330,489. In 1904 the imports amounted to £1,440,351, +and the exports to £1,034,436. 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUCHS, JOHANN NEPOMUK VON<a name="ar20" id="ar20"></a></span> (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.</p> + +<div class="condensed"> +<p>His collected works, including <i>Über den Einfluss der Chemie und +Mineralogie</i> (1824), <i>Die Naturgeschichte des Mineralreichs</i> (1842), +<i>Über die Theorien der Erde</i> (1844), were published at Munich in 1856.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUCHS, LEONHARD<a name="ar21" id="ar21"></a></span> (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 Württemberg he was, in +1535, appointed to the professorship of medicine at the university +of Tübingen, 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 <i>De historia stirpium commentarii insignes</i> (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 <i>Digitalis</i>, which was so named by +him.</p> + + +<hr class="art" /> + +<table class="flt" style="float: right; width: 370px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:315px; height:496px" src="images/img272.jpg" alt="" /></td></tr> +<tr><td class="caption"><i>Fuchsia coccinea</i>.</td></tr> +<tr><td class="caption1">1, Flower cut open after removal of +sepals; 2, fruit; 3, floral diagram.</td></tr></table> + +<p><span class="bold">FUCHSIA,<a name="ar22" id="ar22"></a></span> 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, +<i>F. excorticata, F. Colensoi</i> and <i>F. procumbens</i>, 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 <i>F. verrucosa</i>, being dwarf shrubs; others, as <i>F. arborescens</i> +and <i>F. apetala</i>, attaining a height of 12 to 16 ft., and having +stems several inches in diameter. Plumier, in his <i>Nova plantarum +Americanarum genera</i> (p. 14, tab. 14, Paris, 1703), gave +a description of a species of fuchsia, the first known, under the +name of <i>Fuchsia triphylla, flore coccineo</i>, and a somewhat conventional +outline figure +of the same plant was +published at Amsterdam +in 1757 by Burmann. +In the <i>Histoire +des plantes médicinales</i> +of the South American +traveller Feuillée (p. 64, +pl. XLVII.), written in +1709-1711, and published +by him with his +<i>Journal</i>, Paris, 1725, +the name <i>Thilco</i> is +applied to a species of +fuchsia from Chile, +which is described, +though not evidently +so figured, as having +a pentamerous calyx. +The <i>F. coccinea</i> of Alton +(fig.) (see J.D. Hooker, +in <i>Journal Linnean Soc</i>., +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 <i>F. macrostemma</i> and <i>F. gracilis</i>, 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 <i>F. fulgens</i>, 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. <i>F. Venus-victrix</i>, 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.</p> + +<p>Ripe seed is sown either in autumn or about February or March +in light, rich, well-drained mould, and is thinly covered with +<span class="pagenum"><a name="page273" id="page273"></a>273</span> +sandy soil and watered. A temperature of 70° to 75° 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.</p> + +<p>Among the more hardy or half-hardy plants for inside borders +are varieties of the Chilean species, <i>F. macrostemma</i> (or <i>F. +magellanica</i>), a shrub 6 to 12 ft. high with a scarlet calyx, such +as <i>F. m. globosa, F. m. gracilis</i>; one of the most graceful and +hardy of these, a hybrid <i>F. riccartoni</i>, was raised at Riccarton, +near Edinburgh, in 1830. For inside culture may be mentioned +<i>F. boliviana</i> (Bolivia), 2 to 4 ft. high, with rich crimson flowers +with a trumpet-shaped tube; <i>F. corymbiflora</i> (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; <i>F. microphylla</i> (Central America), +with small leaves and small scarlet funnel-shaped flowers, the +petals deep red; <i>F. procumbens</i> (New Zealand), a pretty little +creeper, the small flowers of which are succeeded by oval magenta-crimson +berries which remain on for months; and <i>F. splendens</i> +(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 <i>Correa</i>, natural +order Rutaceae.</p> + + +<hr class="art" /> +<p><span class="bold">FUCHSINE,<a name="ar23" id="ar23"></a></span> or <span class="sc">Magenta</span>, 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 (<i>Ann</i>., 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, +<i>Dingler’s Poly. Jour</i>., 1860, 158, p. 146); by heating aniline +for red with nitrobenzene, concentrated hydrochloric acid and +iron (Coupier, <i>Ber</i>., 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 <span class="sc"><a href="#artlinks">Dyeing</a></span>). 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.</p> + +<div class="condensed"> +<p>The constitution of the fuchsine bases (pararosaniline and rosaniline) +was determined by E. and O. Fischer in 1878 (<i>Ann</i>., 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)</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">CH<span class="su">3</span>·C<span class="su">6</span>H<span class="su">4</span>NH<span class="su">2</span> + 2C<span class="su">6</span>H<span class="su">5</span>NH<span class="su">2</span> + 3O = HO·C(C<span class="su">6</span>H<span class="su">4</span>NH<span class="su">2</span>)<span class="su">3</span> + 2H<span class="su">2</span>O.</td></tr> + +<tr><td class="tcr">Colour base (pararosaniline).</td></tr> + +<tr><td class="tcr">HO·C(C<span class="su">6</span>H<span class="su">4</span>NH<span class="su">2</span>)<span class="su">3</span>·HCl = H<span class="su">2</span>O + (H<span class="su">2</span>N·C<span class="su">6</span>H<span class="su">4</span>)<span class="su">2</span>C : C<span class="su">6</span>H<span class="su">4</span> : NH<span class="su">2</span>Cl.</td></tr> + +<tr><td class="tcr">Pararosaniline hydrochloride.</td></tr> +</table> + +<p class="noind">A. Rosenstiehl (<i>Jahres</i>., 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</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">(H<span class="su">2</span>N·C<span class="su">5</span>H<span class="su">4</span>)<span class="su">2</span>C : C<span class="su">6</span>H<span class="su">4</span>:NH<span class="su">2</span>Cl → </td> <td class="tcc">HC(C<span class="su">6</span>H<span class="su">4</span>NH<span class="su">2</span>·HCl)<span class="su">3</span> →</td> <td class="tcc">HC(C<span class="su">6</span>H<span class="su">4</span>N<span class="su">2</span>Cl<span class="su">3</span>) + + → </td> <td class="tcc">HC(C<span class="su">6</span>H<span class="su">5</span>)<span class="su">3</span>.</td></tr> + +<tr><td class="tcl">Pararosaniline hydrochloride.</td> <td class="tcc">Paraleucaniline.</td> <td class="tcc"> </td> <td class="tcc">Triphenylmethane.</td></tr> +</table> + +<p class="noind">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.</p> + +<p>The free pararosaniline, C<span class="su">19</span>H<span class="su">19</span>N<span class="su">3</span>O, and rosaniline, +C<span class="su">20</span>H<span class="su">21</span>N<span class="su">3</span>O, +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 (<i>Ber</i>., 1878, II, p. 1348) and of E. and O. +Fischer (<i>Ber.</i>, 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,</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc">C<span class="su">6</span>H<span class="su">5</span>CHO → C<span class="su">6</span>H<span class="su">5</span>CH(C<span class="su">6</span>H<span class="su">4</span>NH<span class="su">2</span>)<span class="su">2</span> → C<span class="su">6</span>H<span class="su">5</span>CH(C<span class="su">6</span>H<span class="su">4</span>OH)<span class="su">2</span> + + → CO(C<span class="su">6</span>H<span class="su">4</span>OH)<span class="su">2</span>;</td></tr> +</table> + +<p class="noind">but if para-nitrobenzaldehyde be used in the above reaction and the +resulting nitro compound NO<span class="su">2</span>.C<span class="su">6</span>H<span class="su">4</span>.CH(C<span class="su">6</span>H<span class="su">4</span>NH<span class="su">2</span>)<span class="su">2</span> 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 <span class="sc"><a href="#artlinks">Dyeing</a></span>).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUCINO, LAGO DI<a name="ar24" id="ar24"></a></span> [Lat. <i>Lacus Fucinus</i>], 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 <span class="scs">A.D.</span> 52 the emperor Claudius, realizing a project of +Julius Caesar, constructed a tunnel 3½ 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 £1,700,000. The +<span class="pagenum"><a name="page274" id="page274"></a>274</span> +reclaimed area is 12½ 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.</p> + +<div class="condensed"> +<p>See A. Brisse and L. de Rotron, <i>Le Desséchement du lac Fucin, +exécuté par S.E. le Prince A. Torlonia</i> (Rome, 1876).</p> +</div> +<div class="author">(T. As.)</div> + + +<hr class="art" /> +<p><span class="bold">FUEL<a name="ar25" id="ar25"></a></span> (O. Fr. <i>feuaile</i>, popular Lat. <i>focalia</i>, from <i>focus</i>, 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.</p> + +<p class="pt2 center"><i>Solid Fuels</i>.</p> + +<p>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 +<span class="sidenote">Wood.</span> +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°, 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 <i>fumerons</i>. 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>i.e</i>. containing 25% of water, at 2800 +to 3000 units. Their evaporative values, <i>i.e</i>. the quantities of +water evaporated by unit weight, are 3.68 and 4.44.</p> + +<p>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.</p> + +<p>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 +<span class="sidenote">Peat.</span> +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° 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° 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.</p> + +<p>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.</p> + +<p>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 +<span class="sidenote">Lignite.</span> +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° 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.</p> + +<p>Of the other natural fuels, apart from coal (<i>q.v.</i>), 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 +<span class="sidenote">Other natural fuels.</span> +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, <i>e.g.</i> +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 (<i>Proc. Inst. of Civil Engineers</i>, +vol. xlviii. p. 75), the evaporative values of 1 ℔ of these +different articles when burnt in a tubular boiler are—coal, 8 ℔; +dry peat, 4 ℔; dry wood, 3.58-3.52 ℔; cotton stalks or +megasse, 3.2-2.7 ℔; straw, 2.46-2.30 ℔. Owing to the +siliceous nature of the ash of <span class="correction" title="amended from sraw">straw</span>, 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.</p> + +<p>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.</p> + +<p>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.</p> + +<p><span class="pagenum"><a name="page275" id="page275"></a>275</span></p> + +<p>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 (<i>q.v.</i>) containing iron pyrites, FeS<span class="su">2</span>, +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.</p> + +<div class="condensed"> +<p><i>Assay and Valuation of Carbonaceous Fuels</i>.—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. +<span class="sidenote">Calorific power.</span> +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 34½ 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.</p> + +<p>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.</p> + +<p>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 <span class="correction" title="amended from form">from</span> different localities.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb">Slag left<br />in Grate.</td> <td class="tccm allb">Ashes in<br />Ashpit.</td> <td class="tccm allb">Soot in<br />Flues.</td> <td class="tccm allb">Water<br />evaporated by<br />1 ℔ of Coal</td></tr> + +<tr><td class="tcl lb rb">Westphalian gas coals</td> <td class="tcl rb">0.33-6.42</td> <td class="tcl rb">2.83-6.53</td> <td class="tcl rb">0.32-0.46</td> <td class="tcl rb">6.60-7.45 ℔</td></tr> +<tr><td class="tcl lb rb">   Do. bituminous coals</td> <td class="tcl rb">0.98-9.10</td> <td class="tcl rb">1.97-9.63</td> <td class="tcl rb">0.24-0.88</td> <td class="tcl rb">7.30-8.66</td></tr> +<tr><td class="tcl lb rb">   Do. dry coals</td> <td class="tcl rb">1.93-5.70</td> <td class="tcl rb">4.37-10.63</td> <td class="tcl rb">0.24-0.48</td> <td class="tcl rb">7.03-8.51</td></tr> +<tr><td class="tcl lb rb">Silesian coals</td> <td class="tcl rb">0.92-1.30</td> <td class="tcl rb">3.15-3.50</td> <td class="tcl rb">0.24-0.30</td> <td class="tcl rb">6.73-7.10</td></tr> +<tr><td class="tcl lb rb">Welsh steam coals</td> <td class="tcl rb">1.20-4.07</td> <td class="tcl rb">4.07</td> <td class="tcl rb">0.32</td> <td class="tcl rb">8.41</td></tr> +<tr><td class="tcl lb rb bb">Newcastle coals</td> <td class="tcl rb bb">1.92</td> <td class="tcl rb bb">2.57</td> <td class="tcl rb bb">0.35</td> <td class="tcl rb bb">7.28</td></tr> + +</table> + +<p>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 <i>Annuaire du Bureau des Longitudes</i>, +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° C. in temperature by the combustion +of 1 gramme of the substance, when it is oxidized to the condition +shown in the second column:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">Element.</td> <td class="tcc allb">Product of Combustion.</td> <td class="tcc allb">Calories.</td></tr> + +<tr><td class="tclm lb rb cl" rowspan="2">Hydrogen</td> <td class="tcl rb">Water, H<span class="su">2</span>O, condensed to liquid</td> <td class="tcr rb">34,500</td></tr> +<tr><td class="tcl rb">   ”    as vapour</td> <td class="tcr rb">29,650</td></tr> +<tr><td class="tcl lb rb">Carbon—</td> <td class="tcl rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb">   Diamond</td> <td class="tcl rb">Carbon Dioxide, CO<span class="su">2</span></td> <td class="tcr rb">7,868</td></tr> +<tr><td class="tcl lb rb">   Graphite</td> <td class="tcl rb">  ”     ”</td> <td class="tcr rb">7,900</td></tr> +<tr><td class="tcl lb rb">   Amorphous</td> <td class="tcl rb">  ”     ”</td> <td class="tcr rb">8,133</td></tr> +<tr><td class="tcl lb rb">Silicon—</td> <td class="tcl rb"> </td> <td class="tcr rb"> </td></tr> +<tr><td class="tcl lb rb">   Amorphous</td> <td class="tcl rb">Silicon Dioxide, SiO<span class="su">2</span></td> <td class="tcr rb">6,414</td></tr> +<tr><td class="tcl lb rb">   Crystallized</td> <td class="tcl rb">  ”     ”</td> <td class="tcr rb">6,570</td></tr> +<tr><td class="tcl lb rb"> Phosphorus</td> <td class="tcl rb">Phosphoric pentoxide, P<span class="su">2</span>O<span class="su">5</span></td> <td class="tcr rb">5,958</td></tr> +<tr><td class="tcl lb rb bb"> Sulphur</td> <td class="tcl rb bb">Sulphur dioxide, SO<span class="su">2</span>, gaseous</td> <td class="tcr rb bb">2,165</td></tr> + +</table> + +<p class="noind">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.</p> + +<p>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:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl"> </td> <td class="tcc" colspan="2">Calorific Value.</td></tr> +<tr><td class="tcl"> </td> <td class="tcr">Higher.</td> <td class="tcr">Lower.</td></tr> +<tr><td class="tcl">Acetylene, C<span class="su">2</span>H<span class="su">2</span></td> <td class="tcr">11,920</td> <td class="tcr">11,500</td></tr> +<tr><td class="tcl">Ethylene, C<span class="su">2</span>H<span class="su">4</span></td> <td class="tcr">11,880</td> <td class="tcr">11,120</td></tr> +<tr><td class="tcl">Methane, CH<span class="su">4</span></td> <td class="tcr">13,240</td> <td class="tcr">11,910</td></tr> +<tr><td class="tcl">Carbon monoxide, CO</td> <td class="tcr">2,440</td> <td class="tcr">2,440</td></tr> +</table> + +<p>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 +<span class="sidenote">Caloric intensity.</span> +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° or 2000°. 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°-2300° C.; this is +realized in the bright spots or “eyes” of the tuyeres of blast furnaces.</p> + +<p>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 +<span class="pagenum"><a name="page276" id="page276"></a>276</span> +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° F. (3000° 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.</p> + +<p>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° +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 1½ 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.</p> +</div> +<div class="author">(H. B.)</div> + +<p class="pt2 center"><i>Liquid Fuel.</i></p> + +<p>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.</p> + +<p>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.</p> + +<p>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:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Carbon</td> <td class="tcr">88.00 %</td></tr> +<tr><td class="tcl">Hydrogen</td> <td class="tcr">10.75 %</td></tr> +<tr><td class="tcl">Oxygen</td> <td class="tcr">1.25 %</td></tr> +<tr><td class="tcl"> </td> <td class="tcr">———  </td></tr> +<tr><td class="tcl">      Total</td> <td class="tcr">100  </td></tr> +</table> + +<p>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° 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° 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° 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°. 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.</p> + +<p>The process of refining by distillation is also necessary to +reduce two impurities which greatly retard storage and combustion, +<i>i.e.</i> 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.</p> + +<p><span class="pagenum"><a name="page277" id="page277"></a>277</span></p> + +<p>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.</p> + +<p>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 +<span class="sidenote">Progress of liquid fuel.</span> +petroleum for the production of kerosene or lamp oil, +large quantities of refuse were produced—known by +the Russian name of <i>astatki</i>—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 <i>astatki</i> fuel.</p> + +<p>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>i.e.</i> 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.</p> + +<p>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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:826px; height:410px" src="images/img277.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 1.</span>—Holden Burner.</td></tr></table> + +<p>The chief factors of economy are the greater calorific value +of oil than coal (about 16 ℔ of water per ℔ of oil fuel evaporated +from a temperature of 212° F.), not only in laboratory practice, +but in actual use on a large scale, and the saving of labour both +<span class="sidenote">Economy of liquid fuel.</span> +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 +<span class="pagenum"><a name="page278" id="page278"></a>278</span> +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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:700px; height:540px" src="images/img278a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 2.</span>—Rusden and Eeles Burner.</td></tr></table> + +<p>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 ℔ oil is +to 35 ℔ coal; all, however, did not realize so high a result.</p> + +<p>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 +<span class="sidenote"><b>Liquid fuel in locomotives.</b></span> +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.</p> + +<p>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 +<span class="sidenote"><b>Liquid fuel at sea.</b></span> +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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:900px; height:280px" src="images/img278b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 3.</span>—Storage of Liquid Fuel on Oil-carrying Steamers +(Flannery-Boyd System).</td></tr></table> + +<p>The British admiralty have undertaken experiments with +liquid fuel at sea, and at the same time investigations of the +<span class="pagenum"><a name="page279" id="page279"></a>279</span> +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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:571px; height:319px" src="images/img279a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 4.</span>—Installation on ss. “Trochas.”</td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:657px; height:250px" src="images/img279b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 5.</span>—Details of Furnace, Meyer System.</td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:458px; height:246px" src="images/img279c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 6.</span>—Details of Exterior Elongation of Furnace, Meyer System.</td></tr></table> + +<p>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 +<span class="sidenote">Advantages in warships.</span> +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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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 +<span class="sidenote">Advantages in merchant ships.</span> +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.</p> + +<p>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 +<span class="pagenum"><a name="page280" id="page280"></a>280</span> +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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:706px; height:389px" src="images/img280a.jpg" alt="" /></td></tr> +<tr><td class="tcl f90"><span class="sc">Fig. 7.</span>—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.</td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:897px; height:162px" src="images/img280b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 8.</span>—Fuel Tanks, &c., of ss. “Murex.”</td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:850px; height:540px" src="images/img280c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 9.</span>—Furnace Gear of ss. “Murex.”</td></tr></table> + +<table class="flt" style="float: right; width: 330px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:278px; height:251px" src="images/img281.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 10.</span>—Section through Furnace +of ss. “Murex.”</td></tr></table> + +<div class="condensed"> +<p>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 +<span class="pagenum"><a name="page281" id="page281"></a>281</span> +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°C. before entering +the pump, and further heated to 90°C. after leaving the pump. It +is then filtered, and passes +to the furnace injector C at +about 30-℔ 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 Körting +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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> +</div> +<div class="author">(F. F.*)</div> + +<p class="pt2 center"><i>Gaseous Fuel.</i></p> + +<p>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.</p> + +<div class="condensed"> +<p>The various descriptions of gaseous fuel employed in practice +may be classified under the following heads:</p> + +<div class="list"> +<p>I. Natural Gas.</p> + +<p>II. Combustible Gases obtained as by-products in various +technical operations.</p> + +<p>III. Coal Gas (Illuminating Gas).</p> + +<p>IV. Combustible Gases obtained by the partial combustion of +coal, &c.</p> +</div></div> + +<p>I. <i>Natural Gas.</i>—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 ℔ 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 ℔ 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.</p> + +<p>The principal constituent of natural gas is always methane, +CH<span class="su">4</span>, 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 ℔, on the average 100 ℔, 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.</p> + +<p>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 <i>diatomaceae</i>, &c., and +even an inorganic origin of both petroleum and natural gas has +been assumed by chemists of the rank of D.I. Mendeléeff and +H. Moissan.</p> + +<p>II. <i>Gases obtained as By-products.</i>—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.</p> + +<p>(<i>a</i>) <i>Blast-furnace Gases.</i>—The gases issuing from the mouths +of blast-furnaces (see <span class="sc"><a href="#artlinks">Iron and Steel</a></span>) 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:</p> + +<table class="ws f90" summary="Contents"> +<tr><td class="tcl">Carbon monoxide</td> <td class="tcl">21.6</td> <td class="tcc">to</td> <td class="tcl">29.0,</td> <td class="tcc">mostly</td> <td class="tcc">about</td> <td class="tcl">26</td> <td class="tcr">%</td></tr> +<tr><td class="tcl">Hydrogen</td> <td class="tcl"> 1.8</td> <td class="tcc">”</td> <td class="tcl"> 6.3,</td> <td class="tcc">”</td> <td class="tcc">”</td> <td class="tcl"> 3</td> <td class="tcr">%</td></tr> +<tr><td class="tcl">Methane</td> <td class="tcl"> 0.1</td> <td class="tcc">”</td> <td class="tcl"> 0.8,</td> <td class="tcc">”</td> <td class="tcc">”</td> <td class="tcl"> 0.5</td> <td class="tcr">%</td></tr> +<tr><td class="tcl">Carbon dioxide</td> <td class="tcl"> 6</td> <td class="tcc">”</td> <td class="tcl">12,</td> <td class="tcc">”</td> <td class="tcc">”</td> <td class="tcl"> 9.5</td> <td class="tcr">%</td></tr> +<tr><td class="tcl">Nitrogen</td> <td class="tcl">51</td> <td class="tcc">”</td> <td class="tcl">60,</td> <td class="tcc">”</td> <td class="tcc">”</td> <td class="tcl">56</td> <td class="tcr">%</td></tr> +<tr><td class="tcl">Steam</td> <td class="tcl"> 5</td> <td class="tcc">”</td> <td class="tcl">12,</td> <td class="tcc">”</td> <td class="tcc">”</td> <td class="tcl"> 5</td> <td class="tcr">%</td></tr> +<tr><td class="tcl" colspan="6"> </td> <td class="tcl" colspan="2">———</td></tr> +<tr><td class="tcl" colspan="6"> </td> <td class="tcl">100</td> <td class="tcr">%</td></tr> +</table> + +<p class="noind">There is always a large amount of mechanically suspended +<span class="pagenum"><a name="page282" id="page282"></a>282</span> +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.</p> + +<p>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.</p> + +<p>(<i>b</i>) <i>Coke-oven Gases.</i>—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 <i>Coal-Tar and +Ammonia</i> (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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>III. <i>Coal Gas</i> (<i>Illuminating Gas</i>).—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 <span class="sc"><a href="#artlinks">Gas</a></span>, § <i>Manufacture</i>), 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).</p> + +<table class="ws f90" summary="Contents"> +<tr><td class="tccm allb">Constituents.</td> <td class="tccm allb">Volume<br />per cent.</td> <td class="tccm allb">Weight<br />per cent.</td> <td class="tccm allb">Heat-value<br />per Cubic<br />Metre<br />Calories. + </td> <td class="tccm allb">Heat-value<br />per Quantity<br />contained in<br />1 Cub. Met.</td> <td class="tccm allb">Heat-value<br />per cent.<br />of Total.</td></tr> + +<tr><td class="tcl lb rb">Hydrogen, H<span class="su">2</span></td> <td class="tcr rb">47 </td> <td class="tcr rb">7.4</td> <td class="tcr rb">2,582</td> <td class="tcr rb">1213</td> <td class="tcr rb">22.8</td></tr> +<tr><td class="tcl lb rb">Methane, CH<span class="su">4</span></td> <td class="tcr rb">34 </td> <td class="tcr rb">42.8</td> <td class="tcr rb">8,524</td> <td class="tcr rb">2898</td> <td class="tcr rb">54.5</td></tr> +<tr><td class="tcl lb rb">Carbon monoxide, CO</td> <td class="tcr rb">9 </td> <td class="tcr rb">19.9</td> <td class="tcr rb">3,043</td> <td class="tcr rb">273</td> <td class="tcr rb">5.1</td></tr> +<tr><td class="tcl lb rb">Benzene vapour, C<span class="su">6</span>H<span class="su">6</span></td> <td class="tcr rb">1.2</td> <td class="tcr rb">7.4</td> <td class="tcr rb">33,815</td> <td class="tcr rb">405</td> <td class="tcr rb">7.7</td></tr> +<tr><td class="tcl lb rb">Ethylene, C<span class="su">2</span>H<span class="su">4</span></td> <td class="tcr rb">3.8</td> <td class="tcr rb">8.4</td> <td class="tcr rb">13,960</td> <td class="tcr rb">530</td> <td class="tcr rb">9.9</td></tr> +<tr><td class="tcl lb rb">Carbon dioxide, CO<span class="su">2</span></td> <td class="tcr rb">2.5</td> <td class="tcr rb">8.6</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> +<tr><td class="tcl lb rb">Nitrogen, N<span class="su">2</span></td> <td class="tcr rb">2.5</td> <td class="tcr rb">5.5</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td> <td class="tcc rb">..</td></tr> + +<tr><td class="tcc lb rb bb">Total</td> <td class="tcr allb">100.0</td> <td class="tcr allb">100.0</td> <td class="tcc allb">..</td> <td class="tcr allb">5319</td> <td class="tcr allb">100.0</td></tr> + +</table> + +<p>One cubic metre of such gas weighs 568 grammes. <i>Rich gas</i>, +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).</p> + +<p>IV. <i>Combustible Gases produced by the Partial Combustion of +Coal, &c.</i>—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.</p> + +<p>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</p> + +<p class="center">C + O = CO    </p> +<div class="author1">(<i>a</i>);</div> + +<p class="noind">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 +<span class="pagenum"><a name="page283" id="page283"></a>283</span> +products of the destructive distillation of the coal, &c., are known +as <i>producer gas</i> or <i>Siemens gas</i>. In the second case the chemical +reaction is mainly</p> + +<p class="center">C + H<span class="su">2</span>O = CO + H<span class="su">2</span></p> +<div class="author1">(<i>b</i>);</div> + +<p class="noind">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 (<i>water gas</i>) is, apart from a few impurities, +entirely combustible. Another kind of water gas is formed by +the reaction</p> + +<p class="center">C + 2H<span class="su">2</span>O = CO<span class="su">2</span> <span class="correction" title="amended from =">+</span> 2H<span class="su">2</span></p> +<div class="author1">(<i>c</i>),</div> + +<p class="noind">but this reaction, which converts all the carbon into the incombustible +form of CO<span class="su">2</span>, is considered as an unwelcome, although +never entirely avoidable, concomitant of (<i>b</i>).</p> + +<p>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 (<i>a</i>) and (<i>b</i>) in such proportions that the heat +generated by (<i>a</i>) at least equals the heat absorbed by (<i>b</i>). 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.</p> + +<p>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 CO<span class="su">2</span> and H<span class="su">2</span>O, the amount of CO<span class="su">2</span> +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% +CO<span class="su">2</span>. But the best smoke gases of steam boilers show only 12 +or 13%, much more frequently only 10% CO<span class="su">2</span>, 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 1½ 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%.</p> + +<p>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.”</p> + +<p>We shall now describe the principal classes of gaseous fuel, +produced by the partial combustion of coal.</p> + +<p>A. <i>Producer Gas, Siemens Gas.</i>—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, CO<span class="su">2</span>, the latter, in rising through the high +column of incandescent fuel, must be reduced to monoxide: +CO<span class="su">2</span> + 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° 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 +CO<span class="su">2</span>, 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 CO<span class="su">2</span>, thus leaving only 70.5% for +the stage CO + O = CO<span class="su">2</span>. 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>i.e.</i> +first with “primary air” in the producer (C + O = CO), then with +“secondary air” in the furnace (CO + O =CO<span class="su">2</span>), 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 CO<span class="su">2</span>, +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, <i>e.g.</i> 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.</p> + +<p>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 +<span class="pagenum"><a name="page284" id="page284"></a>284</span> +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 <span class="sc"><a href="#artlinks">Gas</a></span>, II. <i>For Fuel and Power.</i></p> + +<p>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.</p> + +<p>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 <i>sub</i> C) by volume show 22 to 33% CO, +1 to 7% CO<span class="su">2</span>, 0.5 to 2% H<span class="su">2</span>, 0.5 to 3% hydrocarbons, and +64 to 68% N<span class="su">2</span>.</p> + +<p>B. <i>Water Gas.</i>—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, Tessié 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.</p> + +<p>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 + H<span class="su">2</span>O = CO + H<span class="su">2</span>, +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 + H<span class="su">2</span>O = CO + H<span class="su">2</span> 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 CO<span class="su">2</span>, 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 + H<span class="su">2</span>, 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 CO<span class="su">2</span>, 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 H<span class="su">2</span>O 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 (<i>b</i>) +C + H<span class="su">2</span>O = CO + H<span class="su">2</span> 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° C., that is, for a very few minutes. Even at 1100° C. +a little CO<span class="su">2</span> can be proved to exist in the gas, and at 900° its +proportion becomes too high to allow the process to go on. +About 650° C. the CO has fallen to a minimum, and the reaction +is now essentially (<i>c</i>) C + 2H<span class="su">2</span>O = CO<span class="su">2</span> + 2H<span class="su">2</span>; 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 (<i>c</i>) contains only two-thirds of +combustible matter, and is on that account less valuable than +proper water gas formed by reaction (<i>b</i>); 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 +(<i>c</i>) 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.</p> + +<p>The process of making water gas consequently comprises +<span class="pagenum"><a name="page285" id="page285"></a>285</span> +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° C.; and, secondly “steaming,” by injecting a +current of (preferably superheated) steam until the temperature +of the fuel had fallen to about 900° 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 CO<span class="su">2</span>. The proportion of +hydrogen may exceed 50%, in consequence of reaction (<i>c</i>) +setting in at the close of the steaming. Ordinary “blue” water +gas, if, as usual, made from coke or anthracite, contains 48-52% +H<span class="su">2</span>, 40-41% CO, 1-5% CO<span class="su">2</span>, 4-5% N<span class="su">2</span>, 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:</p> + +<table class="ws f90" summary="Contents"> +<tr><td class="tcc">0.5</td> <td class="tcc">cub. met.</td> <td class="tcl">H<span class="su">2</span></td> <td class="tcr">= 1291</td> <td class="tcc">calories</td></tr> +<tr><td class="tcc">0.5</td> <td class="tcc">  ”   ”  </td> <td class="tcl">CO</td> <td class="tcr">= 1522</td> <td class="tcc">”</td></tr> +<tr><td class="tcc" colspan="3"> </td> <td class="tcr"><span class="ov">2813</span></td> <td class="tcc">”</td></tr> +</table> + +<p class="noind">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.</p> + +<p>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 CO<span class="su">2</span> +formed at first would be reduced to CO; and it was further +assumed that hardly any CO<span class="su">2</span> 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 CO<span class="su">2</span>, +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 1½ 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.</p> + +<p>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.</p> + +<p>C. <i>Mixed Gas</i> (<i>Semi-Water Gas</i>).—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 + H<span class="su">2</span>O = CO + H<span class="su">2</span>, 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.</p> + +<p>Roughly speaking, this gas may be said to be produced by +the combination of the reactions, described <i>sub</i> A and B, to the +joint reaction: 2C + O + H<span class="su">2</span>O = 2CO + H<span class="su">2</span>. The decomposition +of H<span class="su">2</span>O (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>i.e.</i> 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.</p> + +<p><span class="pagenum"><a name="page286" id="page286"></a>286</span></p> + +<p>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 CO<span class="su">2</span> 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.</p> + +<p>A modification of this gas is the <i>Mond gas</i>, 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 + 2H<span class="su">2</span>O = CO<span class="su">2</span> + 2H<span class="su">2</span>; 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.</p> + +<p>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.</p> + +<p><i>Air Gas.</i>—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, <i>e.g.</i> 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.</p> +<div class="author">(G. L.)</div> + + +<hr class="art" /> +<p><span class="bold">FUENTE OVEJUNA<a name="ar26" id="ar26"></a></span> [<i>Fuenteovejuna</i>], 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 (<i>c.</i> 1163-1486).</p> + + +<hr class="art" /> +<p><span class="bold">FUENTERRABIA<a name="ar27" id="ar27"></a></span> (formerly sometimes written <i>Fontarabia</i>; +Lat. <i>Fons Rapidus</i>), a town of northern Spain, in the province +of Guipúzcoa; 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.</p> + +<p>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 +(<i>Paradise Lost</i>, i. 587), is generally understood to have taken +place not here but at Roncesvalles (<i>q.v.</i>), 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 Condé 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 (<i>muy +noble, muy leal, y muy valerosa ciudad</i>, “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.</p> + + +<hr class="art" /> +<p><span class="bold">FUERO,<a name="ar28" id="ar28"></a></span> a Spanish term, derived from the Latin <i>forum</i>. The +Castillan use of the word in the sense of a right, privilege or +charter is most probably to be traced to the Roman <i>conventus +juridici</i>, otherwise known as <i>jurisdictiones</i> or <i>fora</i>, which in +Pliny’s time were already numerous in the Iberian peninsula. In +each of these provincial <i>fora</i> 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 <i>jus italicum</i>. 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.<a name="fa1b" id="fa1b" href="#ft1b"><span class="sp">1</span></a> 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 <i>fora</i>, 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;<a name="fa2b" id="fa2b" href="#ft2b"><span class="sp">2</span></a> and the +mere fact of the preservation of the name <i>alcalde</i>, an official +whose functions corresponded so closely to those of the <i>judex</i> or +<i>defensor civitatis</i>, is fitted to suggest that the old municipal <i>fora</i>, +if much impaired, were not even then in all cases wholly destroyed. +At all events when the word <i>forum</i><a name="fa3b" id="fa3b" href="#ft3b"><span class="sp">3</span></a> begins to appear for the first +time in documents of the 10th century in the sense of a liberty or +<span class="pagenum"><a name="page287" id="page287"></a>287</span> +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.<a name="fa4b" id="fa4b" href="#ft4b"><span class="sp">4</span></a> 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 <i>consuetudines antiquorum jurium</i> 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 (<i>ob.</i> 1035), and +confirmed, in 1076, by Alphonso VI.<a name="fa5b" id="fa5b" href="#ft5b"><span class="sp">5</span></a> Traces of others of perhaps +even an earlier date are occasionally to be met with. In the fuero +of Cardeña, 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.<a name="fa6b" id="fa6b" href="#ft6b"><span class="sp">6</span></a> Among the later fueros of the 11th century, the +most important are those of Jaca (1064) and of Logroño (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 +<i>fora</i> 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;<a name="fa7b" id="fa7b" href="#ft7b"><span class="sp">7</span></a> and among the compilations of Alphonso +the Learned in like manner were an <i>Espejo de Fueros</i> and also the +<i>Fuero de las leyes</i>, better known perhaps as the <i>Fuero Real</i>. The +famous code known as the <i>Ordenamiento Real de Alcalá</i>, or <i>Fuero +Viejo de Castilla</i>, 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 <span class="sc"><a href="#artlinks">Basques</a></span>) 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 <i>señor</i>, 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 Castaños. 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 <i>Foraes</i> of the Portuguese +towns, and of the <i>Fors du Béarn</i>, is precisely analogous to +that of the fueros of Castile.</p> + +<div class="condensed"> +<p>Among the numerous works that more or less expressly deal with +this subject, that of Marina (<i>Ensayo historico-critico sobre la antigua +legislacion y principales cuerpos legales de los reynos de Leon y +Castilla</i>) still continues to hold a high place. Reference may also +be made to Colmeiro’s <i>Curso de derecho político según la historia de +Leon y de Castilla</i> (Madrid, 1873); to Schäfer’s <i>Geschichte von +Spanien</i>, ii. 418-428, iii. 293 seq.; and to Hallam’s <i>Middle Ages</i>, +c. iv.</p> +</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1b" id="ft1b" href="#fa1b"><span class="fn">1</span></a> The nature of the evidence may be gathered from Savigny, <i>Gesch. +d. röm. Rechts</i>. See especially i. pp. 154, 259 seq.</p> + +<p><a name="ft2b" id="ft2b" href="#fa2b"><span class="fn">2</span></a> Compare Lembke u. Schäfer, <i>Geschichte von Spanien</i>, i. 314; ii. 117.</p> + +<p><a name="ft3b" id="ft3b" href="#fa3b"><span class="fn">3</span></a> Or rather <i>forus</i>. See Ducange, <i>s.v.</i></p> + +<p><a name="ft4b" id="ft4b" href="#fa4b"><span class="fn">4</span></a> Cap. xx. begins: “Constituimus etiam ut Legionensis civitas, +quae depopulata fuit a Sarracenis in diebus patris mei Veremundi +regis, repopulatur <i>per hos foros subscriptos</i>.”</p> + +<p><a name="ft5b" id="ft5b" href="#fa5b"><span class="fn">5</span></a> “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.”</p> + +<p><a name="ft6b" id="ft6b" href="#fa6b"><span class="fn">6</span></a> “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.”</p> + +<p><a name="ft7b" id="ft7b" href="#fa7b"><span class="fn">7</span></a> 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.”</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUERTEVENTURA,<a name="ar29" id="ar29"></a></span> an island in the Atlantic Ocean, forming +part of the Spanish archipelago of the Canary Islands (<i>q.v.</i>). +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.</p> + + +<hr class="art" /> +<p><span class="bold">FUGGER,<a name="ar30" id="ar30"></a></span> 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 (<i>Reh</i>) 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 +<span class="pagenum"><a name="page288" id="page288"></a>288</span> +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.</p> + +<p>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 (<i>Pfalzgraf</i>) 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.</p> + +<p>In addition to the bishop, three of Raimund Fugger’s sons +attained some degree of celebrity. Johann Jakob (1516-1575), +was the author of <i>Wahrhaftigen Beschreibung des österreichischen +und habsburgischen Nahmens</i>, which was largely used by S. von +Bircken in his <i>Spiegel der Ehren des Erzhauses Österreich</i> (Nuremberg, +1668), and of a <i>Geheim Ernbuch des Fuggerischen Geschlechtes</i>. +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).</p> + +<p>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, <i>Wie und +wo man ein Gestüt von guten edeln Kriegsrossen aufrichten soll</i> +(1578), and of a German translation of the <i>Historia ecclesiastica</i> +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.</p> + +<p>Johann Fugger had three sons, Christoph (d. 1615) and +Marcus (d. 1614), who founded the families of Fugger-Glött 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 Glött 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 +Glött (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 Württemberg. The heads of the three existing branches +of the Fuggers are all hereditary members of the Bavarian +Upper House.</p> + +<p>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.</p> + +<div class="condensed"> +<p>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 <i>Pinacotheca Fuggerorum</i>, published at Vienna in 1754, +includes 139 portraits. See <i>Chronik der Familie Fugger vom Jahre +1599</i>, edited by C. Meyer (Munich, 1902); A. Geiger, <i>Jakob Fugger, +1459-1525</i> (Regensburg, 1895); A. Schulte, <i>Die Fugger in Rom, +1495-1523</i> (Leipzig, 1904); R. Ehrenberg, <i>Das Zeitalter der Fugger</i> +(Jena, 1896); K. Häbler, <i>Die Geschichte der Fuggerschen Handlung +in Spanien</i> (Weimar, 1897); A. Stauber, <i>Das Haus Fugger</i> (Augsburg, +1900); and M. Jansen, <i>Die Anfänge der Fugger</i> (Leipzig, +1907).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUGITIVE SLAVE LAWS<a name="ar31" id="ar31"></a></span>, 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 +<span class="pagenum"><a name="page289" id="page289"></a>289</span> +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.”</p> + +<p>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.”<a name="fa1c" id="fa1c" href="#ft1c"><span class="sp">1</span></a> The decision of the +Supreme Court of the United States in the case of <i>Prigg</i> v. +<i>Pennsylvania</i> 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 <i>posse +comitatus</i>; 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 <i>ex parte</i> 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 <i>habeas +corpus</i> 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.</p> + +<p>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 +<i>ipso facto</i> 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.</p> + +<div class="condensed"> +<p>See J.F. Rhodes, <i>History of the United States from the Compromise +of 1850</i>, vols. i. and ii. (New York, 1893); and M.G. M’Dougall, +<i>Fugitive Slaves, 1619-1865</i> (Boston, 1891).</p> +</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1c" id="ft1c" href="#fa1c"><span class="fn">1</span></a> 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, <i>The Underground Railroad</i> (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, <i>The Underground +Railroad from Slavery to Freedom</i> (New York, 1898).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUGLEMAN<a name="ar32" id="ar32"></a></span> (from the Ger. <i>Flügelmann</i>, the man on the +<i>Flügel</i> 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUGUE<a name="ar33" id="ar33"></a></span> (Lat. <i>fuga</i>, 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) <i>Fuga +ligata</i> was the exact reproduction by one or more voices of the +statement of a leading part. The reproducing voice (<i>comes</i>) +was seldom if ever written out, for all differences between it +and the <i>dux</i> were rigidly systematic; <i>e.g.</i> it was an exact inversion, +or exactly twice as slow, or to be sung backwards, &c. &c. +Hence, a rule or <i>canon</i> was given, often in enigmatic form, by +which the <i>comes</i> was deduced from the <i>dux</i>: and so the term +<i>canon</i> 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 <i>fuga ricercata</i> or simply a <i>ricercare</i>, a term which +is still used by Bach as a title for the fugues in <i>Das musikalische +Opfer</i>.</p> + +<p>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 +<span class="pagenum"><a name="page290" id="page290"></a>290</span> +preferred to treat the subject separately under the general +heading of <span class="sc"><a href="#artlinks">Contrapuntal Forms</a></span>, reserving only technical +terms for definition here.</p> + +<p>(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 <i>countersubject</i> (see <span class="sc"><a href="#artlinks">Counterpoint</a></span>, under sub-heading +<i>Double Counterpoint</i>). 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>i.e.</i> +fugues which begin with two parts and two subjects simultaneously, +and so also with <i>triple</i> and <i>quadruple fugues</i>.</p> + +<p>(ii.) <i>Episodes</i> are passages separating the entries of the subject.<a name="fa1d" id="fa1d" href="#ft1d"><span class="sp">1</span></a> +Episodes are usually developed from the material of the subject +and countersubjects; they are very rarely independent, but +then conspicuously so.</p> + +<p>(iii.) <i>Stretto</i>, the overlapping of subject and answer, is a resource +the possibilities of which may be exemplified by the setting of +the words <i>omnes generationes</i> in Bach’s <i>Magnificat</i> (see <span class="sc"><a href="#artlinks">Bach</a></span>).</p> + +<p>(iv.) The distinction between <i>real</i> and <i>tonal</i> 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 <i>on</i> the dominant, not <i>in</i> 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>i.e.</i> 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.</p> + +<p>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 <i>Wohltemp</i>. <i>Klav</i>. is subject, answer, answer, +subject; a departure from the usual rule according to which +subject and answer are strictly alternate in the exposition.</p> + +<p>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 <i>Paradise Lost</i>, book xi.</p> + +<table class="reg f90" summary="poem"><tr><td> <div class="poemr"> + <p class="i6">“His volant touch,</p> +<p>Instinct through all proportions, low and high,</p> +<p>Fled and pursued transverse the resonant fugue.”</p> +</div> </td></tr></table> + +<p class="noind">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.</p> +<div class="author">(D. F. T.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1d" id="ft1d" href="#fa1d"><span class="fn">1</span></a> An episode occurring during the exposition is sometimes called +<i>codetta</i>, a distinction the uselessness of which at once appears on +an analysis of Bach’s 2nd fugue in the <i>Wohltemp</i>. <i>Klav</i>. (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).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FÜHRICH, JOSEPH VON<a name="ar34" id="ar34"></a></span> (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. Führich has +been fairly described as a “Nazarene,” a romantic religious artist +whose pencil did more than any other to restore the old spirit +of Dürer 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. Führich’s +fame extended far beyond the walls of the Austrian capital, +and his illustrations to Tieck’s <i>Genofeva</i>, the Lord’s Prayer, +the Triumph of Christ, the Road to Bethlehem, the Succession +of Christ according to Thomas à 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. Führich became a pupil at the Academy +of Prague in 1816. His first inspiration was derived from the +prints of Dürer 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.</p> + +<div class="condensed"> +<p>His autobiography was published in 1875, and a memoir by his +son Lucas in 1886.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUJI<a name="ar35" id="ar35"></a></span> (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 <span class="sc"><a href="#artlinks">Japan</a></span>).</p> + + +<hr class="art" /> +<p><span class="bold">FU-KIEN<a name="ar36" id="ar36"></a></span> (formerly <span class="sc">Min</span>), 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. +<span class="pagenum"><a name="page291" id="page291"></a>291</span> +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 +£7668 and £278,160 respectively.</p> + + +<hr class="art" /> +<p><span class="bold">FUKUI<a name="ar37" id="ar37"></a></span>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUKUOKA<a name="ar38" id="ar38"></a></span>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FULA<a name="ar39" id="ar39"></a></span> (<span class="sc">Fulbe</span>, <span class="sc">Fellatah</span> or <span class="sc">Peuls</span>), 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,<a name="fa1e" id="fa1e" href="#ft1e"><span class="sp">1</span></a> 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.</p> + +<p>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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>See F. Ratzel, <i>History of Mankind</i> (English ed., London, 1896-1898); +Sir F. Lugard, “Northern Nigeria,” in <i>Geographical Journal</i> +(July 1904); Grimai de Guirodon, <i>Les Puls</i> (1887); E.A. Brackenbury, +<i>A Short Vocabulary of the Fulani Language</i> (Zungeru, 1907); +the articles <span class="sc"><a href="#artlinks">Nigeria</a></span> and <span class="sc"><a href="#artlinks">Sokoto</a></span> and authorities there cited.</p> +</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1e" id="ft1e" href="#fa1e"><span class="fn">1</span></a> 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.</p> +</div> + +<p><span class="pagenum"><a name="page292" id="page292"></a>292</span></p> + + +<hr class="art" /> +<p><span class="bold">FULCHER<a name="ar40" id="ar40"></a></span> (or <span class="sc">Foucher</span>) <b>OF CHARTRES</b> (1058-<i>c.</i> 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 +<i>Historia Hierosolymitana</i> or <i>Gesta Francorum Jerusalem expugnantium</i>, +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-Père-en-Vallée +at Chartres, and also with another person of the same +name who distinguished himself at the siege of Antioch in +1098.</p> + +<div class="condensed"> +<p>The <i>Historia</i>, but in an incomplete form, was first published by +J. Bongars in the <i>Gesta Dei per Francos</i> (Hanover, 1611). The best +edition is in tome iii. of the <i>Recueil des historiens des croisades, +Historiens occidentaux</i> (Paris, 1866); and there is a French translation +in tome xxiv. of Guizot’s <i>Collection des mémoires relatifs à +l’histoire de France</i> (Paris, 1823-1835).</p> + +<p>See H. von Sybel, <i>Geschichte des ersten Kreuzzuges</i> (Leipzig, 1881); +and A. Molinier, <i>Les Sources de l’histoire de France</i>, tome ii. (Paris, +1902).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULDA<a name="ar41" id="ar41"></a></span>, a town and episcopal see of Germany, in the Prussian +province of Hesse-Nassau, between the Rhön 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 <i>Schloss</i>, the <i>Bibliothek</i>, 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.</p> + +<p>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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>For the town see A. Hartmann, <i>Zeitgeschichte von Fulda</i> (Fulda, +1895); J. Schneider, <i>Führer durch die Stadt Fulda</i> (Fulda, 1899); +and <i>Chronik von Fulda und dessen Umgebungen</i> (1839). For the +history of the abbey see Gegenbaur, <i>Das Kloster Fulda im Karolinger +Zeitalter</i> (Fulda, 1871-1874); Arndt, <i>Geschichte des Hochstifts Fulda</i> +(Fulda, 1860); and the <i>Fuldaer Geschichtsblätter</i> (1902 fol.).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULGENTIUS, FABIUS PLANCIADES<a name="ar42" id="ar42"></a></span>, Latin grammarian, +a native of Africa, flourished in the first half of the 6th (or the +last part of the 5th) century <span class="scs">A.D.</span> 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) <i>Mythologiarum libri iii.</i>, 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) <i>Expositio Vergilianae +continentiae</i> (<i>continentia</i> = contents), a sort of appendix to (1), +dedicated to Catus. The poet himself appears to the author and +explains the twelve books of the <i>Aeneid</i> as a picture of human +life. The three words <i>arma</i> (= virtus), <i>vir</i> (= sapientia), <i>primus</i> +(= princeps) in the first line represent respectively <i>substantia +corporalis, sensualis, ornans</i>. 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) <i>Expositio sermonum antiquorum</i>, 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) <i>Liber absque +litteris de aetatibus mundi et hominis</i>. 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 <i>Absque litteris</i> 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 <i>Physiologus</i>, 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.</p> + +<p><span class="pagenum"><a name="page293" id="page293"></a>293</span></p> + +<div class="condensed"> +<p>See the edition of the four works by R. Helm (1898, Teubner +series); also M. Zink, <i>Der Mytholog Fulgentius</i> (1867); E. Jungmann, +“De Fulgentii aetate et scriptis,” in <i>Acta Societatis Philologae +Lipsiensis</i>, i. (1871); A. Ebert, <i>Allgemeine Geschichte der Litt. des +Mittelalters</i>, i.; article “Fulgentius” by C.F. Böhr in Ersch and +Gruber’s <i>Allgemeine Encyklopädie</i>; Teuffel-Schwabe, <i>History of +Roman Literature</i> (Eng. trans.).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULGINIAE<a name="ar43" id="ar43"></a></span> (mod. <i>Foligno</i>), 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 <span class="scs">B.C.</span> (See <span class="sc"><a href="#artlinks">Foligno</a></span> and <span class="sc"><a href="#artlinks">Flaminia</a></span>, +<span class="sc">Via</span>.)</p> +<div class="author">(T. As.)</div> + + +<hr class="art" /> +<p><span class="bold">FULGURITE<a name="ar44" id="ar44"></a></span> (from Lat. <i>fulgur</i>, 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.</p> + +<p>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.</p> +<div class="author">(J. S. F.)</div> + + +<hr class="art" /> +<p><span class="bold">FULHAM<a name="ar45" id="ar45"></a></span>, 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, <span class="correction" title="amended from coverging">converging</span> 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.</p> + +<p>Fulham, or in its earliest form <i>Fullanham</i>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FULK<a name="ar46" id="ar46"></a></span>, 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 +<span class="pagenum"><a name="page294" id="page294"></a>294</span> +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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>His reign in Jerusalem is narrated by R. Röhricht (<i>Geschichte des +Königreichs Jerusalem</i>, Innsbruck, 1898), and has been made the +subject of a monograph by G. Dodu (<i>De Fulconis Hierosolymitani +regno</i>, Paris, 1894).</p> +</div> +<div class="author">(E. Br.)</div> + + +<hr class="art" /> +<p><span class="bold">FULK<a name="ar47" id="ar47"></a></span> (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.</p> + +<div class="condensed"> +<p>Fulk left some letters, which are collected in Migne, <i>Patrologia +Latina</i>, vol. cxxxi. 11-14.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULKE, WILLIAM<a name="ar48" id="ar48"></a></span> (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 <i>A Defense of the sincere +true Translations of the holie Scriptures into the English +tong</i> (London, 1583), and confutations of Thomas Stapleton +(1535-1598), Cardinal Allen and other Roman Catholic +controversialists.</p> + + +<hr class="art" /> +<p><span class="bold">FULK NERRA<a name="ar49" id="ar49"></a></span> (<i>c.</i> 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 <span class="sc"><a href="#artlinks">Anjou</a></span>). 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 (<i>c.</i> 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, <i>c.</i> 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 Vénérable, count of Vendôme, he had a daughter, +Adela, who married Boon of Nevers and transmitted to her +children the countship of Vendôme. Elizabeth having died in +1000, Fulk married Hildegarde of Lorraine, by whom he had a +son, Geoffrey Martel (<i>q.v.</i>), and a daughter Ermengarde, who +married Geoffrey, count of Gâtinais, and was the mother of +Geoffrey “le Barbu” (the Bearded) and of Fulk “le Réchin” +(see <span class="sc"><a href="#artlinks">Anjou</a></span>).</p> + +<div class="condensed"> +<p>See Louis Halphen, <i>Le Comté d’Anjou au XI<span class="sp">e</span> siècle</i> (Paris, 1906). +The biography of Fulk Nerra by Alexandre de Salies, <i>Histoire de +Foulques Nerra</i> (Angers, 1874) is confused and uncritical. A very +summary biography is given by Célestin Port, <i>Dictionnaire historique, +géographique et biographique de Maine-et-Loire</i> (3 vols., Paris-Angers, +1874-1878), vol. ii. pp. 189-192, and there is also a sketch in Kate +Norgate, <i>England under the Angevin Kings</i> (2 vols., London, 1887), +vol. i. ch. iii.</p> +</div> +<div class="author">(L. H.*)</div> + + +<hr class="art" /> +<p><span class="bold">FÜLLEBORN, GEORG GUSTAV<a name="ar50" id="ar50"></a></span> (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 <i>De Xenophane, Zenone et Gorgia</i>. 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 <i>Politics</i> of Aristotle (1799-1800), and a large +share in the <i>Beiträge zur Geschichte der Philosophie</i> (published in +twelve parts between 1791 and 1799), in which he collaborated +with Forberg, Reinhold and Niethammer. In philology he +wrote <i>Encyclopaedia philologica sive primae lineae Isagoges in +antiquorum studia</i> (1798; 2nd ed., 1805); <i>Kurze Theorie des +lateinischen Stils</i> (1793); <i>Leitfaden der Rhetorik</i> (1802); and an +annotated edition of the <i>Satires</i> of Persius. Under the pseudonym +“Edelwald Justus” he published several collections of popular +tales—<i>Bunte Blätter</i> (1795); <i>Kleine Schriften zur Unterhaltung</i> +(1798); <i>Nebenstunden</i> (1799). After his death were published +<i>Taschenbuch für Brunnengäste</i> (1806) and <i>Kanzelreden</i> (1807). +He was a frequent contributor to the press, where his writings +were very popular.</p> + +<div class="condensed"> +<p>See Schummel, <i>Gedächtnisrede</i> (1803) and <i>Garve und Fülleborn</i>; +Meusel, <i>Gelehrtes Teutschland</i>, vol. ii.</p> +</div> + +<p><span class="pagenum"><a name="page295" id="page295"></a>295</span></p> + + +<hr class="art" /> +<p><span class="bold">FULLER, ANDREW<a name="ar51" id="ar51"></a></span> (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 <i>The Gospel worthy of all Acceptation</i>, +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, <i>The Calvinistic and Socinian +systems examined and compared as to their moral tendency</i>, 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 <i>The Gospel its own Witness</i>, is regarded as the production +on which his reputation as a theologian mainly rests. Fuller +also published an admirable <i>Memoir of the Rev. Samuel Pearce</i>, +of Birmingham, and a volume of <i>Expository Lectures in Genesis</i>, +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.</p> + +<div class="condensed"> +<p>Several editions of his collected works have appeared, and a +<i>Memoir</i>, 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).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULLER, GEORGE<a name="ar52" id="ar52"></a></span> (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.”</p> + + +<hr class="art" /> +<p><span class="bold">FULLER, MARGARET<a name="ar53" id="ar53"></a></span>, 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 Molière. 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, Körner, Novalis +and Schiller.</p> + +<p>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 <i>Conversations with +Goethe</i>, which was followed in 1842 by a translation of the correspondence +between Karoline von Günderode and Bettina von +Arnim, entitled <i>Günderode</i>. Aided by R.W. Emerson and +George Ripley, she in 1840 started <i>The Dial</i>, 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 <i>The Tribune</i>, and in 1846 she published a selection +from her articles on contemporary authors in Europe and +America, under the title <i>Papers on Literature and Art</i>. 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 +<span class="pagenum"><a name="page296" id="page296"></a>296</span> +Island beach on the 16th of June, and the Ossolis were among +the passengers who perished.</p> + +<div class="condensed"> +<p><i>Life Without and Life Within</i> (Boston, 1860) is a collection of +essays, poems, &c., supplementary to her <i>Collected Works</i>, printed +in 1855. See the <i>Autobiography of Margaret Fuller Ossoli</i>, with +additional memoirs by J.F. Clarke, R.W. Emerson and W.H. +Channing (2 vols., Boston, 1852); also <i>Margaret Fuller (Marchesa +Ossoli)</i>, by Julia Ward Howe (1883), in the “Eminent Women” +series; <i>Margaret Fuller Ossoli</i> (Boston, 1884), by Thomas Wentworth +Higginson in the “American Men of Letters” series, which is +based largely on unedited material; and <i>The Love Letters of Margaret +Fuller, 1845-1846</i> (London and New York, 1903), with an introduction +by Julia Ward Howe.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULLER, MELVILLE WESTON<a name="ar54" id="ar54"></a></span> (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, <i>The +Age</i>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FULLER, THOMAS<a name="ar55" id="ar55"></a></span> (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.</p> + +<p>Fuller’s quaint and humorous oratory soon attracted attention. +He published in 1631 a poem on the subject of David and +Bathsheba, entitled <i>David’s Hainous Sinne, Heartie Repentance, +Heavie Punishment</i>. 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 <i>The Historie of the Holy Warre</i> (1639), +a history of the crusades, and <i>The Holy State and the Prophane +State</i> (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 <i>Church +History</i> a valuable account of the proceedings of this synod, +for sitting in which he was fined £200, which, however, was never +exacted. His first published volume of sermons appeared in +1640 under the title of <i>Joseph’s party-coloured Coat</i>, 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.</p> + +<p>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 <span class="correction" title="added rector">rector</span> of Sidney Sussex College, edited +the <i>Worthies of England</i>, 1662, and became rector of Great +Wakering, Essex, where he died in 1687.</p> + +<p>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 <i>Appeal of Injured Innocence</i> +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, <i>Life of Thomas Fuller</i>, pp. 245 <i>et seq.</i>). +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.</p> + +<p>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 +<i>Jacob’s Vow</i>.</p> + +<p>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 <i>Church History</i>, “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, +<span class="pagenum"><a name="page297" id="page297"></a>297</span> +from churches, old buildings, and the conversation of ancient +gossips, for his <i>Church-History</i> and <i>Worthies of England</i>. He +compiled in 1645 a small volume of prayers and meditations,—the +<i>Good Thoughts in Bad Times</i>,—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. <i>The Fear +of losing the Old Light</i> (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 +<i>Andronicus, or the Unfortunate Politician</i> (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 <i>Good Thoughts in Worse +Times</i>, 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 <i>Cause and Cure of a Wounded Conscience</i> (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.</p> + +<p>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 <i>Annales</i> 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 <i>A Pisgah-Sight of +Palestine</i> (1650), and his <i>Church-History of Britain</i> (1655), from +the birth of Jesus Christ until the year 1648. With the <i>Church-History</i> +was printed <i>The History of the University of Cambridge +since the Conquest</i> and <i>The History of Waltham Abbey</i>. 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 +<i>Church-History</i> 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 <i>Terrae filius</i>, +lampooned Fuller, whom he described in this <i>Oratio</i> as living +in London, ever scribbling and each year bringing forth new +<i>folia</i> like a tree. At length, continues South, the <i>Church-History</i> +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.</p> + +<p>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 <i>Examen Historicum</i>, called <i>The +Appeal of Injured Innocence</i> (1659), was inscribed. At the end +of the <i>Appeal</i> 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 +<i>Peter</i>,” he asked, “fall out with <i>Thomas</i>, 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.”</p> + +<p>In <i>An Alarum to the Counties of England and Wales</i> (1660) +Fuller argued for a free and full parliament—free from force, +as he expressed it, as well as from abjurations or previous +engagements. <i>Mixt Contemplations in Better Times</i> (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. <i>A Panegyrick to His Majesty on his +Happy Return</i> 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 <i>The Worthies</i>) +to give immortality to others, he himself attained it.</p> + +<p>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 (<i>Hist. of +England</i>, 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.</p> + +<p>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” (<i>Literary Remains</i>, vol. ii. (1836), pp. 389-390). This +opinion was formed after the perusal of the <i>Church-History</i>. +That work and <i>The History of the Worthies of England</i> 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 <i>Holy State</i> 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 +<span class="pagenum"><a name="page298" id="page298"></a>298</span> +Lamb’s time, mainly through the appreciative criticisms of +S.T. Coleridge, Robert Southey and others, Fuller’s works have +received much attention.</p> + +<div class="condensed"> +<p>There is an elaborate account of the life and writings of Fuller +by William Oldys in the <i>Biographia Britannica</i>, vol. iii. (1750), based +on Fuller’s own works and the anonymous <i>Life of ... Dr Thomas +Fuller</i> (1661; reprinted in a volume of selections by A.L.J. Gosset, +1893). The completest account of him is <i>The Life of Thomas Fuller, +with Notices of his Books, his Kinsmen and his Friends</i> (1874), by +J.E. Bailey, who gives a detailed bibliography (pp. 713-762) of his +works. <i>The Worthies of England</i> was reprinted by John Nichols +(1811) and by P.A. Nuttall (1840). His <i>Collected Sermons</i> 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.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULLER, WILLIAM<a name="ar56" id="ar56"></a></span> (1670-<i>c.</i> 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 <i>Life of +William Fuller</i> and some <i>Original Letters of the late King James</i>. +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 <i>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</i> +(1704); and <i>An humble appeal to the impartial judgment of all +parties in Great Britain</i> (1716).</p> + +<div class="condensed"> +<p>He must be distinguished from <span class="sc">William Fuller</span> (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 (<i>c.</i> 1580-1659), dean of Ely and later dean of Durham.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULLER’S EARTH<a name="ar57" id="ar57"></a></span> (Ger. <i>Walkererde</i>, Fr. <i>terre à foulon</i>, <i>argile +smectique</i>)—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.</p> + +<div class="condensed"> +<p>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:—</p> + +<p class="pt2 center">Blue Earth (dried at 100° C.).</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Insoluble residue</td> <td class="tcr rb">69.96</td> <td class="tcl">Insoluble residue—</td> <td class="tcr"> </td></tr> +<tr><td class="tcl">Fe<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr rb">2.48</td> <td class="tcl">SiO<span class="su">2</span></td> <td class="tcr">62.81</td></tr> +<tr><td class="tcl">Al<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr rb">3.46</td> <td class="tcl">Al<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr">3.46</td></tr> +<tr><td class="tcl">CaO</td> <td class="tcr rb">5.87</td> <td class="tcl">Fe<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr">1.30</td></tr> +<tr><td class="tcl">MgO</td> <td class="tcr rb">1.41</td> <td class="tcl">CaO</td> <td class="tcr">1.53</td></tr> +<tr><td class="tcl">P<span class="su">2</span>O<span class="su">5</span></td> <td class="tcr rb">0.27</td> <td class="tcl">MgO</td> <td class="tcr">0.86</td></tr> +<tr><td class="tcl">SO<span class="su">3</span></td> <td class="tcr rb">0.05</td> <td class="tcl"> </td> <td class="tcr">———</td></tr> +<tr><td class="tcl">NaCl</td> <td class="tcr rb">0.05</td> <td class="tcr"> </td> <td class="tcr">69.96</td></tr> +<tr><td class="tcl">K<span class="su">2</span>O</td> <td class="tcr rb">0.74</td> <td class="tcl"> </td> <td class="tcr">———</td></tr> +<tr><td class="tcl">H<span class="su">2</span>O (combined)</td> <td class="tcr rb">15.57</td> <td class="tcl"> </td></tr> +<tr><td class="tcl"> </td> <td class="tcr rb">———</td> <td colspan="2"> </td></tr> +<tr><td class="tcl"> </td> <td class="tcr rb">99.86</td> <td colspan="2"> </td></tr> +<tr><td class="tcl"> </td> <td class="tcr rb">———</td> <td colspan="2"> </td></tr> +</table> + +<p class="pt1 center">Yellow Earth (dried at 100° C.).</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Insoluble residue</td> <td class="tcr rb">76.13</td> <td class="tcl">Insoluble residue—</td> <td class="tcr"> </td></tr> +<tr><td class="tcl">Fe<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr rb">2.41</td> <td class="tcl">SiO<span class="su">2</span></td> <td class="tcr">59.37</td></tr> +<tr><td class="tcl">Al<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr rb">1.77</td> <td class="tcl">Al<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr">10.05</td></tr> +<tr><td class="tcl">CaO</td> <td class="tcr rb">4.31</td> <td class="tcl">Fe<span class="su">2</span>O<span class="su">3</span></td> <td class="tcr">3.86</td></tr> +<tr><td class="tcl">MgO</td> <td class="tcr rb">1.05</td> <td class="tcl">CaO</td> <td class="tcr">1.86</td></tr> +<tr><td class="tcl">P<span class="su">2</span>O<span class="su">5</span></td> <td class="tcr rb">0.14</td> <td class="tcl">MgO</td> <td class="tcr">1.04</td></tr> +<tr><td class="tcl">SO<span class="su">3</span></td> <td class="tcr rb">0.07</td> <td class="tcl"> </td> <td class="tcr">———</td></tr> +<tr><td class="tcl">NaCl</td> <td class="tcr rb">0.14</td> <td class="tcl"> </td> <td class="tcr">76.18</td></tr> +<tr><td class="tcl">K<span class="su">2</span>O</td> <td class="tcr rb">0.84</td> <td class="tcl"> </td> <td class="tcr">———</td></tr> +<tr><td class="tcl">H<span class="su">2</span>O (combined)</td> <td class="tcr rb">13.19</td> <td colspan="2"> </td></tr> +<tr><td class="tcl"> </td> <td class="tcr rb">———</td> <td colspan="2"> </td></tr> +<tr><td class="tcl"> </td> <td class="tcr rb">100.05</td> <td colspan="2"> </td></tr> +<tr><td class="tcl"> </td> <td class="tcr rb">———</td> <td colspan="2"> </td></tr> +</table> + +<p class="noind">(Analysis by P.G. Sanford, <i>Geol. Mag.</i>, 1889, 6, pp. 456, 526.)</p> + +<p>Of other published analyses, not a few show a lower silica content +(44%, 50%), along with a higher proportion of alumina (11%, 23%).</p> +</div> + +<p>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.</p> + +<p>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.</p> + +<p>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 <i>Perisphinctes arbustigerus</i> +and <i>Macrocephalus subcontractus</i> with <i>Ostrea acuminata</i>, +<i>Rhynchonella concinna</i> and <i>Goniomya angulifera</i>. The formation +is in part the equivalent of the “Vesulien” of J. Marcou (Vesoul +in Haute-Saône). 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.</p> + +<div class="condensed"> +<p>See H.B. Woodward, “Jurassic Rocks of Great Britain,” vol. +iv. (1894), <i>Mem. Geol. Survey</i> (London).</p> +</div> +<div class="author">[J. A. H.]</div> + + +<hr class="art" /> +<p><span class="bold">FULLERTON, LADY GEORGIANA CHARLOTTE<a name="ar58" id="ar58"></a></span> (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 +<span class="pagenum"><a name="page299" id="page299"></a>299</span> +her husband entered the Roman Catholic church, and in the +following year Lady Georgiana Fullerton published her first novel, +<i>Ellen Middleton</i>, which attracted W.E. Gladstone’s attention +in the <i>English Review</i>. 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 <i>Lady Georgiana Fullerton, sa +vie et ses œuvres</i> (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 <i>Grantley Manor</i> +(1847), <i>Lady Bird</i> (1852), and <i>Too Strange not to be True</i> (1864).</p> + + +<hr class="art" /> +<p><span class="bold">FULMAR<a name="ar59" id="ar59"></a></span>, from the Gaelic <i>Fulmaire</i>, the <i>Fulmarus glacialis</i> of +modern ornithologists, one of the largest of the petrels (<i>Procellariidae</i>) +of the northern hemisphere, being about the size of the +common gull (<i>Larus canus</i>) 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° N., but on the American side comes +habitually to lat. 45° or even lower. In the Pacific it is represented +by a scarcely separable form, <i>F. glupischa</i>. 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 (<i>Birds of the West of Scotland</i>, 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”<a name="fa1f" id="fa1f" href="#ft1f"><span class="sp">1</span></a> +(corrupted from <i>Mallemuck</i>), 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.</p> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1f" id="ft1f" href="#fa1f"><span class="fn">1</span></a> A name misapplied in the southern hemisphere to <i>Diomedea +melanophrys</i>, one of the albatrosses.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULMINIC ACID<a name="ar60" id="ar60"></a></span>, HCNO or H<span class="su">2</span>C<span class="su">2</span>N<span class="su">2</span>O<span class="su">2</span>, 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, CH<span class="su">2</span> : 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 ½H<span class="su">2</span>O) 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 +(Wöhler & Matter, Brit. Pat. 4468 of 1908), and by lead azoimide +(Hyronimus, Brit. Pat. 1819 of 1908).</p> + +<div class="condensed"> +<p>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)<span class="su">2</span>, since it gave mixed and acid salts. Kekulé, +in 1858, concluded that it was nitroacetonitrile, NO<span class="su">2</span>·CH<span class="su">2</span>·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:</p> + +<div class="center ptb2"><img style="width:500px; height:62px; vertical-align: middle;" src="images/img299a.jpg" alt="" /></div> + +<p class="noind">The formulae of Kekulé, 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. Wöhler and K. Theodorovits (<i>Ber.</i>, 1905, 38, p. 345), that only +compounds containing two carbon atoms yielded fulminates, points +to (HCNO)<span class="su">2</span>; on the other hand, Wöhler (<i>loc. cit.</i> 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</p> + +<div class="center ptb2"><img style="width:600px; height:45px; vertical-align: middle;" src="images/img299b.jpg" alt="" /></div> + +<p class="noind">and also on the production from sodium nitromethane and mercuric +chloride, thus CH<span class="su">2</span> : NO·Ohg → H<span class="su">2</span>O + C : NOhg(hg = ½Hg). H. +Wieland and F.C. Palazzo (1907) support this formula, finding that +methyl nitrolic acid, NO<span class="su">2</span>·CH : N·OH, yielded under certain conditions +fulminic acid, and vice versa (Palazzo, 1907). M.Z. Jowitschitsch +(<i>Ann.</i>, 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 (<i>q.v.</i>), and have shown that a close relationship +exists between the nitrile oxides, furoxane, and fulminic acid (see +<i>Ann. Rep.</i>, London Chem. Soc., 1909, p. 84). <i>Fulminuric acid</i>, +(HCNO)<span class="su">3</span>, obtained by Liebig by boiling mercuric fulminate with +water, was synthesized in 1905 by C. Ulpiani and L. Bernardini +(<i>Gazetta</i>, iii. 35, p. 7), who regard it as NO<span class="su">2</span>·CH(CN)·CO·NH<span class="su">2</span>. It +deflagrates at 145°, and forms a characteristic cuprammonium salt.</p> + +<p>The early history of mercuric fulminate and a critical account of its +application as a detonator is given in <i>The Rise and Progress of the +British Explosives Industry</i> (International Congress of Applied +Chemistry, 1909). The manufacture and modern aspects are treated +in Oscar Guttmann, <i>The Manufacture of Explosives</i>, and <i>Manufacture +of Explosives, Twenty Years’ Progress</i> (1909).</p> +</div> + +<p><span class="pagenum"><a name="page300" id="page300"></a>300</span></p> + + +<hr class="art" /> +<p><span class="bold">FULTON, ROBERT<a name="ar61" id="ar61"></a></span> (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 <i>Treatise on the Improvement of Canal Navigation</i>. 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 +(<i>q.v.</i>), 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.</p> + +<div class="condensed"> +<p>See C.D. Colden, <i>Life of Robert Fulton</i> (New York, 1817); Robert +H. Thurston, <i>History of the Growth of the Steam-Engine</i> (New York, +1878); George H. Preble, <i>Chronological History of Steam Navigation</i> +(Philadelphia, 1883); and Mrs A.C. Sutcliffe, <i>Robert Fulton and the +Clermont</i> (New York, 1909).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FULTON<a name="ar62" id="ar62"></a></span>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FULTON<a name="ar63" id="ar63"></a></span>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUM<a name="ar64" id="ar64"></a></span>, or <span class="sc">Funj Hwang</span>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUMARIC AND MALEIC ACIDS<a name="ar65" id="ar65"></a></span>, two isomeric unsaturated +acids of composition C<span class="su">4</span>H<span class="su">4</span>O<span class="su">4</span>. <i>Fumaric acid</i> is found in fumitory +(<i>Fumaria officinalis</i>), in various fungi (<i>Agaricus piperatus</i>, &c.), +and in Iceland moss. It is obtained by heating malic acid alone +to 150° C., or by heating it with hydrochloric acid (V. Dessaignes, +<i>Jahresb</i>., 1856, p. 463) or with a large quantity of hydrobromic +acids (A. Kekulé, <i>Ann.</i>, 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, +<i>Ann.</i>, 1883, 218, p. 169); by the cyanide synthesis from acetylene +di-iodide; and by heating maleic acid to 210° C. (Z. Skraup, +<i>Monats. f. Chemie</i>, 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° 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. Kekulé and R. Anschutz, <i>Ber.</i>, 1881, 14, +p. 713). By long-continued heating with caustic soda at 100° C. +it is converted into inactive malic acid.</p> + +<p><i>Maleic acid</i> is obtained by distilling malic or fumaric acids; +by heating fumaric acid with acetyl chloride to 100° C; or by +the hydrolysis of trichlorphenomalic acid (β-trichloraceto-acrylic +acid) [A. Kekulé, <i>Ann.</i>, 1884, 223, p. 185]. It crystallizes +in monoclinic prisms, which are easily soluble in water, melt +at 130° C., and boil at 160° 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° C. and +boil at 196° C.</p> + +<div class="condensed"> +<p>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, <i>Ann.</i>, 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 [”Über +die räumliche Anordnung der Atome,” &c., <i>Trans, of the Saxon Acad. +of Sciences</i> (Math. Phys. Section), 1887, p. 14] by an extension of +the van’t Hoff hypothesis (see <span class="sc"><a href="#artlinks">Stereo-Isomerism</a></span>). The formulae +of the acids are written thus:</p> + +<div class="center ptb2"><img style="width:500px; height:43px; vertical-align: middle;" src="images/img300.jpg" alt="" /></div> + +<p class="noind">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.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUMAROLE<a name="ar66" id="ar66"></a></span>, a vent from which volcanic vapours issue, +named indirectly from the Lat. <i>fumariolum</i>, a smoke-hole. +<span class="pagenum"><a name="page301" id="page301"></a>301</span> +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° C., and +consist chiefly of anhydrous chlorides, notably sodium chloride. +The acid fumaroles yield vapours of lower temperature (300° to +400°) containing much water vapour, with hydrogen chloride +and sulphur dioxide. The alkaline fumaroles are still cooler, +though above 100°, and evolve ammonium chloride with other +vapours. Cold fumaroles, below 100°, discharge principally +aqueous vapour, with carbon dioxide, and perhaps hydrogen +sulphide. The fumaroles of Mont Pelé 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°. +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 <span class="sc"><a href="#artlinks">Volcano</a></span>).</p> + + +<hr class="art" /> +<p><span class="bold">FUMIGATION<a name="ar67" id="ar67"></a></span> (from Lat. <i>fumigare</i>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUMITORY,<a name="ar68" id="ar68"></a></span> in botany, the popular name for the British +species of <i>Fumaria</i>, 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. <i>F. capreolata</i> +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 (<i>Corydalis +cava</i> or <i>tuberosa</i>) is known as <i>radix aristolochia</i>, 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 <i>Fumaria officinalis</i> and <i>F. racemosa</i> is +used in China under the name of <i>Tsze-hwa-ti-ting</i> 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, <i>Contrib. towards the +Mat. Medica ... of China</i>, p. 99, 1871). The name fumitory, +Latin <i>fumus terrae</i>, has been supposed to be derived from the +fact that its juice irritates the eyes like smoke (see Fuchs, <i>De +historia stirpium</i>, p. 338, 1542); but <i>The Grete Herball</i>, cap. +clxix., 1529, fol., following the <i>De simplici medicina</i> of Platearius, +fo. xciii. (see in <i>Nicolai Praepositi dispensatorium ad aromatarios</i>, +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.”</p> + + +<hr class="art" /> +<p><span class="bold">FUNCHAL<a name="ar69" id="ar69"></a></span>, the capital of the Portuguese archipelago of the +Madeiras; on the south coast of Madeira, in 32° 37′ N. and +16° 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 (<i>quintas</i>), 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 <span class="sc"><a href="#artlinks">Madeira</a></span>). 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 <span class="sc"><a href="#artlinks">Madeira</a></span>). +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).</p> + + +<hr class="art" /> +<p><span class="bold">FUNCTION<a name="ar70" id="ar70"></a></span>,<a name="fa1ga" id="fa1ga" href="#ft1ga"><span class="sp">1</span></a> 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.</p> + +<p class="pt2 center sc">I. Functions of Real Variables</p> + +<p>1. <i>Historical.</i>—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” +(<i>Leibnizens mathematische Schriften</i>, 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, <i>Opera</i>, 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” (<i>Linea functionis</i>). In 1718 +John Bernoulli (<i>Opera</i>, 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 ƒ(x) for a function of a variable +x was introduced by Leonhard Euler in 1734 (<i>Comm. Acad. +Petropol.</i> 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 ax² + bx + c, or +rational algebraic functions which are not integral, such as</p> + +<table class="math0" summary="math"> + +<tr><td>a<span class="su">1</span>x<span class="sp">n</span> + a<span class="su">2</span>x<span class="sp">n−1</span> + ... + a<span class="su">n</span></td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">b<span class="su">1</span>x<span class="sp">m</span> + b<span class="su">2</span>x<span class="sp">m−1</span> + ... + b<span class="su">m</span></td></tr></table> + +<p class="noind">or irrational algebraic functions, such as √x, or, more generally +the algebraic functions that are determined implicitly by an +algebraic equation, as, for instance,</p> + +<p class="center">ƒ<span class="su">n</span>(x, y) + ƒ<span class="su">n−1</span>(x, y) + ... + ƒ<span class="su">0</span> = 0</p> + +<p><span class="pagenum"><a name="page302" id="page302"></a>302</span></p> + +<p class="noind">where ƒ<span class="su">n</span>(x, y), ... mean homogeneous expressions in x and y +having constant coefficients, and having the degrees indicated +by the suffixes, and ƒ<span class="su">0</span> is a constant. Or again we may have +trigonometrical functions, such as sin x and tan x, or inverse +trigonometrical functions, such as sin<span class="sp">−1</span>x, or exponential functions, +such as e<span class="sp">x</span> and a<span class="sp">x</span>, 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/(x² + y²), 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.</p> + +<p>2. <i>Graphic Representation.</i>—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 § 14). In like manner a function +of two variables defines a surface.</p> + +<p>3. <i>The Variable.</i>—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 <i>ordinal</i> numbers (see <span class="sc"><a href="#artlinks">Number</a></span>). 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.</p> + +<p>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 <i>continuum</i>. In fact the notion of a one-dimensional +continuum first becomes precise in virtue of the +establishment of the system of real numbers.</p> + +<p>4. <i>Domain of a Variable.</i>—<i>Theory of Aggregates.</i>—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 <i>aggregate</i>, 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.”</p> + +<div class="condensed"> +<p>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” (<i>Mengenlehre, Théorie des ensembles, +Theory of sets of points</i>). 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.</p> +</div> + +<p>5. <i>General Notion of Functionality.</i>—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 (§ 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.</p> + +<div class="condensed"> +<p>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>i.e.</i> 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, <i>e.g.</i> 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.</p> + +<p>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 +<span class="pagenum"><a name="page303" id="page303"></a>303</span> +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.</p> +</div> + +<p>6. <i>Limits.</i>—Let ƒ(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 ε, however small, has been +specified, it is possible to find a positive number h, so that +|L − ƒ(x)| < ε for all points x of the domain (other than a) for +which |x − a| < h, then L is the “limit of ƒ(x) at the point a.” +The condition for the existence of L is that, after the positive +number ε has been specified, it must be possible to find a positive +number h, so that |ƒ(x′) − ƒ(x)| < ε for all points x and x′ of +the domain (other than a) for which |x − a| < h and |x′ − a| < h.</p> + +<p>It is a fundamental theorem that, when this condition is +satisfied, there exists a perfectly definite number L which is the +limit of ƒ(x) at the point a as defined above. The limit of ƒ(x) +at the point a is denoted by Lt<span class="su">x=a</span>ƒ(x), or by +lim<span class="su">x=a</span>ƒ(x).</p> + +<div class="condensed"> +<p>If ƒ(x) is a function of one variable x in a domain which extends +to infinite values, and if, after ε has been specified, it is possible to +find a number N, so that |ƒ(x′) − ƒ(x)| < ε 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 |ƒ(x) − L| < ε for all such values of x. +In this case ƒ(x) has a limit L at x = ∞. In like manner ƒ(x) may +have a limit at x = −∞. 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,” u<span class="su">1</span>, u<span class="su">2</span>, ... +u<span class="su">n</span>, ..., and this sequence can have a limit at n = ∞.</p> + +<p>The principle common to the above definitions and theorems is +called, after P. du Bois Reymond, “the general principle of convergence +to a limit.”</p> + +<p>It must be understood that the phrase “x = ∞” 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.”</p> + +<p>A very important type of limits is furnished by <i>infinite series</i>. +When a sequence of numbers u<span class="su">1</span>, u<span class="su">2</span>, ... u<span class="su">n</span>, ... is given, we may +form a new sequence s<span class="su">1</span>, s<span class="su">2</span>, ... s<span class="su">n</span>, ... from it by the rules s<span class="su">1</span> = u<span class="su">1</span>, +s<span class="su">2</span> = u<span class="su">1</span> + u<span class="su">2</span>, ... s<span class="su">n</span> = u<span class="su">1</span> + u<span class="su">2</span> + ... + u<span class="su">n</span> or by the equivalent rules +s<span class="su">1</span> = u, s<span class="su">n</span> − s<span class="su">n−1</span> = u<span class="su">n</span>(n = 2, 3, ...). If the new sequence has a limit +at n = ∞, this limit is called the “sum of the infinite series” +u<span class="su">1</span> + u<span class="su">2</span> + ..., and the series is said to be “convergent” (see +<span class="sc"><a href="#artlinks">Series</a></span>).</p> + +<p>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/π, 1/2π, ... 1/nπ, ... or to the aggregate 1/2π, +2/5π, ... n/(n<span class="sp">2</span> + 1)π, ..., 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 +ƒ(a − 0) and ƒ(a + 0).</p> + +<p>The limit L of ƒ(x) at a stands in no necessary relation to the value +of ƒ(x) at a. If the point a is in the domain of the argument, the +value of ƒ(x) at a is assigned by the rule of calculation, and may be +different from L. In case ƒ(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 ƒ(x) at a. In the case where ƒ(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 ƒ(a) the value L; the function may then be said to +be “extrinsically defined.” The so-called “indeterminate forms” +(see <span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>) are examples.</p> +</div> + +<p>7. <i>Superior and Inferior Limits; Infinities.</i>—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).</p> + +<p>If a number can be found which is greater than every value +of the function, then either (α) there is one value of the function +which exceeds all the others, or (β) there is a number S which +exceeds every value of the function but is such that, however +small a positive number ε we take, there are values of the function +which exceed S − ε. In the case (α) the function has a greatest +value; in case (β) 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 ε. 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 ƒ(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.”</p> + +<div class="condensed"> +<p>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 <i>maximum</i> value or a <i>minimum</i> value +of the function.</p> + +<p>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.”</p> + +<p>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.</p> +</div> + +<p>8. <i>Increasing and Decreasing Functions</i>.—A function ƒ(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 ƒ(x′) > ƒ(x); the function “never +decreases throughout the interval” if, x′ and x being as before, +ƒ(x′) > ƒ(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 <i>mutatis mutandis</i> +if the function never increases. If the function is monotonous +throughout the interval, ƒ(b) is the greatest (or least) value +of ƒ(x) in the interval; and if ƒ(b) is the limit of ƒ(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.</p> + +<div class="condensed"> +<p>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 (§ 6), of which principle they are particular cases. +For example, the function represented by x log (1/x) continually +<span class="pagenum"><a name="page304" id="page304"></a>304</span> +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.</p> +</div> + +<p>9. <i>Continuity of Functions</i>.—A function ƒ(x) of one variable x +is said to be continuous at a point a if (1) ƒ(x) is defined in an +interval containing a; (2) ƒ(x) has a limit at a; (3) ƒ(a) is +equal to this limit. The limit in question must be a limit for +continuous variation, not for a restricted domain. If ƒ(x) has +a limit on the left at a and ƒ(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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>The principal properties of a continuous function are:</p> + +<p>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 ε, 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 ε. There is an obvious modification if a is an +end-point of the interval.</p> + +<p>2. The continuity of the function is “uniform.” This means that the +number h which corresponds to any ε as in (1) may be the same at +all points of the interval, or, in other words, that the numbers h which +correspond to ε for different values of a have a positive +inferior limit.</p> + +<p>3. The function has a greatest value and a least value in the interval, +and these are superior and inferior limits which are attained.</p> + +<p>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.</p> + +<p>5. If the interval is unlimited towards the right (or towards the left), +the function has a limit at ∞ (or at −∞).</p> +</div> + +<p>10. <i>Discontinuity of Functions</i>.—The discontinuities of a +function of one variable, defined in an interval with the possible +exception of isolated points, may be classified as follows:</p> + +<p>(1) The function may become infinite, or tend to become +infinite, at a point.</p> + +<p>(2) The function may be undefined at a point.</p> + +<p>(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.</p> + +<p>(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.</p> + +<div class="condensed"> +<p>In case a function ƒ(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 <span class="ov">ƒ(a + 0)</span> and +<span class="un">ƒ(a + 0)</span>, and they are called the “limits of indefiniteness” on the +right. Similar limits on the left are denoted by <span class="ov">ƒ(a − 0)</span> and <span class="un">ƒ(a − 0)</span>. +Unless ƒ(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 ƒ(a), if ƒ(a) exists. If the first two are equal there +is a limit on the right denoted by ƒ(a + 0); if the second two are +equal, there is a limit on the left denoted by ƒ(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 § 7; and now it is +to be noted that, <i>e.g.</i> 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 ƒ(x) tends to become +negatively infinite when x decreases towards a; other modes of +tending to infinite limits may be described in similar terms.</p> +</div> + +<p>11. <i>Oscillation of Functions</i>.—The difference between the +greatest and least of the numbers ƒ(a), <span class="ov">ƒ(a + 0)</span>, <span class="un">ƒ(a + 0)</span>, <span class="ov">ƒ(a − 0)</span>, +<span class="un">ƒ(a − 0)</span>, when they are all finite, is called the “oscillation” or +“fluctuation” of the function ƒ(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 § 7.</p> + +<div class="condensed"> +<p>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 x<span class="su">1</span>, x<span class="su">2</span> ... x<span class="su">n−1</span>, be taken so that +b > x<span class="su">n−1</span> > x<span class="su">n−2</span> ... > x<span class="su">1</span> > a. We may devise a rule by which, as n +increases indefinitely, all the differences +b − x<span class="su">n−1</span>, x<span class="su">n−1</span> − x<span class="su">n−2</span>, ... x<span class="su">1</span> − a +tend to zero as a limit. The interval is then said to be divided +into “indefinitely small partial intervals.”</p> + +<p>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 (§ 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.</p> +</div> + +<p>12. <i>Differentiable Function</i>.—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 ƒ(x) is defined in an interval +containing the point a, and a − k and a + k are points of the +interval, the expression</p> + +<table class="math0" summary="math"> +<tr><td>ƒ(a + h) − ƒ(a)</td></tr> +<tr><td class="denom">h</td></tr> +</table> + +<div class="author">(1)</div> + +<p class="noind">represents a function of h, which we may call φ(h), defined at all +points of an interval for h between −k and k except the point 0. +Thus the four limits <span class="ov">φ(+0)</span>, <span class="un">φ(+0)</span>, <span class="ov">φ(−0)</span>, <span class="un">φ(−0)</span> 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 +ƒ(x) at the point a; when the last two are equal either of them +is the “regressive differential coefficient” of ƒ(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 ƒ(x) at a, +or the “first derived function” of ƒ(x) at a. It is denoted by +dƒ(x) / dx or by ƒ′(x). In this case φ(h) has a definite limit at h = 0, +or is determinately infinite at h = 0 (§ 7). The four limits here in +question are called, after Dini, the “four derivates” of ƒ(x) at a. +In accordance with the notation for derived functions they may +be denoted by</p> + +<p class="center"><span class="ov">ƒ′ + (a)</span>, <span class="un">ƒ′ + (a)</span>, <span class="ov">ƒ′ − (a)</span>, <span class="un">ƒ′ − (a)</span>.</p> + +<div class="condensed"> +<p>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 Ampère 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 § 24). The most +important theorem in regard to differentiable functions is the +“theorem of intermediate value.” (See <span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>.)</p> +</div> + +<p><span class="pagenum"><a name="page305" id="page305"></a>305</span></p> + +<p>13. <i>Analytic Function</i>.—If ƒ(x) and its first n differential +coefficients, denoted byƒ′(x), ƒ″(x), ... ƒ(<span class="sp">n</span>) (x), are continuous +in the interval between a and a + h, then</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(a + h) = ƒ(a) + hƒ′(a) +</td> <td>h²</td> +<td rowspan="2">ƒ″(a) + ... +</td> <td>h<span class="sp">n−1</span></td> +<td rowspan="2">ƒ<span class="sp">(n−1)</span>(a) + R<span class="su">n</span>,</td></tr> +<tr><td class="denom">2!</td> <td class="denom">(n − 1)!</td></tr></table> + +<p class="noind">where R<span class="su">n</span> may have various forms, some of which are given in +the article <span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>. This result is known as +“Taylor’s theorem.”</p> + +<p>When <span class="correction" title="amended from Talyor’s">Taylor's</span> theorem leads to a representation of the +function by means of an infinite series, the function is said to be +“analytic” (cf. § 21).</p> + +<p>14. <i>Ordinary Function</i>.—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. § 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 +(§2). All analytic functions are ordinary, but not all ordinary +functions are analytic.</p> + +<p>15. <i>Integrable Function</i>.—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 ƒ(x) through the interval between a and b” to be +the limit of the sum</p> + +<p class="center"><span class="f150">Σ</span><span class="sp1">n</span><span class="su1">1</span> ƒ(x′<span class="su">r</span>) (x<span class="su">r</span> − x<span class="su">r−1</span>)</p> + +<p class="noind">when the interval is divided into ultimately indefinitely small +partial intervals by points x<span class="su">1</span>, x<span class="su">2</span>, ... x<span class="su">n−1</span>. Here x′<span class="su">r</span> denotes +any point in the rth partial interval, x<span class="su">0</span> is put for a, and x<span class="su">n</span> 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′<span class="su">r</span>, 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.</p> + +<div class="condensed"> +<p>We have the following theorems:—</p> + +<p>1. Any continuous function is integrable.</p> + +<p>2. Any function with restricted oscillation is integrable.</p> + +<p>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 σ, however small, can be enclosed +in partial intervals the sum of whose breadths can be diminished +indefinitely.</p> + +<p>These partial intervals must be a set chosen out of some complete +set obtained by the process used in the definition of integration.</p> + +<p>4. The sum or product of two integrable functions is integrable.</p> + +<p>As regards integrable functions we have the following theorems:</p> + +<p>1. If S and I are the superior and inferior limits (or greatest and +least values) of ƒ(x) in the interval between a and b, <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ(x)<i>dx</i> is +intermediate between S(b − a) and I(b − a).</p> + +<p>2. The integral is a continuous function of each of the end-values.</p> + +<p>3. If the further end-value b is variable, and if <span class="f150">∫</span> <span class="sp1">x</span><span class="su1">a</span> ƒ(x)<i>dx</i> = F(x), +then if ƒ(x) is continuous at b, F(x) is differentiable at b, and +F′(b) = ƒ(b).</p> + +<p>4. In case ƒ(x) is continuous throughout the interval F(x) is continuous +and differentiable throughout the interval, and F′(x) = ƒ(x) +throughout the interval.</p> + +<p>5. In case ƒ′(x) is continuous throughout the interval between a +and b,</p> + +<p class="center"><span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ′(x)<i>dx</i> = ƒ(b) − ƒ(a).</p> + +<p>6. In case ƒ(x) is discontinuous at one or more points of the interval +between a and b, in which it is integrable,</p> + +<p class="center"><span class="f150">∫</span> <span class="sp1">x</span><span class="su1">a</span> ƒ(x)dx</p> + +<p class="noind">is a function of x, of which the four derivates at any point of the +interval are equal to the limits of indefiniteness of ƒ(x) at the point.</p> + +<p>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</p> + +<p class="center">F(x) = <span class="f150">∫</span> <span class="sp1">x</span><span class="su1">a</span> F′(x)<i>dx</i> + const.,</p> + +<p class="noind">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.</p> + +<p>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.</p> + +<p>We have also two theorems concerning the integral of the product +of two integrable functions ƒ(x) and φ(x); these are known as “the +first and second theorems of the mean.” The first theorem of the +mean is that, if φ(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 ƒ(x) in the interval, +which has the property expressed by the equation</p> + +<p class="center">M <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> φ(x)dx = <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ(x)φ(x)dx</p> + +<p>The second theorem of the mean is that, if ƒ(x) is monotonous +throughout the interval, there is a number ξ between a and b which +has the property expressed by the equation</p> + +<p class="center"><span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ(x) φ(x)dx = ƒ(a) <span class="f150">∫</span> <span class="sp1">ξ</span><span class="su1">a</span> φ(x)<i>dx</i> + ƒ(b) <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">ξ</span> φ(x)dx.</p> + +<p>(<i>See</i> <span class="sc"><a href="#artlinks">Fourier’s Series</a></span>.)</p> +</div> + +<p>16. <i>Improper Definite Integrals</i>.—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 ƒ(x) is not defined, we impose a restriction on +the points x′<span class="su">r</span> of the definition: none of them is to be the point c. +This comes to the same thing as defining <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ(x)<i>dx</i> to be</p> + +<p class="center">Lt <span class="su">ε=0</span><span class="f150">∫</span> <span class="sp1">c−ε</span><span class="su1">a</span> ƒ(x)dx + Lt <span class="su">ε′=0</span><span class="f150">∫</span> <span class="sp1">b</span><span class="su1">c+ε′</span> ƒ(x)dx,</p> +<div class="author">(1)</div> + +<p class="noind">where, to fix ideas, b is taken > a, and ε and ε′ are positive. The +same definition applies to the case where ƒ(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.</p> + +<p>Again, when the interval of integration is unlimited to the +right, or extends to positively infinite values, we have as a +definition</p> + +<p class="center"><span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">a</span> ƒ(x)dx = Lt <span class="su">h=∞</span><span class="f150">∫</span> <span class="sp1">h</span><span class="su1">a</span> ƒ(x)dx,</p> + +<p><span class="pagenum"><a name="page306" id="page306"></a>306</span></p> + +<p class="noind">provided this limit exists. Similar definitions apply to</p> + +<p class="center"><span class="f150">∫</span> <span class="sp1">−∞</span><span class="su1">a</span> ƒ(x)dx, and to <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">−∞</span> ƒ(x)dx.</p> + +<p>All such definite integrals as the above are said to be “improper.” +For example, <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span> sin x / x dx is improper in two ways. It means</p> + +<p class="center">Lt <span class="su">h=∞</span> Lt <span class="su">ε=0</span> <span class="f150">∫</span> <span class="sp1">h</span><span class="su1">ε</span> sinx/x dx,</p> + +<p class="noind">in which the positive number ε is first diminished indefinitely, +and the positive number h is afterwards increased indefinitely.</p> + +<p>The “theorems of the mean” (§ 15) require modification when +the integrals are improper (see <span class="sc"><a href="#artlinks">Fourier’s Series</a></span>).</p> + +<p>When the improper definite integral of a function which +becomes, or tends to become, infinite, exists, the integral is said +to be “convergent.” If ƒ(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 <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ(x)dx, which has no value, is called +a “divergent integral, “and it may happen that there is a definite +value for</p> + +<p class="center">Lt <span class="f150">{</span> <span class="f150">∫</span> <span class="sp1">c−ε</span><span class="su1">a</span> ƒ(x) dx + <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">c+ε′</span> ƒ(x) dx <span class="f150">}</span></p> + +<p class="noind">provided that ε and ε′ 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 ε′ = ε, +<i>i.e.</i> 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.”</p> + +<p>17. <i>Domain of a Set of Variables.</i>—The numerical continuum +of n dimensions (C<span class="su">n</span>) is the aggregate that is arrived at by attributing +simultaneous values to each of n variables x<span class="su">1</span>, x<span class="su">2</span>, ... x<span class="su">n</span>, +these values being any real numbers. The elements of such an +aggregate are called “points,” and the numbers x<span class="su">1</span>, x<span class="su">2</span> ... x<span class="su">n</span> +the “co-ordinates” of a point. Denoting in general the points +(x<span class="su">1</span>, x<span class="su">2</span>, ... x<span class="su">n</span>) and (x′<span class="su">1</span>, x′<span class="su">2</span> ... x′<span class="su">n</span>) by x and x′, the sum of +the differences |x<span class="su">1</span> − x′<span class="su">1</span>| + |x<span class="su">2</span> − x′<span class="su">2</span>| + ... + |x<span class="su">n</span> − x′<span class="su">n</span>| 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 ε, +however small, a number m and points x′, x″, ... x<span class="sp">(m)</span> of the +aggregate can be found so that all the differences denoted by +|x′ − a|, |x″ − x′|, ... |b − x<span class="sp">(m)</span>| are less than ε. A perfect connected +aggregate is a <i>continuum</i>. This is G. Cantor’s definition.</p> + +<div class="condensed"> +<p>The definition of a continuum in C<span class="su">n</span> 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<span class="su">n</span> of C<span class="su">n</span>. Such a part would correspond to +an interval in C<span class="su">1</span>, or to an area bounded by a simple closed contour +in C<span class="su">2</span>; and, besides being perfect and connected, it would have the +following properties: (1) There are points of C<span class="su">n</span>, which are not points +of H<span class="su">n</span>; these form a complementary aggregate H′<span class="su">n</span>. (2) There are +points “within” H<span class="su">n</span>; this means that for any such point there is +a neighbourhood consisting exclusively of points of H<span class="su">n</span>. (3) The +points of H<span class="su">n</span> which do not lie “within” H<span class="su">n</span> are limiting points of +H′<span class="su">n</span>; they are not points of H′<span class="su">n</span>, but the neighbourhood of any such +point for any number h, however small, contains points within H<span class="su">n</span> +and points of H′<span class="su">n</span>: the aggregate of these points is called the +“boundary” of H<span class="su">n</span>. (4) When any two points a, b within H<span class="su">n</span> are +taken, it is possible to find a number ε and a corresponding number +m, and to choose points x′, x″, ... x<span class="sp">(m)</span>, so that the neighbourhood +of a for ε contains x′, and consists exclusively of points within H<span class="su">n</span>, +and similarly for x′ and x″, x″ and x″′, ... x<span class="sp">(m)</span> 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.</p> +</div> + +<p>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<span class="su">n</span> 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<span class="su">n</span> of C<span class="su">n</span> 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.</p> + +<div class="condensed"> +<p>We take a positive fraction ε 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 ƒ has a limit at a if for any positive ε, +however small, there is a corresponding h which has the property +that |ƒ(x′) − ƒ(x)| < ε, whatever points x, x′ in the neighbourhood +of a for h we take (a excluded). For example, when there are two +variables x<span class="su">1</span>, x<span class="su">2</span>, 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 ε, however small, has been +specified, a number N can be found which is such that |ƒ(x′) − ƒ(x)| < ε, +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 +(§ 6). For example, a function of x<span class="su">1</span> and x<span class="su">2</span> may have a limit at +(x<span class="su">1</span> = 0, x<span class="su">2</span> = 0) if we first diminish x<span class="su">1</span> without limit, keeping x<span class="su">2</span> constant, +and afterwards diminish x<span class="su">2</span> without limit. Expressed in +geometrical language, this process amounts to approaching the +origin along the axis of x<span class="su">2</span>. 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 (§ 7). The definition of a continuous +function (§ 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 {4 tan<span class="sp">−1</span> (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).</p> +</div> + +<p>19. <i>Differentiation and Integration.</i>—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 +<span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>).</p> + +<p>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| < ½h, where h is a chosen positive number; +the extent of this domain is h<span class="sp">n</span>, 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 +<span class="pagenum"><a name="page307" id="page307"></a>307</span> +numerical continuum of n dimensions, and its boundary consists +of all the points for which |x − a| = ½h. 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 <i>C<span class="su">n</span></i>, 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 <i>C<span class="su">n</span></i>.</p> + +<div class="condensed"> +<p>The integral of a function ƒ(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<span class="su">n</span>, 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.</p> +</div> + +<p>20. <i>Representation of Functions in General</i>.—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 (§ 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<span class="su">y</span> = ∞xy / (x²y + 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. § 6). +When the terms u<span class="su">1</span>, u<span class="su">2</span>, ... of a series are functions of a variable +x, the sum s<span class="su">n</span> 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<span class="su">n</span> = ∞ s<span class="su">n</span>, 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.</p> + +<div class="condensed"> +<p>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. Brodén. 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 Brodén’s method. As an example of these two latter processes +we may cite the Gamma function [Γ(x)] defined for positive values +of x by the definite integral</p> + +<p class="center"><span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span> e<span class="sp">−t</span> t<span class="sp">x−1</span> dt,</p> + +<p class="noind">or by the infinite product</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">Lt<span class="su">n=∞</span> n<span class="sp">x</span>/x(1 + x)(1 + ½x) ... <span class="f150">(</span> 1 +</td> <td>x</td> +<td rowspan="2"><span class="f150">)</span>.</td></tr> +<tr><td class="denom">n − 1</td></tr></table> + +<p class="noind">The second of these expressions avails for the representation of the +function at all points at which x is not a negative integer.</p> +</div> + +<p>21. <i>Power Series</i>.—Taylor’s theorem leads in certain cases +to a representation of a function by an infinite series. We have +under certain conditions (§ 13)</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(x) = ƒ(a) + <span class="f150">Σ</span><span class="sp1">n−1</span><span class="su1">r=1</span></td> <td>(x − a)<span class="sp">r</span></td> +<td rowspan="2">ƒ<span class="sp">(r)</span>(a) + R<span class="su">n</span>;</td></tr> +<tr><td class="denom">r!</td></tr></table> + +<p class="noind">and this becomes</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(x) = ƒ(a) + <span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">r=1</span></td> <td>(x − a)<span class="sp">r</span></td> +<td rowspan="2">ƒ<span class="sp">(r)</span>(a),</td></tr> +<tr><td class="denom">r!</td></tr></table> + +<p class="noind">provided that (α) a positive number k can be found so that at +all points in the interval between a and a + k (except these points) +ƒ(x) has continuous differential coefficients of all finite orders, +and at a has progressive differential coefficients of all finite +orders; (β) Cauchy’s form of the remainder <i>R<span class="su">n</span></i>, viz. +[(x − a) / (n − 1)!] (1 − θ)<span class="sp">n−1</span> ƒ<span class="sp">n</span> {a + θ(x − a)}, has the limit zero when n increases +indefinitely, for all values of θ 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” (§ 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.”</p> + +<p>Taylor’s series is a power series, <i>i.e</i>. a series of the form</p> + +<p class="center"><span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">n=0</span> a<span class="su">n</span> (x − a)<span class="sp">n</span>.</p> + +<div class="condensed"> +<p class="noind">As regards power series we have the following theorems:</p> + +<p>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.</p> + +<p>2. The power series represents a continuous function in its domain +of convergence (the end-points may have to be excluded).</p> + +<p>3. This function is analytic in the domain, and the power series +representing it is the Taylor’s series for the function.</p> + +<p>The theory of power series has been developed chiefly from the +point of view of the theory of functions of complex variables.</p> +</div> + +<p>22. <i>Uniform Convergence</i>.—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 ƒ(x) be the sum +of the series at any point x of the domain, and ƒ<span class="su">n</span>(x) the sum of +the first n + 1 terms. The condition of convergence at a point +a is that, after any positive number ε, however small, has been +specified, it must be possible to find a number n so that +|ƒ<span class="su">m</span>(a) − ƒ<span class="su">p</span>(a)| < ε for all values of m and p which exceed n. +The sum, ƒ(a), is the limit of the sequence of numbers ƒ<span class="su">n</span>(a) at +<span class="pagenum"><a name="page308" id="page308"></a>308</span> +n = ∞. The convergence is said to be “uniform” in an interval +if, after specification of ε, the same number n suffices at all +points of the interval to make |ƒ(x) − ƒ<span class="su">m</span>(x)| < ε for all values of +m which exceed n. The numbers n corresponding to any ε, +however small, are all finite, but, when ε is less than some fixed +finite number, they may have an infinite superior limit (§ 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, ε can be taken so small that, at some point in the neighbourhood +of a, n must be taken > N to make |ƒ(x) − f<span class="su">m</span>(x)| < ε +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 ε afterwards, then the number n is +finite; choose ε 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, however small ε may be, the convergence is uniform.</p> + +<div class="condensed"> +<p>For example, the series sin x − ½ sin 2x + <span class="spp">1</span>⁄<span class="suu">3</span> sin 3x − ... is convergent +for all real values of x, and, when π > x > −π its sum is ½x; +but, when x is but a little less than π, the number of terms which +must be taken in order to bring the sum at all near to the value of +½x is very large, and this number tends to increase indefinitely as +x approaches π. This series does not converge uniformly in the +neighbourhood of x = π. Another example is afforded by the series</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">n=0</span></td> <td>nx</td> +<td rowspan="2">−</td> <td>(n + 1)x</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">n²x² + 1</td> <td class="denom">(n + 1)²x² + 1</td></tr></table> + +<p class="noind">of which the remainder after n terms +is nx/(n²x² + 1). If we put x = 1/n, for any value of n, however +great, the remainder is ½; 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.</p> +</div> + +<p>As regards series whose terms represent continuous functions +we have the following theorems:</p> + +<p>(1) If the series converges uniformly in an interval it represents +a function which is continuous throughout the interval.</p> + +<p>(2) If the series represents a function which is discontinuous +in an interval it cannot converge uniformly in the interval.</p> + +<p>(3) A series which does not converge uniformly in an interval +may nevertheless represent a function which is continuous +throughout the interval.</p> + +<p>(4) A power series converges uniformly in any interval contained +within its domain of convergence, the end-points being +excluded.</p> + +<p>(5) If <span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">r=0</span> ƒ<span class="su">r</span>(x) = ƒ(x) converges uniformly in the interval +between a and b</p> + +<p class="center"><span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ(x)dx = <span class="f150">Σ</span><span class="sp1">b</span><span class="su1">r=0</span> <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ<span class="su">r</span>(x)dx,</p> + +<p class="noind">or a series which converges <span class="correction" title="amended from unformly">uniformly</span> may be integrated term by +term.</p> + +<p>(6) If <span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">r=0</span> ƒ′<span class="su">r</span>(x) converges uniformly in an interval, then +<span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">r=0</span> ƒ<span class="su">r</span>(x) converges in the interval, and represents a continuous +differentiable function, φ(x); in fact we have</p> + +<p class="center">φ′(x) = <span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">r=0</span> ƒ′<span class="su">r</span>(x),</p> + +<p class="noind">or a series can be differentiated term by term if the series of +derived functions converges uniformly.</p> + +<p>A series whose terms represent functions which are not continuous +throughout an interval may converge uniformly in the +interval. If <span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">r=0</span> ƒ<span class="su">r</span>(x) = ƒ(x), is such a series, and if all the +functions ƒ<span class="su">r</span>(x) have limits at a, then ƒ(x) has a limit at a, which +is <span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">r=0</span> Lt <span class="su">x=a</span> ƒ<span class="su">r</span>(x). A similar theorem holds for limits on the left +or on the right.</p> + +<p>23. Fourier’s Series.—An extensive class of functions admit +of being represented by series of the form</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">a<span class="su">0</span> + <span class="f150">Σ</span><span class="sp1">∞</span><span class="su1">n=1</span> <span class="f150">(</span> a<span class="su">n</span> cos</td> <td>nπx</td> +<td rowspan="2">+ b<span class="su">n</span> sin</td> <td>nπx</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">c</td> <td class="denom">c</td></tr></table> + +<p class="noind">and the rule for determining the coefficients a<span class="su">n</span>, b<span class="su">n</span> of such a +series, in order that it may represent a given function ƒ(x) in +the interval between −c and c, was given by Fourier, viz. we +have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">a<span class="su">0</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> <span class="sp1">c</span><span class="su1">−c</span> ƒ(x)dx,   a<span class="su">n</span>=</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> <span class="sp1">c</span><span class="su1">−c</span> ƒ(x)cos</td> <td>nπx</td> +<td rowspan="2">dx,   b<span class="su">n</span>=</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> <span class="sp1">c</span><span class="su1">−c</span> sin</td> <td>nπx</td> +<td rowspan="2">dx.</td></tr> +<tr><td class="denom">2c</td> <td class="denom">c</td> +<td class="denom">c</td> <td class="denom">c</td> <td class="denom">c</td></tr></table> + +<p class="noind">The interval between −c and c may be called the “periodic +interval,” and we may replace it by any other interval, <i>e.g.</i> that +between 0 and 1, without any restriction of generality. When +this is done the sum of the series takes the form</p> + +<p class="center">Lt <span class="su">n=∞</span> <span class="f150">∫</span> <span class="sp1">1</span><span class="su1">0</span> <span class="f150">Σ</span><span class="sp1">r = n</span><span class="su1">r = −n</span> ƒ(z) cos {2rπ(z − x)}dz,</p> + +<p class="noind">and this is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">Lt <span class="su">n=∞</span> <span class="f150">∫</span> <span class="sp1">1</span><span class="su1">0</span> ƒ(z)</td> <td>sin {(2n + 1) (z − x)π}</td> +<td rowspan="2">dz.</td></tr> +<tr><td class="denom">sin {(z − x)π}</td></tr></table> +<div class="author1">(ii.)</div> + +<p>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 (§ 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.</p> + +<div class="condensed"> +<p>By a discussion of the integral (ii.) based on the Second Theorem +of the Mean (§ 15) it can be shown that, if ƒ(x) has restricted oscillation +in the interval (§ 11), the sum of the series is equal to ½{ƒ(x + 0) + ƒ(x − 0)} +at any point x within the interval, and that it is equal to +½ {ƒ(+0) + ƒ(1 − 0} at each end of the interval. (See the article +<span class="sc"><a href="#artlinks">Fourier’s Series</a></span>.) 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 ƒ(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 ƒ(x) +which tends to become infinite at a finite number of points a of the +interval, provided (1) ƒ(x) tends to become determinately infinite +at each of the points a, (2) the improper definite integral of ƒ(x) +through the interval is convergent, (3) ƒ(x) has not an infinite number +of discontinuities or of maxima or minima in the interval.</p> +</div> + +<p>24. <i>Representation of Continuous Functions by Series</i>.—If the +series for ƒ(x) formed by Fourier’s rule converges at the point +a of the periodic interval, and if ƒ(x) is continuous at a, the +sum of the series is ƒ(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:</p> + +<p>(1) If ƒ(x) is continuous throughout the interval between 0 and +2π, and if any positive number ε however small is specified, +it is possible to find an integer n, so that the difference between +the value of ƒ(x) and the sum of the first n terms of the series +for ƒ(x), formed by Fourier’s rule with periodic interval from +0 to 2π, shall be less than ε at all points of the interval. This +result can be extended to a function which is continuous in any +given interval.</p> + +<p>(2) If ƒ(x) is continuous throughout an interval, and any +positive number ε 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 ƒ(x) and the value of the +polynomial shall be less than ε at all points of the interval.</p> + +<p>Again it can be proved that, if ƒ(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 ƒ(x) at all points of the interval. Methods of representation +of continuous functions by infinite series of rational fractional +functions have also been devised.</p> + +<div class="condensed"> +<p>Particular interest attaches to continuous functions which are +not differentiable. Weierstrass gave as an example the function +represented by the series <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">0</span> a<span class="sp">n</span> cos (b<span class="sp">n</span> xπ), where a is positive and less +than unity, and b is an odd integer exceeding (1 + <span class="spp">3</span>⁄<span class="suu">2</span>π)/a. It can be +shown that this series is uniformly convergent in every interval, +<span class="pagenum"><a name="page309" id="page309"></a>309</span> +and that the continuous function ƒ(x) represented by it has the +property that there is, in the neighbourhood of any point x<span class="su">0</span>, an +infinite aggregate of points x′, having x<span class="su">0</span> as a limiting point, for +which {ƒ(x′) − ƒ(x<span class="su">0</span>)} / (x′ − x<span class="su">0</span>) tends to become infinite with one +sign when x′ − x<span class="su">0</span> approaches zero through positive values, and +infinite with the opposite sign when x′ − x<span class="su">0</span> approaches zero through +negative values. Accordingly the function is not differentiable at +any point. The definite integral of such a function ƒ(x) through the +interval between a fixed point and a variable point x, is a continuous +differentiable function F(x), for which F′(x) = ƒ(x); and, if ƒ(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 Σa<span class="su">n</span>φ<span class="su">n</span>(x), where Σa<span class="su">n</span> is an absolutely convergent series of +numbers, and φ<span class="su">n</span>(x) is an analytic function whose absolute value +never exceeds unity.</p> +</div> + +<p>25. <i>Calculations with Divergent Series</i>.—When the series +described in (1) and (2) of § 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 − ½ + <span class="spp">1</span>⁄<span class="suu">3</span> − ...</p> + +<div class="condensed"> +<p>In general, let ƒ<span class="su">0</span>(x) + ƒ<span class="su">1</span>(x) + ... be a series of functions which +does not converge in a certain domain. It may happen that, if any +number ε, however small, is first specified, a number n can afterwards +be found so that, at a point a of the domain, the value ƒ(a) of +a certain function ƒ(x) is connected with the sum of the first n + 1 +terms of the series by the relation |ƒ(a) − <span class="f150">Σ</span> <span class="sp1">n</span><span class="su1">r = 0</span> ƒ<span class="su">r</span>(a)| < ε. 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′, |<span class="f150">Σ</span> <span class="sp1">m</span><span class="su1">r = 0</span> ƒ<span class="su">r</span>(a)| > N. The divergent series ƒ<span class="su">0</span>(x) + ƒ<span class="su">1</span>(x) + ... is then an +asymptotic expansion for the function f(x) in the domain.</p> + +<p>The best known example of an asymptotic expansion is Stirling’s +formula for n! when n is large, viz.</p> + +<p class="center">n! = √<span class="ov">(2π)</span> ½n<span class="sp">n + ½</span> e<span class="sp">−n + θ/12n</span>,</p> + +<p class="noind">where θ is some number lying between 0 and 1. This formula is +included in the asymptotic expansion for the Gamma function. +We have in fact</p> + +<p class="center">log {Γ(x)} = (x − ½) log x − x + ½ log 2π + <span class="ov">ω</span>(x),</p> + +<p class="noind">where <span class="ov">ω</span>(x) is the function defined by the definite integral</p> + +<p class="center"><span class="ov">ω</span>(x) = <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span> {(1 − e<span class="sp">−t</span>)<span class="sp">−1</span> − t<span class="sp">−1</span> − ½} t<span class="sp">−1</span> e<span class="sp">−tx</span> dt.</p> + +<p class="noind">The multiplier of e<span class="sp">−tx</span> under the sign of integration can be expanded +in the power series</p> + +<table class="math0" summary="math"> +<tr> <td>B<span class="su">1</span></td> +<td rowspan="2">−</td> <td>B<span class="su">2</span></td> +<td rowspan="2">t<span class="sp">2</span> +</td> <td>B<span class="su">3</span></td> +<td rowspan="2">t<span class="sp">4</span> − ...,</td></tr> +<tr><td class="denom">2!</td> <td class="denom">4!</td> <td class="denom">6!</td></tr></table> + +<p class="noind">where B<span class="su">1</span>, B<span class="su">2</span>, ... are “Bernoulli’s numbers” given by the formula</p> + +<p class="center">B<span class="su">m</span> = 2.2m! (2π)<span class="sp">−2m</span> <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">r = 1</span> (r<span class="sp">−2m</span>).</p> + +<p class="noind">When the series is integrated term by term, the right-hand member +of the equation for <span class="ov">ω</span>(x) takes the form</p> + +<table class="math0" summary="math"> +<tr><td>B<span class="su">1</span></td> +<td rowspan="2"> </td> <td>1</td> +<td rowspan="2">−</td> <td>B<span class="su">2</span></td> +<td rowspan="2"> </td> <td>1</td> +<td rowspan="2">+</td> <td>B<span class="su">3</span></td> +<td rowspan="2"> </td> <td>1</td> +<td rowspan="2">− ...,</td></tr> +<tr><td class="denom">1·2</td> <td class="denom">x</td> +<td class="denom">3·4</td> <td class="denom">x<span class="sp">3</span></td> +<td class="denom">5·6</td> <td class="denom">x<span class="sp">5</span></td></tr></table> + +<p class="noind">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 <span class="ov">ω</span>(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 (<i>Math. and Phys. Papers</i>, vol. ii. p. 329), and a general +theory of asymptotic expansions has been developed by H. Poincaré. +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 É. 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.</p> +</div> + +<p>26. <i>Interchange of the Order of Limiting Operations</i>.—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.</p> + +<div class="condensed"> +<p>Among examples of such interchanges we have the differentiation +and integration of an infinite series term by term (§ 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 (§ 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 <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">0</span> ƒ<span class="su">r</span>(x) represents a function +(ƒx) in an interval containing a point a, and that each of the functions +ƒ<span class="su">r</span>(x) has a limit at a. If we first put x=a, and then sum the series, +we have the value ƒ(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 +ƒ(x) at a; if we first replace each function ƒ<span class="su">r</span>(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 ƒ(x) is continuous at a, the first and +second results are equal; if the functions ƒ<span class="su">r</span>(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.</p> + +<p><span class="sc">Authorities.</span>—Among the more important treatises and memoirs +connected with the subject are: R. Baire, <i>Fonctions discontinues</i> +(Paris, 1905); O. Biermann, <i>Analytische Functionen</i> (Leipzig, 1887); +É. Borel, <i>Théorie des fonctions</i> (Paris, 1898) (containing an introductory +account of the Theory of Aggregates), and <i>Séries divergentes</i> +(Paris, 1901), also <i>Fonctions de variables réelles</i> (Paris, 1905); T.J. +I’A. Bromwich, <i>Introduction to the Theory of Infinite Series</i> (London, +1908); H.S. Carslaw, <i>Introduction to the Theory of Fourier’s Series +and Integrals</i> (London, 1906); U. Dini, <i>Functionen e. reellen Grösse</i> +(Leipzig, 1892), and <i>Serie di Fourier</i> (Pisa, 1880); A. Genocchi +u. G. Peano, <i>Diff.- u. Int.-Rechnung</i> (Leipzig, 1899); J. Harkness +and F. Morley, <i>Introduction to the Theory of Analytic Functions</i> +(London, 1898); A. Harnack, <i>Diff. and Int. Calculus</i> (London, 1891); +E.W. Hobson, <i>The Theory of Functions of a real Variable and the +Theory of Fourier’s Series</i> (Cambridge, 1907); C. Jordan, <i>Cours +d’analyse</i> (Paris, 1893-1896); L. Kronecker, <i>Theorie d. einfachen +u. vielfachen Integrale</i> (Leipzig, 1894); H. Lebesgue, <i>Leçons sur +l’intégration</i> (Paris, 1904); M. Pasch, <i>Diff.- u. Int.-Rechnung</i> +(Leipzig, 1882); E. Picard, <i>Traité d’analyse</i> (Paris, 1891); O. +Stolz, <i>Allgemeine Arithmetik</i> (Leipzig, 1885), and <i>Diff.- u. Int.-Rechnung</i> +(Leipzig, 1893-1899); J. Tannery, <i>Théorie des fonctions</i> +(Paris, 1886); W.H. and G.C. Young, <i>The Theory of Sets of Points</i> +(Cambridge, 1906); Brodén, “Stetige Functionen e. reellen Veränderlichen,” +<i>Crelle</i>, Bd. cxviii.; G. Cantor, A series of memoirs on the +“Theory of Aggregates” and on “Trigonometric series” in <i>Acta +Math</i>. tt. ii., vii., and <i>Math. Ann</i>. Bde. iv.-xxiii.; Darboux, “Fonctions +discontinues,” <i>Ann. Sci. École normale sup</i>. (2), t. iv.; Dedekind, +<i>Was sind u. was sollen d. Zahlen</i>? (Brunswick, 1887), and <i>Stetigkeit +u. irrationale Zahlen</i> (Brunswick, 1872); Dirichlet, “Convergence +des séries trigonométriques,” <i>Crelle</i>, Bd. iv.; P. Du Bois Reymond, +<i>Allgemeine Functionentheorie</i> (Tübingen, 1882), and many memoirs +in <i>Crelle</i> and in <i>Math. Ann</i>.; Heine, “Functionenlehre,” <i>Crelle</i>, +Bd. lxxiv.; J. Pierpont, <i>The Theory of Functions of a real Variable</i> +(Boston, 1905); F. Klein, “Allgemeine Functionsbegriff,” <i>Math. +Ann</i>. Bd. xxii.; W.F. Osgood, “On Uniform Convergence,” <i>Amer. +J. of Math</i>. vol. xix.; Pincherle, “Funzioni analitiche secondo +Weierstrass,” <i>Giorn. di mat</i>. t. xviii.; Pringsheim, “Bedingungen +d. Taylorschen Lehrsatzes,” <i>Math. Ann</i>. Bd. xliv.; Riemann, +“Trigonometrische Reihe,” <i>Ges. Werke</i> (Leipzig, 1876); Schoenflies, +“Entwickelung d. Lehre v. d. Punktmannigfaltigkeiten,” <i>Jahresber. +d. deutschen Math.-Vereinigung</i>, Bd. viii.; Study, Memoir on +“Functions with Restricted Oscillation,” <i>Math. Ann</i>. Bd. xlvii.; +Weierstrass, Memoir on “Continuous Functions that are not Differentiable,” +<i>Ges. math. Werke</i>, 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,” <i>Proc. London +Math. Soc.</i> (Ser. 2) t. 6. Further information and very full references +will be found in the articles by Pringsheim, Schoenflies and Voss in +the <i>Encyclopädie der math. Wissenschaften</i>, Bde. i., ii. (Leipzig, 1898, +1899).</p> +</div> +<div class="author">(A. E. H. L.)</div> + +<p><span class="pagenum"><a name="page310" id="page310"></a>310</span></p> + +<p class="pt2 center sc">II—Functions of Complex Variables</p> + +<p>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: (§ 1), <i>Complex numbers</i>, states what a +complex number is; (§ 2), <i>Plotting of simple expressions involving +complex numbers</i>, 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; (§ 3), +<i>Limiting operations</i>, 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 λ(z); (§ 4), <i>Functions of a complex variable in general</i>, +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; +(§ 5), <i>Applications</i>, 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 (§ 6), <i>Singular points</i>, to +a definition of the region of existence and singular points of a +function of a complex variable, and thence, in (§ 7), <i>Monogenic +Functions</i>, to what the writer believes to be the simplest definition +of a function of a complex variable, that of Weierstrass; (§ 8), +<i>Some elementary properties of single valued functions</i>, 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 +(§ 9), <i>The construction of a monogenic function with a given region of +existence</i>, with which is connected (§10), <i>Expression of a monogenic +function by rational functions in a given region</i>, of which the +method is applied in (§ 11), <i>Expression of</i> (1 − z)<span class="sp">−1</span> <i>by polynomials</i>, +to a definite example, used here to obtain (§ 12), <i>An expansion +of an arbitrary function by means of a series of polynomials, over +a star region</i>, also obtained in the original manner of Mittag-Leffler; +(§ 13), <i>Application of Cauchy’s theorem to the determination +of definite integrals</i>, gives two examples of this method; (§ 14), +<i>Doubly Periodic Functions</i>, 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 (§ 20) below, +dealing with <i>Elliptic Integrals</i>; (§ 15), <i>Potential Functions, +Conformal representation in general</i>, 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; (§ 16), <i>Multiple-valued +Functions, Algebraic Functions</i>, deals for the most part with +algebraic functions, proving the residue theorem, and establishing +that an algebraic function has a definite Order; (§ 17), <i>Integrals +of Algebraic Functions</i>, enunciating Abel’s theorem; (§ 18), +<i>Indeterminateness of Algebraic Integrals</i>, deals with the periods +associated with an algebraic integral, establishing that for an +elliptic integral the number of these is two; (§ 19), <i>Reversion of +an algebraic integral</i>, mentions a problem considered below in +detail for an elliptic integral; (§ 20), <i>Elliptic Integrals</i>, 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; (§ 21), <i>Modular Functions</i>, 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; (§ 22), <i>A property of integral functions, +deduced from the theory of modular functions</i>, 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; (§ 23), <i>Geometrical applications of Elliptic Functions</i>, +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 ℜ(u); (§ 24), <i>Integrals of Algebraic +Functions in connexion with the theory of plane curves</i>, discusses +the generalization to curves of any deficiency; (§ 25), <i>Monogenic +Functions of several independent variables</i>, describes briefly the +beginnings of this theory, with a mention of some fundamental +theorems: (§ 26), <i>Multiply-Periodic Functions and the Theory +of Surfaces</i>, attempts to show the nature of some problems now +being actively pursued.</p> + +<p>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 <i>area</i> 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.</p> + +<p>§ 1. <i>Complex Numbers</i>.—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 i² = −1.</p> + +<div class="condensed"> +<p>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 2π between Ox and OP be called θ, we may write +z = r (cos θ + i sin θ); then r is called the modulus or absolute value +of z and often denoted by |z| and θ is called the phase or amplitude +of z, and often denoted by ph (z); strictly the phase is ambiguous +by additive multiples of 2π. 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(iθ) for cos θ + i sin θ; it is +at once verified that E(iα). E(iβ) = E[i(α + β)], so that the phase of a +product of complex quantities is obtained by addition of their +respective phases.</p> +</div> + +<p>§ 2. <i>Plotting and Properties of Simple Expressions involving +a Complex Number</i>.—If we put ζ = (z-i)/(z + i), and, putting +ζ = ξ + iη, take a new plane upon which ξ, η are rectangular +co-ordinates, the equations ξ= (x² + y²− 1)/[x² + (y + 1)²], +η = −2xy/[x² + (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 ζ describes +once a circle of radius unity, with centre at ζ = 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 +ζ-plane which are interior to this circle; in particular z = i +corresponds to ζ = 0.</p> + +<div class="condensed"> +<p>Moreover, ζ being a rational function of z, both ξ and η are continuous +differentiable functions of x and y, save when ζ is infinite; +<span class="pagenum"><a name="page311" id="page311"></a>311</span> +writing ζ = ƒ(x, y) = ƒ(z − iy, y), the fact that this is really independent +of y leads at once to ∂f/∂x + i∂ƒ/∂y = 0, and hence to</p> + +<table class="math0" summary="math"> +<tr><td>∂ξ</td> +<td rowspan="2">=</td> <td>∂η</td> +<td rowspan="2">,</td> <td>∂ξ</td> +<td rowspan="2">= −</td> <td>∂η</td> +<td rowspan="2">,</td> <td>∂²ξ</td> +<td rowspan="2">+</td> <td>∂²ξ</td> +<td rowspan="2">= 0;</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂x′</td> +<td class="denom">∂y</td> <td class="denom">∂x′</td> +<td class="denom">∂x²</td> <td class="denom">∂y²</td></tr></table> + +<p class="noind">so that ξ is not any arbitrary function of x, y, and when ξ is known +η is determinate save for an additive constant. Also, in virtue of +these equations, if ζ, ζ′ be the values of ζ corresponding to two +near values of z, say z and z′, the ratio (ζ′ − ζ)/(z′ − z) has a definite +limit when z′ = z, independent of the ultimate phase of z′ − z, this +limit being therefore equal to ∂ζ/∂x, that is, ∂ξ/∂x + i∂η)/∂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 +∂ξ/∂x, ∂η/∂x are not zero, and P′, P″ be two points near +to P on these curves respectively, and the corresponding points of the +ζ-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 ξ, η 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 ζ-plane with the same angles; the magnification, +however, which is equal to [(∂ξ/∂x)² + (∂ξ/∂y)² ]<span class="sp">1/2</span> 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.</p> + +<p>As another illustration consider the case when ζ is a polynomial +in z,</p> + +<p class="center">ζ = p<span class="su">0</span>z<span class="sp">n</span> + p<span class="su">1</span>z<span class="sp">n−1</span> + ... + p<span class="su">n</span>;</p> + +<p class="noind">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 |ζ| > H; +consider the lower limit of |ζ| for |z| < R; as ξ² + η² is a real +continuous function of x, y for |z| < R, there is a point (x, y), +say (x<span class="su">0</span>, y<span class="su">0</span>), at which |ζ| is least, say equal to ρ, and therefore +within a circle in the ζ-plane whose centre is the origin, of radius ρ, +there are no points ζ representing values corresponding to |z| < R. +But if ζ<span class="su">0</span> be the value of ζ corresponding to (x<span class="su">0</span>, y<span class="su">0</span>), and the expression +of ζ − ζ<span class="su">0</span> near z<span class="su">0</span> = x<span class="su">0</span> + iy<span class="su">0</span>, in terms of z − z<span class="su">0</span>, be A(z − z<span class="su">0</span>)<span class="sp">m</span> + +B(z − z<span class="su">0</span>)<span class="sp">m + 1</span> + ..., where A is not zero, to two points near to (x<span class="su">0</span>, y<span class="su">0</span>), +say (x<span class="su">1</span>, y<span class="su">1</span>) or z<span class="su">1</span> and +z<span class="su">2</span> = z<span class="su">0</span> + (z<span class="su">1</span> − z<span class="su">0</span>) (cos π/m + i sin π/m), will correspond +two points near to ζ<span class="su">0</span>, say ζ<span class="su">1</span>, and 2ζ<span class="su">0</span> − ζ′<span class="su">1</span>, situated so that ζ<span class="su">0</span> +is between them. One of these must be within the circle (ρ). We +infer then that ρ = 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 − α, where α denotes a complex +number. This proposition alone suffices to suggest the importance +of complex numbers.</p> +</div> + +<p>§ 3. <i>Limiting Operations</i>.—In order that a complex number +ζ = ξ + iη may have a limit it is necessary and sufficient that each +of ξ and η has a limit. Thus an infinite series w<span class="su">0</span> + w<span class="su">1</span> + w<span class="su">2</span> + ..., +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 ε, a number m exists such that for every +n > m, and every positive p, the batch of terms w<span class="su">n</span> + w<span class="su">n+1</span> + + ... + w<span class="su">n+p</span> is less than ε in absolute value. If the terms depend +upon a complex variable z, the convergence is called <i>uniform</i> +for a range of values of z, when the inequality holds, for the +same ε and m, for all the points z of this range.</p> + +<div class="condensed"> +<p>The infinite series of most importance are those of which the +general term is a<span class="su">n</span>z<span class="sp">n</span>, wherein a<span class="su">n</span> 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 = z<span class="su">0</span> and every +n, |a<span class="su">n</span>z<span class="su">0</span><span class="sp">n</span>| < M, a condition which is satisfied, for instance, if the +series converges for z = z<span class="su">0</span>, then it is at once proved that the series +converges absolutely for every z for which |z| < |z<span class="su">0</span>|, and converges +uniformly over every range |z| < r′ for which r′ < |z<span class="su">0</span>|. +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 z<span class="su">0</span> on the circumference, as z +approaches to z<span class="su">0</span> provided the series converges for z = z<span class="su">0</span>, as can be +shown without much difficulty. Within a common circle of convergence +two power series Σ a<span class="su">n</span>z<span class="sp">n</span>, Σ b<span class="su">n</span>z<span class="sp">n</span> can be multiplied together +according to the ordinary rule, this being a consequence of a theorem +for absolutely convergent series. If r<span class="su">1</span> be less than the radius of +convergence of a series Σ a<span class="su">n</span> z<span class="sp">n</span> and for |z| = r<span class="su">1</span>, the sum of the series +be in absolute value less than a real positive quantity M, it can be +shown that for |z| = r<span class="su">1</span> every term is also less than M in absolute value, +namely, |a<span class="su">n</span>| < Mr<span class="su">1</span><span class="sp">−n</span>. If in every arbitrarily small neighbourhood of +z=0 there be a point for which two converging power series Σa<span class="su">n</span> z<span class="sp">n</span>, +Σb<span class="su">n</span>z<span class="sp">n</span> agree in value, then the series are identical, or a<span class="su">n</span> = b<span class="su">n</span>; thus also +if Σa<span class="su">n</span>z<span class="sp">n</span> 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 ƒ(z) = Σa<span class="su">n</span>z<span class="sp">n</span> of radius of +convergence R, if |z<span class="su">0</span>| < R and we put z = z<span class="su">0</span> + t with |t| < R-|z<span class="su">0</span>|, +the resulting series Σa<span class="su">n</span>(z<span class="su">0</span> + t)<span class="sp">n</span> may be regarded as a double series +in z<span class="su">0</span> and t, which, since |z<span class="su">0</span>| + t < R, is absolutely convergent; +it may then be arranged according to powers of t. Thus we may +write ƒ(z) = Σ A<span class="su">n</span>t<span class="sp">n</span>; hence A<span class="su">0</span> = ƒ(z<span class="su">0</span>), and we have [ƒ(z<span class="su">0</span> + t) − ƒ(z<span class="su">0</span>)]/t = +Σ<span class="su">n=1</span> A<span class="su">n</span>t<span class="sp">n−l</span>, wherein the continuous series on the right reduces to A<span class="su">1</span> +for t = 0; thus the ratio on the left has a definite limit when t = 0, +equal namely to A<span class="su">1</span> or Σna<span class="su">n</span>z<span class="su">0</span><span class="sp">n − 1</span>. In other words, the original series +may legitimately be differentiated at any interior point z<span class="su">0</span> of its circle +of convergence. Repeating this process we find ƒ(z<span class="su">0</span> + t) = Σt<span class="sp">n</span>ƒ<span class="sp">(n)</span>(z<span class="su">0</span>)/n!, +where ƒ<span class="sp">(n)</span>(z<span class="su">0</span>) is the nth differential coefficient. Repeating for this +power series, in t, the argument applied about z = 0 for Σa<span class="su">n</span>z<span class="sp">n</span>, we +infer that for the series ƒ(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 ƒ(z).</p> + +<p>Perhaps the simplest possible power series is e<span class="sp">z</span> = exp(z) = 1 + z<span class="sp">2</span>/2! + +z<span class="sp">3</span>/3! + ... of which the radius of convergence is infinite. By +multiplication we have exp(z)·exp(z<span class="sp">1</span>) = exp(z + z<span class="sp">1</span>). In particular +when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy). Now the +functions</p> + +<p class="center">U<span class="su">0</span> = sin y, V<span class="su">0</span> = 1 − cos y, U<span class="su">1</span> = y − sin y,</p> + +<p class="center">V<span class="su">1</span> = <span class="spp">1</span>⁄<span class="suu">2</span>y² − 1 + cos y, U<span class="su">2</span> = <span class="spp">1</span>⁄<span class="suu">6</span>y³ − y + sin y, V<span class="su">2</span> = <span class="spp">1</span>⁄<span class="suu">24</span>y<span class="sp">4</span> − <span class="spp">1</span>⁄<span class="suu">2</span>y<span class="sp">2</span> + 1 − cos y, ...</p> + +<p class="noind">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</p> + +<p class="center">cos y = 1 − <span class="spp">1</span>⁄<span class="suu">2</span>y² + <span class="spp">1</span>⁄<span class="suu">24</span>y<span class="sp">4</span> − ...,   sin y = y − <span class="spp">1</span>⁄<span class="suu">6</span>y³ + <span class="spp">1</span>⁄<span class="suu">120</span>y<span class="sp">5</span> − ...,</p> + +<p class="noind">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</p> + +<p class="center">exp(z + 2πi) = exp(x) [cos (y + 2π) + i sin(y + 2π)],</p> + +<p class="noind">which we express by saying that exp (z) has the period 2πi, +and hence also the period 2kπi, 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.</p> + +<p>Taking in the plane of z an infinite strip lying between the lines +y = 0, y = 2π and plotting the function ζ = exp (z) upon a new plane, +it follows at once from what has been said that every complex value +of ζ arises when z takes in turn all positions in this strip, and that +no value arises twice over. The equation ζ = exp(z) thus defines z, +regarded as depending upon ζ, with only an additive ambiguity +2kπi, where k is an integer. We write z = λ(ζ); when ζ is real this +becomes the logarithm of ζ; in general λ(ζ) = log |ζ| + i ph (ζ) + +2kπi, where k is an integer; and when ζ describes a closed circuit +surrounding the origin the phase of ζ increases by 2π, or k increases +by unity. Differentiating the series for ζ we have dζ/dz = ζ, so +that z, regarded as depending upon ζ, is also differentiable, with +dz/dζ = ζ<span class="sp">− 1</span>. On the other hand, consider the series ζ − 1 − ½(ζ− 1)<span class="sp">2</span> + +<span class="spp">1</span>⁄<span class="suu">3</span>(ζ − 1)<span class="sp">3</span> − ...; it converges when ζ = 2 and hence converges for +|ζ − 1| < 1; its differential coefficient is, however, 1 − (ζ − 1) + +(ζ − 1)<span class="sp">2</span> − ..., that is, (1 + ζ − 1)<span class="sp">− 1</span>. Wherefore if φ(ζ) denote this +series, for |ζ − 1| < 1, the difference λ(ζ) − φ(ζ), regarded as a +function of ξ and η, has vanishing differential coefficients; if we +take the value of λ(ζ) which vanishes when ζ = 1 we infer thence +that for |ζ − 1| < 1, λ(ζ) = Σ<span class="su">n=1</span> [(−1)<span class="sp">(n−1)</span>/n (ζ − 1)<span class="sp">n</span>. It is to be remarked +that it is impossible for ζ while subject to |ζ − 1| < 1 to make a +circuit about the origin. For values of ζ for which |ζ − 1| ≮ 1, we +can also calculate λ(ζ) with the help of infinite series, utilizing the +fact that λ(ζζ′) = λ(ζ) + λ(ζ′).</p> + +<p>The function λ(ζ) is required to define ζ<span class="sp">a</span> when ζ and a are complex +numbers; this is defined as exp [aλ(ζ)], that is as Σ<span class="su">n=0</span> a<span class="sp">n</span> [λ(ζ)]<span class="sp">n</span>/n!. +When a is a real integer the ambiguity of λ(ζ) is immaterial here, +since exp [aλ(ζ) + 2kaπi] = exp [aλ(ζ)]; when a is of the form 1/q, +where q is a positive integer, there are q values possible for ζ<span class="sp">1/q</span>, of +the form exp [1/q λ(ζ)] exp (2kπi/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 ζ; when a = p/q, where p, q are integers without +common factor, q being positive, we have ζ<span class="sp">p/q</span> = (ζ<span class="sp">1/q</span>)<span class="sp">p</span>. The +definition of the symbol ζ<span class="sp">a</span> 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<span class="sp">i</span>; the number i is of +modulus unity and phase ½π; thus λ(i) = i (½π + 2kπ); thus</p> + +<p class="center">i<span class="sp">i</span> = exp (−½π − 2kπ) = exp (−½π) exp (−2kπ),</p> + +<p class="noind">is always real, but has an infinite number of values.</p> + +<p><span class="pagenum"><a name="page312" id="page312"></a>312</span></p> + +<p>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 = ½[exp (iz) + exp (−iz)] and sin z = −½i [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.</p> +</div> + +<p>§4. <i>Of Functions of a Complex Variable in General</i>.—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.</p> + +<div class="condensed"> +<p>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 <i>region</i> 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 <i>path</i> 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.</p> + +<p>There is a logical principle also which must be referred to. We +frequently have cases where, about every interior or boundary, +point z<span class="su">0</span> of a certain region a circle can be put, say of radius r<span class="su">0</span>, such +that for all points z of the region which are interior to this circle, +for which, that is, |z − z<span class="su">0</span>| < r<span class="su">0</span>, a certain property holds. Assuming +that to r<span class="su">0</span> is given the value which is the upper limit for z<span class="su">0</span>, of the +possible values, we may call the points |z − z<span class="su">0</span>| < r<span class="su">0</span>, the neighbourhood +belonging to or <i>proper</i> to z<span class="su">0</span>, and may speak of the property +as the property (z, z<span class="su">0</span>). The value of r<span class="su">0</span> will in general vary with z<span class="su">0</span>; +what is in most cases of importance is the question whether the +lower limit of r<span class="su">0</span> for all positions is zero or greater than zero. (A) +This lower limit is certainly greater than zero provided the property +(z, z<span class="su">0</span>) is of a kind which we may call extensive; such, namely, that +if it holds, for some position of z<span class="su">0</span> and all positions of z, within a certain +region, then the property (z, z<span class="su">1</span>) holds within a circle of radius R +about any interior point z<span class="su">1</span> of this region for all points z for which +the circle |z − z<span class="su">1</span>| = R is within the region. Also in this case r<span class="su">0</span> +varies continuously with z<span class="su">0</span>. (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 <i>some</i> point z<span class="su">0</span> within or upon the boundary +of the sub-region, (2) for <i>every</i> point z within or upon the boundary +of the sub-region.</p> + +<p>We prove these statements (A), (B) in reverse order. To prove +(B) let a region for which the property (z, z<span class="su">0</span>) holds for all points z and +some point z<span class="su">0</span> of the region, be called <i>suitable</i>: 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 ζ; after a certain stage +all these will be interior to the proper region of ζ; this, however, is +contrary to the supposition that they are all unsuitable.</p> + +<p>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 ƒ(x, y). It may have a finite upper limit H +for the region, so that no point (x, y) exists for which ƒ(x, y) > H, +but points (x, y) exist for which ƒ(x, y) > H − ε, however small ε 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 ƒ(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 +ƒ(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 ƒ(x, y) is the +radius r<span class="su">0</span> of the neighbourhood proper to any point z<span class="su">0</span>, spoken of +above. We can hence prove the statement (A) above.</p> + +<p>Suppose the property (z, z<span class="su">0</span>) extensive, and, if possible, that the +lower limit of r<span class="su">0</span> is zero. Let then ζ be a point such that the lower +limit of r<span class="su">0</span> is zero for points z<span class="su">0</span> within a circle about ζ however small; +let r be the radius of the neighbourhood proper to ζ; take z<span class="su">0</span> so +that |z<span class="su">0</span>-ζ| < ½r; the property (z, z<span class="su">0</span>), being extensive, holds +within a circle, centre z<span class="su">0</span>, of radius r − |z<span class="su">0</span> − ζ|, which is greater +than |z<span class="su">0</span> − ζ|, and increases to r as |z<span class="su">0</span> − ζ| diminishes; this being +true for all points z<span class="su">0</span> near ζ, the lower limit of r<span class="su">0</span> is not zero for the +neighbourhood of ζ, contrary to what was supposed. This proves +(A). Also, as is here shown that r<span class="su">0</span> ⋝ r − |z<span class="su">0</span> − ζ|, may similarly be +shown that r ⋝ r<span class="su">0</span> − |z<span class="su">0</span> − ζ|. Thus r<span class="su">0</span> differs arbitrarily little from +r when |z<span class="su">0</span> − ζ| is sufficiently small; that is, r<span class="su">0</span> varies continuously +with z<span class="su">0</span>. Next suppose the function ƒ(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 z<span class="su">0</span>, +within or upon the boundary of the region, η 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 +|ƒ(x, y) −ƒ(x<span class="su">0</span>, y<span class="su">0</span>)| < ½η, and therefore (x′, y′) being any other point +interior to this circle, |ƒ(x′, y′) − ƒ(x, y)| < η. We can then apply +the result (A) obtained above, taking for the neighbourhood proper +to any point z<span class="su">0</span> the circular area within which, for any two points +(x, y), (x′, y′), we have |ƒ(x′, x′) − ƒ(x, y)| < η. 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 − z<span class="su">0</span>| = r about any point z<span class="su">0</span>, we have |ƒ(x′, y′) − ƒ(x, y)| < η, +and, in particular, |ƒ(x, y) −ƒ(x<span class="su">0</span>, y<span class="su">0</span>)| < η, where η is an arbitrary +real positive quantity agreed upon beforehand.</p> + +<p>Take now any path in the region, whose extreme points are z<span class="su">0</span>, z, +and let z<span class="su">1</span>, ... z<span class="su">n−1</span> be intermediate points of the path, in order; +denote the continuous function ƒ(x, y) by ƒ(z), and let ƒ<span class="su">r</span> denote any +quantity such that |ƒ<span class="su">r</span> − ƒ(z<span class="su">r</span>)| ⋜ |ƒ(z<span class="su">r+1</span>) − ƒ(z<span class="su">r</span>)|; consider the sum</p> + +<p class="center">(z<span class="su">1</span> − z<span class="su">0</span>)ƒ<span class="su">0</span> + (z<span class="su">2</span> − z<span class="su">1</span>)ƒ<span class="su">1</span> + ... + (z − z<span class="su">n−1</span>)ƒ<span class="su">n−1</span>.</p> + +<p class="noind">By the definition of a path we can suppose, n being large enough, +that the intermediate points z<span class="su">1</span>, ... z<span class="su">n − 1</span> are so taken that if z<span class="su">i</span>, +z<span class="su">i + 1</span> be any two points intermediate, in order, to z<span class="su">r</span> and z<span class="su">r + 1</span>, we have +|z<span class="su">i + i</span>-z<span class="su">i</span>| < |z<span class="su">r+1</span> − z<span class="su">r</span>|; we can thus suppose |z<span class="su">1</span> − z<span class="su">0</span>|, |z<span class="su">2</span> − z<span class="su">1</span>|, ... +|z − z<span class="su">n−1</span>|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<span class="su">r, 1</span>, z<span class="su">r, 2</span>, ... z<span class="su">r, m−1</span> be intermediate in order to z<span class="su">r</span> and z<span class="su">r+1</span>, and +|ƒ<span class="su">r, i</span> − ƒ(z<span class="su">r, i</span>)| < |ƒ(z<span class="su">r, i+1</span>) − ƒ(z<span class="su">r, i</span>)|, the difference between Σ(z<span class="su">r+1</span> − z<span class="su">r</span>)ƒ<span class="su">r</span> +and</p> + +<p class="center"><span class="f150">Σ</span> { (z<span class="su">r, 1</span>-z<span class="su">r</span>)ƒ<span class="su">r, 0</span> + (z<span class="su">r, 2</span> − z<span class="su">r, 1</span>)ƒ<span class="su">r, 1</span> + ... + (z<span class="su">r+1</span> − z<span class="su">r, m−1</span>)ƒ<span class="su">r, m−1</span> },</p> + +<p class="noind">which is equal to</p> + +<p class="center"><span class="f150">Σ</span><span class="su">r</span> <span class="f150">Σ</span><span class="su">i</span> (z<span class="su">r, i+1</span> − z<span class="su">r, i</span>) (ƒ<span class="su">r, i</span> − ƒ<span class="su">r</span>),</p> + +<p class="noind">is, when |z<span class="su">r+1</span> − z<span class="su">r</span>| is small enough, to ensure |ƒ(z<span class="su">r+1</span>) − ƒ(z<span class="su">r</span>)| < η, +less in absolute value than</p> + +<p class="center"><span class="f150">Σ</span>2η <span class="f150">Σ</span> |z<span class="su">r, i+1</span> − z<span class="su">r, i</span>|,</p> + +<p class="noind">which, if S be the upper limit of the perimeter of the polygon from +which the path is generated, is < 2ηS, and is therefore arbitrarily +small.</p> + +<p>The limit in question is called <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> ƒ(z)dz. In particular when +ƒ(z) = 1, it is obvious from the definition that its value is z − z<span class="su">0</span>; +when ƒ(z) = z, by taking ƒ<span class="su">r</span> = ½(z<span class="su">r+1</span> − z<span class="su">r</span>), it is equally clear that its +value is ½(z² − z<span class="su">0</span>²); these results will be applied immediately.</p> + +<p>Suppose now that to every interior and boundary point z<span class="su">0</span> of a +certain region there belong two definite finite numbers ƒ(z<span class="su">0</span>), F(z<span class="su">0</span>), +such that, whatever real positive quantity η may be, a real positive +number ε exists for which the condition</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">|</span></td> <td>ƒ(z) − ƒ(z<span class="su">0</span>)</td> +<td rowspan="2">− F(z<span class="su">0</span>) <span class="f150">|</span> < η,</td></tr> +<tr><td class="denom">z − z<span class="su">0</span></td></tr></table> + +<p class="noind">which we describe as the condition (z, z<span class="su">0</span>), is satisfied for every point z, +within or upon the boundary of the region, satisfying the limitation +|z − z<span class="su">0</span>| < ε. Then ƒ(z<span class="su">0</span>) is called a differentiable function of the +complex variable z<span class="su">0</span> over this region, its differential coefficient being +F(z<span class="su">0</span>). The function ƒ(z<span class="su">0</span>) is thus a continuous function of the real +<span class="pagenum"><a name="page313" id="page313"></a>313</span> +variables x<span class="su">0</span>, y<span class="su">0</span>, where z<span class="su">0</span> = x<span class="su">0</span> + iy<span class="su">0</span>, over the region; it will appear +that F(z<span class="su">0</span>) is also continuous and in fact also a differentiable function +of z<span class="su">0</span>.</p> + +<p>Supposing η to be retained the same for all points z<span class="su">0</span> of the region, +and σ<span class="su">0</span> to be the upper limit of the possible values of ε for the point z<span class="su">0</span>, +it is to be presumed that σ<span class="su">0</span> will vary with z<span class="su">0</span>, and it is not obvious +as yet that the lower limit of the values of σ<span class="su">0</span> as z<span class="su">0</span> 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, z<span class="su">0</span>), above, is satisfied for all points z, within or upon +the boundary of this sub-region, for an appropriate position of z<span class="su">0</span>, +within or upon the boundary of this sub-region. This is proved +above as result (B).</p> + +<p>Hence it can be proved that, for a differentiable function ƒ(z), +the integral <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z1</span> ƒ(z)dz has the same value by whatever path within +the region we pass from z<span class="su">1</span> to z. This we prove by showing that when +taken round a closed path in the region the integral ∫ƒ(z)dz vanishes. +Consider first a triangle over which the condition (z, z<span class="su">0</span>) holds, for +some position of z<span class="su">0</span> and every position of z, within or upon the +boundary of the triangle. Then as</p> + +<p class="center">ƒ(z) = ƒ(z<span class="su">0</span>) + (z − z<span class="su">0</span>) F(z<span class="su">0</span>) + ηθ(z − z<span class="su">0</span>), where |θ| < 1,</p> + +<p class="noind">we have</p> + +<p class="center">∫ƒ(z)dz = [ƒ(z<span class="su">0</span>) − z<span class="su">0</span> F(z<span class="su">0</span>)] ∫dz + F(z<span class="su">0</span>) ∫zdz + η∫θ(z − z<span class="su">0</span>)dz,</p> + +<p class="noind">which, as the path is closed, is η ∫θ(z − z<span class="su">0</span>)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 ηap, +where a is the greatest side of the triangle and p is its perimeter; if +Δ be the area of the triangle, we have Δ = ½ab sin C > (α/π) ba, where +α is the least angle of the triangle, and hence a(a + b + c) < 2a(b + c) +< 4πΔ/α; the integral ∫ƒ(z)dz round the perimeter of the triangle +is thus < 4πηΔ/α. Now consider any region made up of triangles, +as before explained, in each of which the condition (z, z<span class="su">0</span>) holds, as +in the triangle just taken. The integral ∫ƒ(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 +4πηK/α, where K is the whole area of the region, and α is the smallest +angle of the component triangles. However small η 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.</p> + +<p>Hence we can deduce the remarkable result that the value of ƒ(z) +at any interior point of a region is expressible in terms of the value +of ƒ(z) at the boundary points. For consider in the original region +the function ƒ(z)/(z − z<span class="su">0</span>), where z<span class="su">0</span> is an interior point: this satisfies +the same conditions as ƒ(z) except in the immediate neighbourhood +of z<span class="su">0</span>. Taking out then from the original region a small regular +polygonal region with z<span class="su">0</span> as centre, the theorem holds for the remaining +portion. Proceeding to the limit when the polygon becomes a +circle, it appears that the integral <span class="f150">∫</span> dzƒ(z)/(z − z<span class="su">0</span>) round the boundary of +the original region is equal to the same integral taken counter-clockwise +round a small circle having z<span class="su">0</span> as centre; on this circle, +however, if z − z<span class="su">0</span> = rE(iθ), dz/(z − z<span class="su">0</span>) = idθ, and ƒ(z) differs arbitrarily +little from f(z<span class="su">0</span>) if r is sufficiently small; the value of the integral +round this circle is therefore, ultimately, when r vanishes, equal to +2πiƒ(z<span class="su">0</span>). Hence ƒ(z<span class="su">0</span>) = 1/2πi <span class="f150">∫</span> (dtƒ(t)/(t − z<span class="su">0</span>), where this integral is round the +boundary of the original region. From this it appears that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">F(z<span class="su">0</span>) = lim.</td> <td>ƒ(z) − ƒ(z<span class="su">0</span>)</td> +<td rowspan="2">=</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>dtƒ(t)</td></tr> +<tr><td class="denom">z − z<span class="su">0</span></td> <td class="denom">2πi</td> +<td class="denom">(t − z<span class="su">0</span>)²</td></tr></table> + +<p class="noind">also round the boundary of the original region. This form shows, +however, that F(z<span class="su">0</span>) is a continuous, finite, differentiable function of z<span class="su">0</span> +over the whole interior of the original region.</p> +</div> + +<p>§ 5. <i>Applications.</i>—The previous results have manifold applications.</p> + +<div class="condensed"> +<p>(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) = u<span class="su">0</span>(z) + u<span class="su">1</span>(z) + ... + +u<span class="su">n−1</span>(z) + R<span class="su">n</span>(z), where |R<span class="su">n</span>(z)| < ε for all points of the path, we have</p> + +<table class="math0" summary="math"> +<tr><td> +<span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> S(z)dz = <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> u<span class="su">0</span>(z)dz + <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> u<span class="su">1</span>(z)dz + ... + <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> u<span class="su">n−1</span>(z)dz + <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> R<span class="su">n</span>(z)dz, +</td></tr></table> + +<p class="noind">wherein, in absolute value, <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> R<span class="su">n</span>(z)dz < εL, if L be the length of the +path. Thus the series may be integrated, and the resulting series +is also uniformly convergent.</p> + +<p>(2) If ƒ(x, y) be definite, finite and continuous at every point of a +region, and over any closed path in the region ∫ƒ(x, y)dz = 0, then +ψ(z) = <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z0</span> ƒ(x, y)dz, for interior points z<span class="su">0</span>, z, is a differentiable function +of z, having for its differential coefficient the function ƒ(x, y), which +is therefore also a differentiable function of z at interior points.</p> + +<p>(3) Hence if the series u<span class="su">0</span>(z) + u<span class="su">1</span>(z) + ... to ∞ 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 (1/2πi) ∫ 2πi/(t − z)², is obtainable by differentiating the +series. This theorem, unlike (1), does not hold for functions of a +real variable.</p> + +<p>(4) If the region of definition of a differentiable function ƒ(z) +include the region bounded by two concentric circles of radii r, R, +with centre at the origin, and z<span class="su">0</span> be an interior point of this region,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(z<span class="su">0</span>) =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>ƒ(t)dt</td> +<td rowspan="2">−</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>ƒ(t)dt</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">2πi</td> <td class="denom">R<span class="sp">t</span> − z<span class="su">0</span></td> +<td class="denom">2πi</td> <td class="denom">r<span class="sp">t</span> − z<span class="su">0</span></td></tr></table> + +<p class="noind">where the integrals are both counter-clockwise +round the two circumferences respectively; putting in the +first (t − z<span class="su">0</span>)<span class="sp">−1</span> = <span class="f150">Σ</span><span class="su">n=0</span> z<span class="su">0</span><span class="sp">n</span>/t<span class="sp">n+1</span>, and in the second (t − z<span class="su">0</span>)<span class="sp">−1</span> = − <span class="f150">Σ</span><span class="su">n=0</span> t<span class="sp">n</span>/z<span class="su">0</span><span class="sp">n+1</span>, +we find ƒ(z<span class="su">0</span>) = <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">−∞</span> A<span class="su">n</span>z<span class="su">0</span><span class="sp">n</span>, wherein A<span class="su">n</span> = (1/2πi) <span class="f150">∫</span> [ƒ(t)/t<span class="sp">n+1</span>] dt, taken round any +circle, centre the origin, of radius intermediate between r and R. +Particular cases are: (α) 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<span class="su">n</span> for which n < 0 all vanish, and +the function ƒ(z<span class="su">0</span>) is expressed for the whole interior |z<span class="su">0</span>| < R by a +power series <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">0</span> A<span class="su">n</span>z<span class="su">0</span><span class="sp">n</span>. In other words, <i>about every interior point c of +the region of definition a differentiable function of z is expressible by a +power series in z − c</i>; a very important result.</p> + +<p>(β) 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<span class="sp">m</span>ƒ(z) has a finite limit when |z| diminishes +to zero, all the coefficients A<span class="su">n</span> for which n < −m vanish, and we have</p> + +<p class="center">f(z<span class="su">0</span>) = A<span class="su">−m</span>z<span class="su">0</span><span class="sp">−m</span> + A<span class="su">−m+1</span>z<span class="su">0</span><span class="sp">−m+1</span> + ... + A<span class="su">−1</span>z<span class="su">0</span><span class="sp">−1</span> + A<span class="su">0</span> + A<span class="su">1</span>z<span class="su">0</span> ... to ∞.</p> + +<p class="noind">Such a case occurs, for instance, when ƒ(z) = cosec z, the number m +being unity.</p> +</div> + +<p>§ 6. <i>Singular Points.</i>—The <i>region of existence</i> 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 <i>singular point</i> of the function.</p> + +<div class="condensed"> +<p>About every interior point z<span class="su">0</span> of the region of existence the function +may be represented by a power series in z − z<span class="su">0</span>, and the series converges +and represents the function over any circle centre at z<span class="su">0</span> +which contains no singular point in its interior. This has been +proved above. And it can be similarly proved, putting z = 1/ζ, +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<span class="sp">− 1</span> or ζ; in such case we say +that the region of existence of the function contains the point z = ∞. +A series in z<span class="sp">− 1</span> has a finite limit when |z| = ∞; 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 Σa<span class="su">n</span>z<span class="sp">n</span> in z is in absolute value less than M, +we have |a<span class="su">n</span>| < Mr<span class="sp">−n</span>, and therefore, if M remains finite for all values +of r however great, a<span class="su">n</span> = 0. Thus the region of existence of a function +if it contains all finite points of the plane cannot contain the point +z = ∞; such is, for instance, the case of the function exp (z) = Σz<span class="sp">n</span>/n!. +This may be regarded as a particular case of a well-known result +(§ 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 <span class="f150">Σ</span> <span class="su">n=0</span> z<span class="sp">m</span>, where m = n², the series <span class="f150">Σ</span> <span class="su">n=0</span>z<span class="sp">m</span> +where m = n!, and the series <span class="f150">Σ</span> <span class="su">n=1</span> z<span class="sp">m</span>/(m + 1)(m + 2) where m = a<span class="sp">n</span>, +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 − z<span class="su">0</span>; as z<span class="su">0</span> +approaches to a singular point of the function, lying on the circle +of convergence, the radii of convergence of these derived series in +z − z<span class="su">0</span> diminish to zero; when, however, a circle can be put about z<span class="su">0</span>, +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 − z<span class="su">0</span> 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 − z<span class="su">0</span> +converging beyond the original circle gives what is known as an +<i>analytical continuation</i> of the function. It appears from what has +<span class="pagenum"><a name="page314" id="page314"></a>314</span> +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.</p> +</div> + +<p>§ 7. <i>Monogenic Functions</i>.—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.</p> + +<div class="condensed"> +<p>Starting with a convergent power series, say in powers of z, this +series can be arranged in powers of z − z<span class="su">0</span>, about any point z<span class="su">0</span> interior +to its circle of convergence, and the new series converges certainly for +|z − z<span class="su">0</span>| < r − |z<span class="su">0</span>|, if r be the original radius of convergence. If for +every position of z<span class="su">0</span> 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 z<span class="su">0</span> +the derived series converges for |z − z<span class="su">0</span>| < r − |z<span class="su">0</span>| + D, then it can be +shown that for points z, interior to the original circle, lying in the +annulus r − |z<span class="su">0</span>| < |z − z<span class="su">0</span>| < r − |z<span class="su">0</span>| + D, the value represented by the +derived series agrees with that represented by the original series. +If for another point z<span class="su">1</span> interior to the original circle the derived series +converges for |z − z<span class="su">1</span>| < r − |z<span class="su">1</span>| + E, and the two circles |z − z<span class="su">0</span>| = +r − |z<span class="su">0</span>| + D, |z − z<span class="su">1</span>| = r − |z<span class="su">1</span>| + 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 <i>monogenic</i> 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 <i>monogenic +analytical</i> function, or simply an <i>analytical</i> 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 ∫ƒ(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 <i>Théorie +des fonctions analytiques</i>. 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.</p> +</div> + +<p>§ 8. <i>Some Elementary Properties of Single Valued Functions</i>.—A +<i>pole</i> is a singular point of the function ƒ(z) which is not a +singularity of the function 1/ƒ(z); this latter function is therefore, +by the definition, capable of representation about this point, +z<span class="su">0</span>, by a series [ƒ(z)]<span class="sp">−1</span> = Σa<span class="su">n</span>(z − z<span class="su">0</span>)<span class="sp">n</span>. If herein a<span class="su">0</span> is not zero we +can hence derive a representation for ƒ(z) as a power series about +z<span class="su">0</span>, contrary to the hypothesis that z<span class="su">0</span> is a singular point for this +function. Hence a<span class="su">0</span> = 0; suppose also a<span class="su">1</span> = 0, a<span class="su">2</span> = 0, ... a<span class="su">m−1</span> = 0, +but a<span class="su">m</span> ± 0. Then [ƒ(z)]<span class="sp">−1</span> = (z − z<span class="su">0</span>)<span class="sp">m</span>[a<span class="su">m</span> + a<span class="su">m+1</span> (z − z<span class="su">0</span>) + ...], and +hence (z − z<span class="su">0</span>)<span class="sp">m</span>ƒ(z) = a<span class="su">m</span><span class="sp">−1</span> + Σb<span class="su">n</span> (z − z<span class="su">0</span>)<span class="sp">n</span>, namely, the expression of +ƒ(z) about z = z<span class="su">0</span> contains a finite number of negative powers +of z − z<span class="su">0</span> and a (finite or) infinite number of positive powers. +Thus a pole is always an isolated singularity.</p> + +<div class="condensed"> +<p>The integral ∫ƒ(z)dz taken by a closed circuit about the pole not +containing any other singularity is at once seen to be 2πiA<span class="su">1</span>, where +A<span class="su">1</span> is the coefficient of (z − z<span class="su">0</span>)<span class="sp">−1</span> in the expansion of ƒ(z) at the pole; +this coefficient has therefore a certain uniqueness, and it is called +the <i>residue of ƒ(z) at the pole</i>. Considering a region in which there +are no other singularities than poles, all these being interior points, +<i>the integral (1/2πi) <span class="f150">∫</span> ƒ(z)dz round the boundary of this region is equal to +the sum of the residues at the included poles</i>, a very important result. +Any singular point of a function which is not a pole is called an +<i>essential singularity</i>; 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 +<span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">−∞</span> a<span class="su">n</span>(z − z<span class="su">0</span>)<span class="sp">n</span>; it thus cannot remain finite in the immediate neighbourhood +of the point. The point is necessarily an isolated essential +singularity also of the function {ƒ(z) − A}<span class="sp">−1</span> for if this were expressible +by a power series about the point, so would also the function ƒ(z) +be; as {ƒ(z) − A}<span class="sp">− 1</span> approaches infinity, so does ƒ(z) approach the +arbitrary value A. Similar remarks apply to the point z = ∞, the +function being regarded as a function of ζ = z<span class="sp">−1</span>. 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.</p> +</div> + +<p>A single valued function is said to be an <i>integral</i> function +when it has no singular points except z = ∞. Such is, for +instance, an integral polynomial, which has z = ∞ for a pole, and +the functions exp (z) which has z = ∞ as an essential singularity. +A function which has no singular points for finite values of +z other than poles is called a <i>meromorphic</i> function. If it also +have a pole at z = ∞ it is a <i>rational</i> function; for then, if +a<span class="su">1</span>, ... a<span class="su">s</span> be its finite poles, of orders m<span class="su">1</span>; m<span class="su">2</span>, ... m<span class="su">s</span>, the +product (z − a<span class="su">1</span>)<span class="sp">m<span class="su">1</span></span> ... (z − a<span class="su">s</span>) <span class="sp">m<span class="su">s</span></span>ƒ(z) is an integral function with +a pole at infinity, capable therefore, for large values of z, of an +expression (z<span class="sp">−1</span>)<span class="sp">−m</span> <span class="f150">Σ</span> <span class="su">r=0</span> a<span class="su">r</span>(z<span class="sp">−1</span>)<span class="sp">r</span>; thus (z − a<span class="su">1</span>)<span class="sp">m1</span> ... (z − a<span class="su">s</span>)<span class="sp">m<span class="su">s</span></span>ƒ(z) +is capable of a form <span class="f150">Σ</span> <span class="su">r=0</span> b<span class="su">r</span>z<span class="sp">r</span>, but z<span class="sp">−m</span> <span class="f150">Σ</span> <span class="su">r=0</span> b<span class="su">r</span>z<span class="sp">r</span> remains finite for +z = ∞. Therefore b<span class="su">r+1</span> = b<span class="su">r+2</span> = ... = 0, andƒ(z) is a rational +function.</p> + +<div class="condensed"> +<p>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 a<span class="su">1</span>, a<span class="su">2</span>, a<span class="su">3</span>, ... with |a<span class="su">1</span>| ⋜ |a<span class="su">2</span>| ⋜ |a<span class="su">3</span>| ... and limit +|a<span class="su">n</span>| = ∞. About a<span class="su">s</span> the function is expressible as <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">−∞</span> A<span class="su">n</span>(z − a<span class="su">s</span>)<span class="sp">n</span>; +let ƒ<span class="su">s</span>(z) = <span class="f150">Σ</span> <span class="sp1">1</span><span class="su1">−∞</span> A<span class="sp">n</span>(z − a<span class="su">s</span>)<span class="sp">n</span> be the sum of the negative powers in this +expansion. Assuming z = 0 not to be a singular point, let ƒ<span class="su">s</span>(z) be +expanded in powers of z, in the form <span class="f150">Σ</span> <span class="su">n=0</span> C<span class="su">n</span>z<span class="sp">n</span>, and μ<span class="su">s</span> be chosen so +that F<span class="su">s</span>(z) = ƒ<span class="su">s</span>(z) − <span class="f150">Σ</span> <span class="sp1">μ<span class="su">s</span>−1</span><span class="su1">1</span> C<span class="su">n</span>z<span class="sp">n</span> = <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">μ<span class="su">s</span></span> C<span class="su">n</span>z<span class="sp">n</span> is, for |z| < r<span class="su">s</span> < |a<span class="su">s</span>|, less in absolute +value than the general term ε<span class="su">s</span> of a fore-agreed convergent series of +real positive terms. Then the series φ(z) = <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">s=1</span> F<span class="su">s</span>(z) converges uniformly +in any finite region of the plane, other than at the points a<span class="su">s</span>, +and is expressible about any point by a power series, and near +a<span class="su">s</span>, φ(z) − f<span class="su">s</span>(z) is expressible by a power series in z − a<span class="su">s</span>. Thus +F(z) − φ(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<span class="su">s</span>(z)| < ε<span class="su">s</span> is imposed only to render the series ΣF<span class="su">s</span>(z) +uniformly convergent; this condition may in particular cases be +satisfied by a series <span class="f150">Σ</span> G<span class="su">s</span>(z) where G<span class="su">s</span>(z) = ƒ<span class="su">s</span>(z) − <span class="f150">Σ</span> <span class="sp1">ν<span class="su">s</span>−1</span><span class="su1">1</span> C<span class="su">n</span>z<span class="sp">n</span> and ν<span class="su">s</span> < μ<span class="su">s</span>. +An example of the theorem is the function π cot πz − z<span class="sp">− 1</span> for which, +taking at first only half the poles, ƒ<span class="su">s</span>(z) = 1/(z − s); in this case the +series <span class="f150">Σ</span> F<span class="su">s</span>(z) where F<span class="su">s</span>(z) = (z − s)<span class="sp">−1</span> + s<span class="sp">−1</span> is uniformly convergent; +thus π cot πz − z<span class="sp">−1</span> − <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">−∞</span> [(z − s)<span class="sp">−1</span> + s<span class="sp">−1</span>], where s = 0 is excluded from +the summation, is an integral function. It can be proved that this +integral function vanishes.</p> + +<p>Considering an integral function ƒ(z), if there be no finite positions +of z for which this function vanishes, the function λ[ƒ(z)] is at once +seen to be an integral function, φ(z), or ƒ(z) = exp[φ(z)]; if however +great R may be there be only a finite number of values of z for which +ƒ(z) vanishes, say z = a<span class="su">1</span>, ... a<span class="su">m</span>, then it is at once seen that ƒ(z) = +exp [φ(z)]. (z − a<span class="su">1</span>)<span class="sp">h1</span>...(z − a<span class="su">m</span>)<span class="sp">h<span class="su">m</span></span>, where φ(z) is an integral function, +and h<span class="su">1</span>, ... h<span class="su">m</span> are positive integers. If, however, ƒ(z) vanish for z = a<span class="su">1</span>, +a<span class="su">2</span> ... where |a<span class="su">1</span>| ⋜ |a2| ⋜ ... and limit |a<span class="su">n</span>| = ∞, and if for simplicity +we assume that z − 0 is not a zero and all the zeros a<span class="su">1</span>, a<span class="su">2</span>, ... are +of the first order, we find, by applying the preceding theorem to +the function [1 / ƒ(z)] [dƒ(z) / dz], that ƒ(z) = exp [φ(z)] <span class="f150">Π</span> <span class="sp1">∞</span><span class="su1">n=1</span> {(1 − z/a<span class="su">n</span>) exp φ<span class="su">n</span>(z)}, +where φ(z) is an integral function, and φ<span class="su">n</span>(z) is an integral polynomial +of the form φ<span class="su">n</span>(z) = z/a<span class="su">n</span> + z<span class="sp">2</span>/2a<span class="su">n</span><span class="sp">2</span> + ... + z<span class="sp">s</span>/sa<span class="su">n</span><span class="sp">s</span>. 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 +<span class="pagenum"><a name="page315" id="page315"></a>315</span> +sinπx/πx, we have</p> + +<table class="math0" summary="math"> +<tr><td>sin πx</td> +<td rowspan="2">= <span class="f150">Π</span> <span class="sp1">∞</span><span class="su1">−∞</span> <span class="f150">{ (</span>1 −</td> <td>x</td> +<td rowspan="2"><span class="f150">)</span> exp <span class="f150">(</span></td> <td>x</td> +<td rowspan="2"><span class="f150">) }</span>,</td></tr> +<tr><td class="denom">πx</td> <td class="denom">n</td> +<td class="denom">n</td></tr></table> + +<p class="noind">where n = 0 is excluded from the product. Or again we have</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2">= xe<span class="sp">C<span class="su">x</span></span> <span class="f150">Π</span> <span class="sp1">∞</span><span class="su1">n=1</span> <span class="f150">{ (</span>1 +</td> <td>x</td> +<td rowspan="2"><span class="f150">)</span> exp <span class="f150">(</span> −</td> <td>x</td> +<td rowspan="2"><span class="f150">) }</span>,</td></tr> +<tr><td class="denom">Γ(x)</td> <td class="denom">n</td> +<td class="denom">n</td></tr></table> + +<p class="noind">where C is a constant, and Γ(x) is a function expressible when x is +real and positive by the integral <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span> e<span class="sp">−t</span> t<span class="sp">x−1</span>dt.</p> + +<p>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 ƒ(z), and the law of increase of the coefficients in the series +ƒ(z) = <span class="f150">Σ</span> a<span class="su">n</span>z<span class="sp">n</span> as n increases (see the bibliography below under <i>Integral +Functions</i>). 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 § 21, <i>Modular +Functions</i>).</p> +</div> + +<p>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 = ∞ 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:—</p> + +<div class="condensed"> +<p>I. If a<span class="su">1</span>, a<span class="su">2</span>, a<span class="su">3</span>, ... 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 a<span class="su">1</span>, a<span class="su">2</span>, ..., and with every point a<span class="su">i</span> there be associated +a polynomial in (z − a<span class="su">i</span>)<span class="sp">−1</span>, say g<span class="su">i</span>; 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<span class="su">i</span> a pole whereat the expansion +consists of the terms g<span class="su">i</span>, together with a power series in z − a<span class="su">i</span>; +the function is expressible as an infinite series of terms g<span class="su">i</span> − γ<span class="su">i</span>, +where γ<span class="su">i</span> is also a rational function.</p> + +<p>II. With a similar aggregate (a), with limiting points (c), suppose +with every point a<span class="su">i</span> there is associated a positive integer r<span class="su">i</span>. Then +there exists a single valued function whose region of existence +excludes only the points (c), vanishing to order r<span class="su">i</span> at the point a<span class="su">i</span>, +but not elsewhere, expressible in the form</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">Π</span> <span class="sp1">∞</span><span class="su1">n=1</span> <span class="f150">(</span> 1 −</td> <td>a<span class="su">n</span> − c<span class="su">n</span></td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">r <span class="su">n</span></span> exp (g<span class="su">n</span>),</td></tr> +<tr><td class="denom">z − c<span class="su">n</span></td></tr></table> + +<p class="noind">where with every point a<span class="su">n</span> is associated a proper point c<span class="su">n</span> of (c), and</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">g<span class="su">n</span> = r<span class="su">n</span> <span class="f150">Σ</span> <span class="sp1">μ <span class="su">n</span></span><span class="su1">s=1</span></td> <td>1</td> +<td rowspan="2"><span class="f150">(</span></td> <td>a<span class="su">n</span> − c<span class="su">n</span></td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">s</span>,</td></tr> +<tr><td class="denom">s</td> <td class="denom">z − c<span class="su">n</span></td></tr></table> + +<p class="noind">μ<span class="su">n</span> being a properly chosen positive integer.</p> + +<p>If it should happen that the points (c) determine a path dividing +the plane into separated regions, as, for instance, if a<span class="su">n</span> = R(1 − n<span class="sp">−1</span>) exp (iπ √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.</p> +</div> + +<p>§ 9. <i>Construction of a Monogenic Function with a given Region +of Existence.</i>—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.</p> + +<p>§ 10. <i>Expression of a Monogenic Function by means of Rational +Functions in a given Region.</i>—Suppose that we have a region R<span class="su">0</span> +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 C<span class="su">0</span>. 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 C<span class="su">0</span>; 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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">|</span></td> <td>a − b</td> +<td rowspan="2"><span class="f150">|</span> < ε, for |a − b| < η;</td></tr> +<tr><td class="denom">z − b</td></tr></table> + +<p class="noind">the rational function of the complex variable t,</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"><span class="f150">[</span> 1 − <span class="f150">(</span></td> <td>a − b</td> +<td rowspan="2"><span class="f150">)</span><span class="sp1">n</span> <span class="f150">]</span>,</td></tr> +<tr><td class="denom">t − a</td> <td class="denom">t − a</td></tr></table> + +<p class="noind">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 R<span class="su">0</span>, differs as little as may be +desired from (t − a)<span class="sp">−1</span>. By taking a sum of terms such as</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">F = <span class="f150">Σ</span> A<span class="su">p</span> <span class="f150">{</span></td> <td>1</td> +<td rowspan="2"><span class="f150">[</span> 1 − <span class="f150">(</span></td> <td>a − b</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">n</span> <span class="f150">] }</span> <span class="sp1">p</span>,</td></tr> +<tr><td class="denom">t − a</td> <td class="denom">t − b</td></tr></table> + +<p class="noind">we can thus build a rational function differing, in value, in +R<span class="su">0</span>, as little as may be desired from a given rational function</p> + +<p class="center">ƒ = <span class="f150">Σ</span> A<span class="su">p</span>(t − a)<span class="sp">−p</span>,</p> + +<p class="noind">and differing, outside R or upon the boundary of R, from ƒ, +in the fact that while ƒ 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 R<span class="su">0</span> +arbitrarily little from the former, whose poles are all at the +point c.</p> + +<div class="condensed"> +<p>Now any monogenic function ƒ(t) whose region of definition includes +C and the interior of R can be represented at all points z in R<span class="su">0</span> by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(z) =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>ƒ(t)dt</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">2πi</td> <td class="denom">t − z</td></tr></table> + +<p class="noind">where the path of integration is C. This integral is the limit of a +sum</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">S =</td> <td>1</td> +<td rowspan="2"><span class="f150">Σ</span></td> <td>ƒ(t<span class="su">i</span>) (t<span class="su">i+1</span> − t<span class="su">i</span>)</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">2πi</td> <td class="denom">t<span class="su">i</span> − z</td></tr></table> + +<p class="noind">where the points t<span class="su">i</span> are upon C; and the proof we have given of the +existence of the limit shows that the sum S converges to ƒ(z) uniformly +in regard to z, when z is in R<span class="su">0</span>, so that we can suppose, when +the subdivision of C into intervals t<span class="su">i+1</span> − t<span class="su">i</span>, has been carried sufficiently +far, that</p> + +<p class="center">|S − ƒ(z)| < ε,</p> + +<p class="noind">for all points z of R<span class="su">0</span>, where ε is arbitrary and agreed upon beforehand. +The function S is, however, a rational function of z with poles upon C, +that is external to R<span class="su">0</span>. We can thus find a rational function differing +arbitrarily little from S, and therefore arbitrarily little from ƒ(z), +for all points z of R<span class="su">0</span>, with poles at arbitrary positions outside R<span class="su">0</span> +which can be reached by finite continuous curves lying outside R +from the points of C.</p> + +<p>In particular, to take the simplest case, if C<span class="su">0</span>, C be simple closed +polygons, and Γ 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 ƒ(z) for all points of R<span class="su">0</span> whose +poles are at one finite point c external to Γ. By a transformation +of the form t − c = r<span class="sp">−1</span>, 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 ε<span class="su">1</span>, ε<span class="su">2</span>, ... +to be an indefinitely continued sequence of real positive numbers, +converging to zero, and P<span class="su">r</span> to be the polynomial such that, within +C<span class="su">0</span>, |P<span class="su">r</span> − ƒ(z)| < ε<span class="su">r</span>; then the infinite series of polynomials</p> + +<p class="center">P<span class="su">1</span>(z) + {P<span class="su">2</span>(z) − P<span class="su">1</span>(z)} + {P<span class="su">3</span>(z) − P<span class="su">2</span>(z)} + ...,</p> + +<p class="noind">whose sum to n terms is P<span class="su">n</span>(z), converges for all finite values of z and +represents ƒ(z) within C<span class="su">0</span>.</p> + +<p>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 +Γ of a region, we can suppose the poles of the rational function, +constructed to approximate to ƒ(z) within R<span class="su">0</span>, to be at points of Γ. +A series of rational functions of the form</p> + +<p class="center">H<span class="su">1</span>(z) + {H<span class="su">2</span>(z) − H<span class="su">1</span>(z)} + {H<span class="su">3</span>(z) − H<span class="su">2</span>(z)} + ...</p> + +<p class="noind">then, as before, represents ƒ(z) within R<span class="su">0</span>. And R<span class="su">0</span> may be taken to +coincide as nearly as desired with the interior of the region bounded +by Γ.</p> +</div> + +<p>§ 11. <i>Expression of</i> (1 − z)<span class="sp">−1</span> <i>by means of Polynomials. Applications.</i>—We +pursue the ideas just cursorily explained in some +further detail.</p> + +<div class="condensed"> +<p>Let c be an arbitrary real positive quantity; putting the complex +variable ζ = ξ + iη, enclose the points ζ = l, ζ = 1 + c by means +of (i.) the straight lines η = ±a, from ξ = l to ξ = 1 + c, (ii.) a semicircle +convex to ζ = 0 of equation (ξ − 1)<span class="sp">2</span> + η<span class="sp">2</span> = a<span class="sp">2</span>, (iii.) a semicircle +concave to ζ = 0 of equation (ξ − 1 − c)<span class="sp">2</span> + η<span class="sp">2</span> = a<span class="sp">2</span>. 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 σ = 1/r (c/a). Now take</p> + +<p class="center">c<span class="su">1</span> = 1 + c/r, c<span class="su">2</span> = 1 + 2c/r, ... c<span class="su">r</span> = 1 + c;</p> + +<p><span class="pagenum"><a name="page316" id="page316"></a>316</span></p> + +<p class="noind">if n<span class="su">1</span>, n<span class="su">2</span>, ... n<span class="su">r</span>, be positive integers, the rational function</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"><span class="f150">{</span> 1 − <span class="f150">(</span></td> <td>c<span class="su">1</span> − 1</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">n<span class="su">1</span></span> <span class="f150">}</span></td></tr> +<tr><td class="denom">1 − ζ</td> <td class="denom">c<span class="su">1</span> − ζ</td></tr></table> + +<p class="noind">is finite at ζ = 1, and has a pole of order n<span class="su">1</span> at ζ = c<span class="su">1</span>; the rational +function</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"><span class="f150">{</span> 1 − <span class="f150">(</span></td> <td>c<span class="su">1</span> − 1</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">n<span class="su">1</span></span> <span class="f150">} {</span> 1 − <span class="f150">(</span></td> <td>c<span class="su">2</span> − c<span class="su">1</span></td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">n<span class="su">2</span></span> <span class="f150">}</span> <span class="sp1">n<span class="su">1</span></span></td></tr> +<tr><td class="denom">1 − ζ</td> <td class="denom">c<span class="su">1</span> − ζ</td> +<td class="denom">c<span class="su">2</span> − ζ</td></tr></table> + +<p class="noind">is thus finite except for ζ = c<span class="su">2</span>, where it has a pole of order n<span class="su">1</span>n<span class="su">2</span>; +finally, writing</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x<span class="su">s</span> = <span class="f150">(</span></td> <td>c<span class="su">s</span> − c<span class="su">s−1</span></td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">n<span class="su">s</span></span>,</td></tr> +<tr><td class="denom">c<span class="su">s</span> − ζ</td></tr></table> + +<p class="noind">the rational function</p> + +<table class="math0" summary="math"><tr><td> +U = (1 − ζ)<span class="sp">−1</span> (1 − x<span class="su">1</span>) (1 − x<span class="su">2</span>)<span class="sp">n<span class="su">1</span></span> (1 − x<span class="su">3</span>)<span class="sp">n<span class="su">1</span>n<span class="su">2</span></span> ... (1 − x<span class="su">r</span>)<span class="sp">n<span class="su">1</span>n<span class="su">2</span> ... n<span class="su">r − 1</span></span> +</td></tr></table> + +<p class="noind">has a pole only at ζ = 1 + c, of order n<span class="su">1</span>n<span class="su">2</span> ... n<span class="su">r</span>.</p> + +<p>The difference (1 − ζ)<span class="sp">−1</span> − U is of the form (1 − ζ)<span class="sp">−1</span>P, where P, of +the form</p> + +<p class="center">1 − (1 − ρ<span class="su">1</span>) (1 − ρ<span class="su">2</span>)...(1 − ρ<span class="su">k</span>),</p> + +<p class="noind">in which there are equalities among ρ<span class="su">1</span>, ρ<span class="su">2</span>, ... ρ<span class="su">k</span>, is of the form</p> + +<p class="center">Σρ<span class="su">1</span> − Σρ<span class="su">1</span>ρ<span class="su">2</span> + Σρ<span class="su">1</span>ρ<span class="su">2</span>ρ<span class="su">3</span> − ...;</p> + +<p class="noind">therefore, if |r<span class="su">i</span>| = |ρ<span class="su">i</span>|, we have</p> + +<table class="math0" summary="math"><tr><td> +|P| < Σ r<span class="su">1</span> + Σ r<span class="su">1</span>r<span class="su">2</span> + Σ r<span class="su">1</span>r<span class="su">2</span>r<span class="su">3</span> + ... < (1 + r<span class="su">1</span>) (1 + r<span class="su">2</span>)...(1 + r<span class="su">k</span>) − 1; +</td></tr></table> + +<p class="noind">now, so long as ζ is without the closed curve above described round +ζ = 1, ζ = 1 + c, we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">|</span></td> <td>1</td> +<td rowspan="2"><span class="f150">|</span> <</td> <td>1</td> +<td rowspan="2">, <span class="f150">|</span></td> <td>c<span class="su">m</span> − c<span class="su">m−1</span></td> +<td rowspan="2"><span class="f150">|</span> <</td> <td>c/r</td> +<td rowspan="2">< σ,</td></tr> +<tr><td class="denom">1 − ζ</td> <td class="denom">a</td> +<td class="denom">c<span class="su">m</span> − ζ</td> <td class="denom">a</td></tr></table> + +<p class="noind">and hence</p> + +<table class="math0" summary="math"><tr><td> +|(1 − ζ)<span class="sp">−1</span> − U| < a<span class="sp">−1</span> {(1 + σ<span class="sp">n<span class="su">1</span></span>) (1 + σ<span class="sp">n<span class="su">2</span></span>)<span class="sp">n<span class="su">1</span></span> (1 + σ<span class="sp">n<span class="su">3</span></span>)<span class="sp">n<span class="su">1</span>n<span class="su">2</span></span> ... (1 + σ<span class="sp">n<span class="su">r</span></span>)<span class="sp">n<span class="su">1</span>n<span class="su">2</span> ... n<span class="su">r−1</span> − 1</span>}. +</td></tr></table> + +<p>Take an arbitrary real positive ε, and μ, a positive number, so that +ε<span class="sp">mu</span> − 1 < εa, then a value of n<span class="su">1</span> such that σ<span class="sp">n<span class="su">1</span></span> < μ/(1 + μ) and therefore +σ<span class="sp">n<span class="su">1</span></span>/(1 − σ<span class="sp">n<span class="su">1</span></span> < μ, and values for n<span class="su">2</span>, n<span class="su">3</span> ... such that σ<span class="sp">n<span class="su">2</span></span> < 1/n<span class="su">1</span> σ<span class="sp">2n<span class="su">1</span></span>, +σ<span class="sp">n<span class="su">3</span></span> < 1/n<span class="su">1</span>n<span class="su">2</span> σ<span class="sp">3n<span class="su">1</span></span>, ... σ<span class="sp">n</span><span class="su">r</span> < 1/(n<span class="su">1</span> ... n<span class="su">r−1</span>) σ<span class="sp">n<span class="su">r</span> n<span class="su">1</span></span>; then, as 1 + x < e<span class="sp">x</span>, we have</p> + +<table class="math0" summary="math"><tr><td> +|(−ζ)<span class="sp">−1</span> − U| < a<span class="sp">−1</span> {exp (σ<span class="sp">n<span class="su">1</span></span> + n<span class="su">1</span>σ<span class="sp">n<span class="su">2</span></span> + n<span class="su">1</span>n<span class="su">2</span>σ<span class="sp">n<span class="su">3</span></span> + ... + n<span class="su">1</span>n<span class="su">2</span> ... n<span class="su">r−1</span>σ<span class="sp">n<span class="su">r</span></span>) − 1}, +</td></tr></table> + +<p class="noind">and therefore less than</p> + +<p class="center">a<span class="sp">−1</span> {exp (σ<span class="sp">n<span class="su">1</span></span> + σ<span class="sp">2n<span class="su">1</span></span> + ... + σ<span class="sp">n<span class="su">r</span> n<span class="su">1</span></span>) − 1},</p> + +<p class="noind">which is less than</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"><span class="f150">[</span> exp <span class="f150">(</span></td> <td>σ<span class="sp">n<span class="su">1</span></span></td> +<td rowspan="2"><span class="f150">)</span> − 1 <span class="f150">]</span></td></tr> +<tr><td class="denom">a</td> <td class="denom">1 − σ<span class="sp">n<span class="su">1</span></span></td></tr></table> + +<p class="noind">and therefore less than ε.</p> + +<p>The rational function U, with a pole at ζ = 1 + c, differs therefore +from (1 − ζ)<span class="sp">−1</span>, for all points outside the closed region put about +ζ = 1, ζ = l + c, by a quantity numerically less than ε. So long as +a remains the same, r and σ will remain the same, and a less value +of ε will require at most an increase of the numbers n<span class="su">1</span>, n<span class="su">2</span>, ... n<span class="su">r</span>; but +if a be taken smaller it may be necessary to increase r, and with this +the complexity of the function U.</p> + +<p>Now put</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">z =</td> <td>cζ</td> +<td rowspan="2">,   ζ =</td> <td>(c + 1)z</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">c + 1 − ζ</td> <td class="denom">c + z</td></tr></table> + +<p class="noind">thereby the points ζ = 0, 1, 1 + c become the points z = 0, 1, ∞, the +function (1 − z)<span class="sp">−1</span> being given by (1 − z)<span class="sp">−1</span> = c(c + 1)<span class="sp">−1</span> (1 − ζ)<span class="sp">−1</span> + (c + 1)<span class="sp">−1</span>; +the function U becomes a rational function of z with a pole only at +z = ∞, that is, it becomes a polynomial in z, say [(c + 1)/c] H − 1/c, where H +is also a polynomial in z, and</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2">− H =</td> <td>c</td> +<td rowspan="2"><span class="f150">[</span></td> <td>1</td> +<td rowspan="2">− U <span class="f150">]</span>;</td></tr> +<tr><td class="denom">1 − z</td> <td class="denom">c + 1</td> +<td class="denom">1 − ζ</td></tr></table> + +<p class="noind">the lines η = ±a become the two circles expressed, if z = x + iy, by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">(x + c)² + y² = ±</td> <td>c(c + 1)</td> +<td rowspan="2">y,</td></tr> +<tr><td class="denom">a</td></tr></table> + +<p class="noind">the points (η = 0, ξ = 1 − a), (η = 0, ξ = 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 = −∞). The circle (x + c)² + y² = c(c + 1)y/a can be written</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">y =</td> <td>(x + c)²</td> +<td rowspan="2">+</td> <td>(x + c)<span class="sp">4</span></td> +<td rowspan="2">{μ + √[μ² − (x + c)²]}<span class="sp">−2</span>,</td></tr> +<tr><td class="denom">2μ</td> <td class="denom">2μ</td></tr></table> + +<p class="noind">where μ = ½c(c + 1)/a; its ordinate y, for a given value of x, can +therefore be supposed arbitrarily small by taking a sufficiently small.</p> + +<p>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 ⋜ ∞, we can take a quantity a, and hence, with an +arbitrary c, determine a number r; then corresponding to an arbitrary +ε<span class="su">s</span>, we can determine a polynomial P<span class="su">s</span>, such that, for all points +interior to the region, we have</p> + +<p class="center">|(1 − z<span class="sp">−1</span>) − P<span class="su">s</span>| < ε<span class="su">s</span>;</p> + +<p class="noind">thus the series of polynomials</p> + +<p class="center">P<span class="su">1</span> + (P<span class="su">2</span> − P<span class="su">1</span>) + (P<span class="su">3</span> − P<span class="su">2</span>) + ...,</p> + +<p class="noind">constructed with an arbitrary aggregate of real positive numbers +ε<span class="su">1</span>, ε<span class="su">2</span>, ε<span class="su">3</span>, ... with zero as their limit, converges uniformly and +represents (1 − z)<span class="sp">−1</span> for the whole region considered.</p> + +<p>§ 12. <i>Expansion of a Monogenic Function in Polynomials, over a +Star Region.</i>—Now consider any monogenic function ƒ(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 = ∞, +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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">2πiƒ(x) = <span class="f150">∫</span></td> <td>ƒ(t)</td> +<td rowspan="2"> </td> <td>ƒ(t)</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">t</td> <td class="denom">1 − xt<span class="sp">−1</span></td></tr></table> + +<p class="noind">where ƒ(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 R<span class="su">0</span> lying entirely within R, +and let x be restricted to be within R<span class="su">0</span> or upon its boundary; then +for any point t on the boundary of R, the points z of the plane for +which zt<span class="sp">− 1</span> is real and positive and equal to or greater than 1, being +points for which |z| = |t| or |z| > |t|, are without the region R<span class="su">0</span>, and +not infinitely near to its boundary points. Taking then an arbitrary +real positive ε we can determine a polynomial in xt<span class="sp">− 1</span>, say P(xt<span class="sp">−1</span>), +such that for all points x in R<span class="su">0</span> we have</p> + +<p class="center">|(1 − xt<span class="sp">−1</span>)<span class="sp">−1</span> − P(xt<span class="sp">−1</span>)| < ε;</p> + +<p class="noind">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 ε,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">|</span> 2πiƒ(x) − <span class="f150">∫</span></td> <td>dt</td> +<td rowspan="2">ƒ(t)P(xt<span class="sp">−1</span>) <span class="f150">|</span> = <span class="f150">|</span> <span class="f150">∫</span></td> <td>dt</td> +<td rowspan="2">ƒ(t)E <span class="f150">|</span> ⋜ εLM,</td></tr> +<tr><td class="denom">t</td> <td class="denom">t</td></tr></table> + +<p class="noind">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<span class="sp">−1</span> ƒ(t)| < M. If now</p> + +<p class="center">P (xt<span class="sp">−1</span>) = c<span class="su">0</span> + c<span class="su">1</span>xt<span class="sp">−1</span> + ... + c<span class="su">m</span> x<span class="sp">m</span> t<span class="sp">−m</span>,</p> + +<p class="noind">and</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"><span class="f150">∫</span> t<span class="sp">−r−1</span> ƒ(t)dt = μ<span class="su">r</span>,</td></tr> +<tr><td class="denom">2πi</td></tr></table> + +<p class="noind">this gives</p> + +<p class="center">|ƒ(x) − {c<span class="su">0</span>μ<span class="su">0</span> + c<span class="su">1</span>μ<span class="su">1</span>x + ... + c<span class="su">m</span>μ<span class="su">m</span>x<span class="sp">m</span>}| ⋜ εLM/2π,</p> + +<p class="noind">where the quantities μ<span class="su">0</span>, μ<span class="su">1</span>, μ<span class="su">2</span>, ... are the coefficients in the expansion +of ƒ(x) about the origin.</p> + +<p>If then an arbitrary finite region be constructed of the kind +explained, excluding the barriers joining the singular points of ƒ(x) +to x = ∞, it is possible, corresponding to an arbitrary real positive +number σ, to determine a number m, and a polynomial Q(x), of +order m, such that for all interior points of this region</p> + +<p class="center">|ƒ(x) − Q(x)| < σ.</p> + +<p>Hence as before, within this region ƒ(x) can be represented by a +series of polynomials, converging uniformly; when ƒ(x) is not a +single valued function the series represents one branch of the function.</p> + +<p>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, φ(t) be capable of expression as a power +series in t − x about a point x, for |t − x| ⋜ ρ, and for all points of this +circle |φ(t)| < g, we know that |φ<span class="sp">(n)</span>(x)| < gρ<span class="sp">−n</span>(n!). Hence, taking +|z| < <span class="spp">1</span>⁄<span class="suu">3</span>ρ, and, for any assigned positive integer μ, taking m so that +for n > m we have (μ + n)<span class="sp">μ</span> < (<span class="spp">3</span>⁄<span class="suu">2</span>)<span class="sp">n</span>, we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">|</span></td> <td>φ<span class="sp">(μ + n)</span>(x)·z<span class="sp">n</span></td> +<td rowspan="2"><span class="f150">|</span> <</td> <td>φ<span class="sp">(μ + n)</span>(x)</td> +<td rowspan="2">(μ + n)<span class="sp">μ</span> |z|<span class="sp">n</span> <</td> <td>g</td> +<td rowspan="2"><span class="f150">(</span></td> <td>3</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">n</span> <span class="f150">(</span></td> <td>ρ</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">n</span> <</td> <td>g</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">n!</td> +<td class="denom">(μ + n)!</td> <td class="denom">ρ<span class="sp">μ + n</span></td> +<td class="denom">2</td> <td class="denom">3</td> <td class="denom">ρ<span class="sp">μ</span> 2<span class="sp">n</span></td></tr></table> + +<p class="noind">and therefore</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ<span class="sp">μ</span> (x + z) = <span class="f150">Σ</span> <span class="sp1">m</span><span class="su1">n=0</span></td> <td>φ<span class="sp">(μ + n)</span> (x)</td> +<td rowspan="2">z<span class="sp">n</span> + ε<span class="su">μ</span>,</td></tr> +<tr><td class="denom">n!</td></tr></table> + +<p class="noind">where</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">|</span>ε<span class="su">μ</span><span class="f150">|</span> <</td> <td>g</td> +<td rowspan="2"><span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">n=m+1</span></td> <td>1</td> +<td rowspan="2"><</td> <td>g</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">ρ<span class="sp">μ</span></td> <td class="denom">2<span class="sp">n</span></td> +<td class="denom">ρ<span class="sp">μ</span> 2<span class="sp">m</span></td></tr></table> + +<p>Now draw barriers as before, directed from the origin, joining the +singular point of φ(z) to z = ∞, take a finite region excluding all +these barriers, let ρ be a quantity less than the radii of convergence +of all the power series developments of φ(z) about interior points of +this region, so chosen moreover that no circle of radius ρ with centre +at an interior point of the region includes any singular point of φ(z), +let g be such that |φ(z)| < g for all circles of radius ρ 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| < <span class="spp">1</span>⁄<span class="suu">3</span>ρ; then take the +points a<span class="su">1</span> = x/n, a<span class="su">2</span> = 2x/n, a<span class="su">3</span> = 3x/n, ... a<span class="su">n</span> = 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</p> + +<p><span class="pagenum"><a name="page317" id="page317"></a>317</span></p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ<span class="sp">(μ)</span> (a<span class="su">1</span>) = <span class="f150">Σ</span> <span class="sp1">m1</span><span class="su1">λ1=0</span></td> <td>φ<span class="sp">(μ + λ1)</span> (0)</td> +<td rowspan="2"><span class="f150">(</span></td> <td>x</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">λ1</span> + α<span class="su">μ</span></td></tr> +<tr><td class="denom">λ<span class="su">1</span>!</td> +<td class="denom">n</td></tr></table> + +<p class="noind">with</p> + +<p class="center">α<span class="su">μ</span> < g/ρ<span class="sp">μ</span> 2<span class="sp">m1</span>,</p> + +<p class="noind">provided (μ + m<span class="su">1</span> + 1)<span class="su">μ</span> < (<span class="spp">2</span>⁄<span class="suu">3</span>)<span class="sp">m1 + 1</span>; in fact for μ ⋜ 2n<span class="sp">2n−2</span> it is sufficient +to take m<span class="su">1</span> = n<span class="sp">2n</span>; by another application of the same inequality, +replacing x, z respectively by a<span class="su">1</span> and x/n, we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ<span class="sp">(μ)</span> (a<span class="su">2</span>) = <span class="f150">Σ</span> <span class="sp1">m<span class="su">2</span></span><span class="su1">λ<span class="su">2</span>=0</span></td> + <td>φ<span class="sp">(μ + λ<span class="su">2</span>)</span> (a<span class="su">1</span>)</td> +<td rowspan="2"><span class="f150">(</span></td> <td>x</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">λ<span class="su">2</span></span> + β′<span class="su">μ</span> ,</td></tr> +<tr><td class="denom">λ<span class="su">2</span>!</td> <td class="denom">n</td></tr></table> + +<p class="noind">where</p> + +<p class="center">|β′<span class="su">μ</span>| < g / ρ<span class="sp">μ</span> 2<span class="sp">m</span><span class="su">2</span></p> + +<p class="noind">provided (μ + m<span class="su">2</span> + 1)<span class="sp">μ</span> < (<span class="spp">3</span>⁄<span class="suu">2</span>)<span class="sp">m<span class="su">2</span></span> + 1; we take m<span class="su">2</span> = n<span class="sp">2n − 2</span>, supposing +μ < 2n<span class="sp">2n−4</span>. So long as λ<span class="su">2</span> ⋜ m<span class="su">2</span> ⋜ n<span class="sp">2n−2</span> and μ < 2n<span class="sp">2n−4</span> we have +μ + λ<span class="su">2</span> < 2n<span class="sp">2n−2</span>, and we can use the previous inequality to substitute +here for φ<span class="sp">(μ + λ<span class="su">2</span>)</span> (a<span class="su">1</span>). When this is done we find</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ<span class="sp">(μ)</span> (a<span class="su">2</span>) = + <span class="f150">Σ</span> <span class="sp1">m<span class="su">2</span></span><span class="su1">λ<span class="su">2</span>=0</span> + <span class="f150">Σ</span> <span class="sp1">m<span class="su">1</span></span><span class="su1">λ<span class="su">1</span>=0</span></td> + <td>φ<span class="sp">(μ + λ<span class="su">1</span> + λ<span class="su">2</span>)</span> (0)</td> +<td rowspan="2"><span class="f150">(</span></td> <td>x</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">λ<span class="su">1</span> + λ<span class="su">2</span></span> + β<span class="su">μ</span> ,</td></tr> +<tr><td class="denom">λ<span class="su">1</span>! λ<span class="su">2</span>!</td> <td class="denom">n</td></tr></table> + +<p class="noind">where |βμ| < 2g/ρ<span class="sp">μ</span> 2<span class="sp">m<span class="su">2</span></span>, the numbers m<span class="su">1</span>, m<span class="su">2</span> being respectively n<span class="sp">2n</span> +and n<span class="sp">2n−2</span>.</p> + +<p>Applying then the original inequality to φ<span class="sp">(μ)</span> (a<span class="su">3</span>) = φ<span class="sp">(μ)</span> (a<span class="su">2</span> + x/n), +and then using the series just obtained, we find a series for φ<span class="sp">(μ)</span> (a<span class="su">3</span>). +This process being continued, we finally obtain</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ(x) = + <span class="f150">Σ</span> <span class="sp1">m<span class="su">1</span></span><span class="su1">λ<span class="su">1</span>=0</span> + <span class="f150">Σ</span> <span class="sp1">m<span class="su">2</span></span><span class="su1">λ<span class="su">2</span>=0</span> ... + <span class="f150">Σ</span> <span class="sp1">m<span class="su">n</span></span><span class="su1">λ<span class="su">n</span>=0</span></td> + <td>φ<span class="sp">h</span> (0)</td> +<td rowspan="2"><span class="f150">(</span></td> <td>x</td> +<td rowspan="2"><span class="f150">)</span> <span class="sp1">h</span> + ε ,</td></tr> +<tr><td class="denom">K</td> <td class="denom">n</td></tr></table> + +<p class="noind">where h = λ<span class="su">1</span> + λ<span class="su">2</span> + ... + λ<span class="su">n</span>, K = λ<span class="su">1</span>! λ<span class="su">2</span>! ... λ<span class="su">n</span>!, +m<span class="su">1</span> = n<span class="sp">2n</span>, m<span class="su">2</span> = n<span class="sp">2n−2</span>, ..., m<span class="su">n</span>= n², |ε| < 2g/2<span class="sp">m</span><span class="su">n</span>.</p> + +<p>By this formula φ(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.</p> +</div> + +<p>§ 13. <i>Application of Cauchy’s Theorem to the Determination of +Definite Integrals.</i>—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.</p> + +<div class="condensed"> +<p>We are to evaluate an integral <span class="f150">∫</span> <span class="sp1">b</span><span class="su1">a</span> ƒ(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 ∫ƒ(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, ƒ(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 ∫ƒ(z)dz taken +round these points. Two instances will suffice to explain the +method. (1) The integral <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span> [(tan x)/x] dx is convergent if it be understood +to mean the limit when ε, ζ, σ, ... all vanish of the sum of the +integrals</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span> <span class="sp1">1/2π−ε</span><span class="su1">0</span></td> <td>tan x</td> +<td rowspan="2">dx,  <span class="f150">∫</span> <span class="sp1">3/2π−ζ</span><span class="su1">1/2π+ε</span></td> <td>tan x</td> +<td rowspan="2">dx,  <span class="f150">∫</span> <span class="sp1">5/2π−σ</span><span class="su1">3/2π+ζ</span></td> <td>tan x</td> +<td rowspan="2">dx, ...</td></tr> +<tr><td class="denom">x</td> <td class="denom">x</td> +<td class="denom">x</td></tr></table> + +<p class="noind">Now draw a contour consisting in part of the whole of the positive +and negative real axis from x = −nπ to x = +nπ, where n is a positive +integer, broken by semicircles of small radius whose centres are the +points x = ±½π, x = ±¾π, ... , the contour containing also the lines +x = nπ and x = −nπ for values of y between 0 and nπ tan α, where α +is a small fixed angle, the contour being completed by the portion +of a semicircle of radius nπ sec α which lies in the upper half of the +plane and is terminated at the points x = ±nπ, y = nπ tan α. Round +this contour the integral <span class="f150">∫</span> [(tan z / z)] dz has the value zero. The contributions +to this contour integral arising from the semicircles of centres +−½(2s − 1)π, + ½(2s − 1)π, 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 <span class="f150">∫</span> <span class="sp1">nπ</span><span class="su1">0</span> [(tan x / x)] dx. The contribution +to the contour integral from the two straight portions at +x = ±nπ is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span> <span class="sp1">nπ tan α</span><span class="su1">0</span> idy <span class="f150">(</span></td> <td>tan iy</td> +<td rowspan="2">−</td> <td>tan iy</td> +<td rowspan="2"><span class="f150">)</span></td></tr> +<tr><td class="denom">nπ + iy</td> <td class="denom">−nπ + iy</td></tr></table> + +<p class="noind">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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span> <span class="sp1">nπ tan α</span><span class="su1">0</span> dy</td> <td>2nπ</td> +<td rowspan="2">, that is than 2α.</td></tr> +<tr><td class="denom">n²π² + y²</td></tr></table> + +<p>Finally, for the remaining part of the contour, for which, with +R = nπ sec α, we have z = R(cos θ + i sin θ) = RE(iθ), we have</p> + +<table class="math0" summary="math"> +<tr><td>dz</td> +<td rowspan="2">= idθ, i tan z =</td> <td>exp(−R sin θ) E(iR cos θ) − exp(R sin θ) E(−iR cos θ)</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">z</td> <td class="denom">exp(−R sin θ) E(iR cos θ) + exp(R sin θ) E(−iR cos θ)</td></tr></table> + +<p class="noind">when n and therefore R is very large, the limit of this contribution +to the contour integral is thus</p> + +<p class="center">− <span class="f150">∫</span> <span class="sp1">π−α</span><span class="su1">α</span> dθ = − (π − 2α).</p> + +<p class="noind">Making n very large the result obtained for the whole contour is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">2 <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span></td> <td>tan x</td> +<td rowspan="2">dx − (π − 2α) − 2αε = 0,</td></tr> +<tr><td class="denom">x</td></tr></table> + +<p class="noind">where ε is numerically less than unity. Now supposing α to diminish +to zero we finally obtain</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span></td> <td>tan x</td> +<td rowspan="2">dx =</td> <td>π</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">x</td> <td class="denom">2</td></tr></table> + +<p class="noind">(2) For another case, to illustrate a different point, we may take the +integral</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span></td> <td>z<span class="sp">a−1</span></td> +<td rowspan="2">dz,</td></tr> +<tr><td class="denom">1 + z</td></tr></table> + +<p class="noind">wherein a is real quantity such that 0 < a < 1, and the contour consists +of a small circle, z = rE(iθ), terminated at the points x = r cos α, +y = ± r sin α, where α is small, of the two lines y = ± r sin α for +r cos α ⋜ x ⋜ R cos β, where R sin β = r sin α, and finally of a large +circle z = RE(iφ), terminated at the points x = R cos β, y = ±R sin β. +We suppose α and β both zero, and that the phase of z is zero for +r cos a ⋜ x ⋜ R cos β, y = r sin α = R sin β. Then on r cos α ⋜ x ⋜ R cos β, +y = −r sin α, the phase of z will be 2π, and z<span class="sp">α − 1</span> will be equal to +x<span class="sp">α − 1</span> exp [2πi(a − 1)], where x is real and positive. The two straight +portions of the contour will thus together give a contribution</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">[1 − exp(2πiα)] <span class="f150">∫</span> <span class="sp1">R cos β</span><span class="su1">r cos α</span></td> <td>x<span class="sp">a−1</span></td> +<td rowspan="2">dx.</td></tr> +<tr><td class="denom">1 + x</td></tr></table> + +<p class="noind">It can easily be shown that if the limit of zƒ(z) for z = 0 is zero, the +integral ∫ƒ(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 zƒ(z) for z = ∞ 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 ƒ(z) = z<span class="sp">α−1</span>/(1 + z), we have +zƒ(z) = z<span class="sp">a</span>/(1 + z), which, for 0 < a < 1, diminishes to zero both for z = 0 +and for z = ∞. Thus, finally the limit of the contour integral when +r = 0, R = ∞ is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">[1 − exp(2πiα)] <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span></td> <td>x<span class="sp">α−1</span></td> +<td rowspan="2">dx.</td></tr> +<tr><td class="denom">1 + x</td></tr></table> + +<p class="noind">Within the contour ƒ(z) is single valued, and has a pole at z = 1; at +this point the phase of z is π and z<span class="sp">a−1</span> is exp [iπ(a − 1)] or − exp(iπa); +this is then the residue of ƒ(z) at z = −1; we thus have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">[1 − exp (2πia)] <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span></td> <td>x<span class="sp">a−1</span></td> +<td rowspan="2">dx = −2πi exp(iπa),</td></tr> +<tr><td class="denom">1 + x</td></tr></table> + +<p class="noind">that is</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">0</span></td> <td>x<span class="sp">a−1</span></td> +<td rowspan="2">dx = π cosec (aπ).</td></tr> +<tr><td class="denom">1 + x</td></tr></table> +</div> + +<p>§ 14. <i>Doubly Periodic Functions.</i>—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.</p> + +<div class="condensed"> +<p>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 ƒ(z) is periodic is to say that there exists a constant ω +such that for every point z of the interior of the region of existence +of ƒ(z) we have ƒ(z + ω) = ƒ(z). This involves, considering all existing +periods ω = ρ + iσ, that there exists a lower limit of ρ² + σ² other than +zero; for otherwise all the differential coefficients of ƒ(z) would be +zero, and ƒ(z) a constant; we can then suppose that not both ρ +and σ are numerically less than ε, where ε > σ. Hence, if g be any +real quantity, since the range (−g, ... g) contains only a finite +number of intervals of length ε, and there cannot be two periods +ω = ρ + iσ such that με ⋜ ρ < (μ + 1)ε, νε ⋜ σ < (ν + 1)ε, where μ, ν are +integers, it follows that there is only a finite number of periods +for which both ρ and σ are in the interval (−g ... g). Considering +then all the periods of the function which are real multiples of one +period ω, and in particular those periods λω wherein 0 < λ ⋜ 1, there is +a lower limit for λ, 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 λ, say λ<span class="su">0</span>. If +Ω = λ<span class="su">0</span>ω and λ = Mλ<span class="su">0</span> + λ′, where M is an integer and 0 ⋜ λ′ < λ<span class="su">0</span>, any +period λω is of the form MΩ + λ′ω; since, however, Ω, MΩ and λω +are periods, so also is λ′ω, and hence, by the construction of λ<span class="su">0</span>, +we have λ′ = 0; thus all periods which are real multiples of ω are +expressible in the form MΩ where M is an integer, and Ω a period.</p> + +<p>If beside ω the functions have a period ω′ which is not a real +multiple of ω, consider all existing periods of the form μω + νω′ +wherein μ, ν are real, and of these those for which 0 ⋜ μ ⋜ 1, 0 < ν ⋜ 1; +<span class="pagenum"><a name="page318" id="page318"></a>318</span> +as before there is a least value for ν, actually occurring in one or +more periods, say in the period Ω′ = μ<span class="su">0</span>ω + ν<span class="su">0</span>ω′; now take, if μω + νω′ +be a period, ν = N′ν<span class="su">0</span> + ν′, where N′ is an integer, and 0 ⋜ ν′ < ν<span class="su">0</span>; +thence μω + νω′ = μω + N′(Ω′ − μ<span class="su">0</span>ω) + ν′ω′; take then μ − Nμ<span class="su">0</span> = Nλ<span class="su">0</span> + λ′, +where N is an integer and λ<span class="su">0</span> is as above, and 0 ⋜ λ′ < λ<span class="su">0</span>; we +thus have a period NΩ + N′Ω′ + λ′ω + ν′ω′, and hence a period +λ′ω + ν′ω′, wherein λ′ < λ<span class="su">0</span>, ν′ < ν<span class="su">0</span>; hence ν′ = 0 and λ′ = 0. All +periods of the form μω + νω′ are thus expressible in the form +NΩ + N′Ω′, where Ω, Ω′ 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 μ, ν real, in the +form μω + νω′; for if ω = ρ + iσ, ω′ = ρ′ + iσ′, this requires only +P = μρ + νρ′, Q = μσ + νσ′, equations which, since ω′/ω is not real, +always give finite values for μ and ν.</p> + +<p>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 MΩ, where Ω 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 NΩ + N′Ω′, +where Ω, Ω′ are periods, and N, N′ are integers. In the former case, +putting ζ = 2πiz/Ω, and the function ƒ(z) = φ(ζ), the function φ(ζ) +has, like exp (ζ), the period 2πi, and if we take t = exp (ζ) or ζ = λ(t) +the function is a single valued function of t. If then in particular ƒ(z) +is an integral function, regarded as a function of t, it has singularities +only for t = 0 and t = ∞, and may be expanded in the form <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">−∞</span> a<span class="su">n</span> t<span class="sp">n</span>.</p> + +<p>Taking the case when the single valued monogenic function has +two periods ω, ω′ whose ratio is not real, we can form a network +of parallelograms covering the plane of z whose angular points are +the points c + mω + m′ω′, 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, <i>congruent</i> to a definite point, z, of the primary parallelogram, +z′ − z being of the form mω + m′ω′, where m, m′ are integers. +Such a function cannot be an integral function, since then, if, in the +primary parallelogram |ƒ(z)| < M, it would also be the case, on a circle +of centre the origin and radius R, that |ƒ(z)| < M, and therefore, if +Σa<span class="su">n</span> z<span class="sp">n</span> be the expansion of the function, which is valid for an integral +function for all finite values of z, we should have |a<span class="su">n</span>| < MR<span class="sp">−n</span>, which +can be made arbitrarily small by taking R large enough. The +function must then have singularities for finite values of z.</p> + +<p>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/(2πi) <span class="f150">∫</span>ƒ(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 + ω +(or as z, z + ω′) are mutually destructive. This integral is, however, +equal to the sum of the residues of ƒ(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)<span class="sp">−2</span> + (power series in z − a).</p> + +<p>Considering next the function φ(z) = [ƒ(z)]<span class="sp">−1</span> dƒ(z)/dz, it is easily seen +that an ordinary point of ƒ(z) is an ordinary point of φ(z), that a +zero of order m for ƒ(z) in the neighbourhood of which ƒ(z) has a form, +(z − a)<span class="sp">m</span> multiplied by a power series, is a pole of φ(z) of residue m, +and that a pole of ƒ(z) of order n is a pole of φ(z) of residue −n; +manifestly φ(z) has the two periods of ƒ(z). We thus infer, since the +sum of the residues of φ(z) is zero, that for the function ƒ(z), the +sum of the orders of its vanishing at points belonging to one parallelogram, +Σm, is equal to the sum of the orders of its poles, Σn; 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 +ƒ(z) − A, where A is an arbitrary constant, we have the result, that +the function ƒ(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 ƒ(z) in any one of its parallelograms, and in fact +at least twice. The number of times it arises is called the <i>order</i> of the +function; the result suggests a property of rational functions.</p> + +<p>Consider further the integral <span class="f150">∫</span> z [ƒ′(z)/ƒ(z)] dz, where ƒ′(z) = dƒ(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 + ω +is of the form −ω <span class="f150">∫</span> z [ƒ′(z)/ƒ(z)] dz, which, as z increases from z<span class="su">0</span> to z<span class="su">0</span> + ω′, gives, +if λ denote the generalized logarithm, − ω {λ [ƒ(z<span class="su">0</span> + ω′)] − λ[ƒ(z<span class="su">0</span>)]}, that +is, since ƒ(z<span class="su">0</span> + ω′) = ƒ(z<span class="su">0</span>), gives 2πiNω, where N is an integer; similarly +the result of the integration along the other two opposite sides is of +the form 2πiN′ω′, where N′ is an integer. The integral, however, +is equal to 2πi times the sum of the residues of zƒ′(z) / ƒ(z) at the poles +interior to the parallelogram. For a zero, of order m, of ƒ(z) at z = a, +the contribution to this sum is 2πima, for a pole of order n at z = b +the contribution is −2πinb; we thus infer that Σma − Σnb = Nω + N′ω′; +this we express in words by saying that the sum of the values of z +where ƒ(z) = 0 within any parallelogram is equal to the sum of the +values of z where ƒ(z) = ∞ save for integral multiples of the periods. +By considering similarly the function ƒ(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.</p> +</div> + +<p>We pass now to the construction of a function having two +arbitrary periods ω, ω′ of unreal ratio, which has a single pole +of the second order in any one of its parallelograms.</p> + +<div class="condensed"> +<p>For this consider first the network of parallelograms whose corners +are the points Ω = mω + m′ω′, 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 z<span class="su">0</span>, z<span class="su">0</span> + ω, z<span class="su">0</span> + ω + ω′, z<span class="su">0</span> + ω′; we shall denote z<span class="su">0</span>, z<span class="su">0</span> + ω by A<span class="su">0</span>, B<span class="su">0</span>; +this parallelogram Π<span class="su">0</span> is surrounded by eight other parallelograms, +forming with Π<span class="su">0</span> a larger parallelogram Π<span class="su">1</span>, of which one side, for +instance, contains the points z<span class="su">0</span> − ω − ω′, z<span class="su">0</span> − ω′, z<span class="su">0</span> − ω′ + ω, z<span class="su">0</span> − ω′ + 2ω, +which we shall denote by A<span class="su">1</span>, B<span class="su">1</span>, C<span class="su">1</span>, D<span class="su">1</span>. This parallelogram Π<span class="su">1</span> is +surrounded by sixteen of the original parallelograms, forming with +Π<span class="su">1</span> a still larger parallelogram Π<span class="su">2</span> of which one side, for instance, +contains the points z<span class="su">0</span> − 2ω − 2ω′, z<span class="su">0</span> − ω − 2ω′, z<span class="su">0</span> − 2ω′, z<span class="su">0</span> + ω − 2ω′, +z<span class="su">0</span> + 2ω − 2ω′, z<span class="su">0</span> + 3ω − 2ω′, which we shall denote by A<span class="su">2</span>, B<span class="su">2</span>, C<span class="su">2</span>, D<span class="su">2</span>, +E<span class="su">2</span>, F<span class="su">2</span>. 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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">S<span class="su">0</span> =</td> <td>1</td> +<td rowspan="2">+ <span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">)</span> + <span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2">+ ... +</td> <td>1</td> +<td rowspan="2"><span class="f150">)</span> + ...</td></tr> +<tr><td class="denom">PA<span class="su">0</span><span class="sp">3</span></td> <td class="denom">PA<span class="su">1</span><span class="sp">3</span></td> +<td class="denom">PB<span class="su">1</span><span class="sp">3</span></td> <td class="denom">PC<span class="su">1</span><span class="sp">3</span></td> +<td class="denom">PA<span class="su">2</span><span class="sp">3</span></td> <td class="denom">PB<span class="su">2</span><span class="sp">3</span></td> +<td class="denom">PE<span class="su">2</span><span class="sp">3</span></td></tr></table> + +<p class="noind">and three other sets of terms, each infinite in number, formed in a +similar way. If the perpendiculars from P to the sides A<span class="su">0</span>B<span class="su">0</span>, +A<span class="su">1</span>B<span class="su">1</span>C<span class="su">1</span>, A<span class="su">2</span>B<span class="su">2</span>C<span class="su">2</span>D<span class="su">2</span>E<span class="su">2</span>, and so on, be p, p + q, p + 2q and so on, the +sum S<span class="su">0</span> is at most equal to</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2">+</td> <td>3</td> +<td rowspan="2">+</td> <td>5</td> +<td rowspan="2">+ ... +</td> <td>2n + 1</td> +<td rowspan="2">+ ...</td></tr> +<tr><td class="denom">p<span class="sp">3</span></td> <td class="denom">(p + q)<span class="sp">3</span></td> +<td class="denom">(p + 2q)<span class="sp">3</span></td> <td class="denom">(p + nq)<span class="sp">3</span></td></tr></table> + +<p class="noind">of which the general term is ultimately, when n is large, in a ratio of +equality with 2q<span class="sp">−3</span> n<span class="sp">−2</span>, so that the series S<span class="su">0</span> is convergent, as we know +the sum Σn<span class="sp">−2</span> to be; this assumes that p ≠ 0; if P be on A<span class="su">0</span>B<span class="su">0</span> +the proof for the convergence of S<span class="su">0</span> − 1/PA<span class="su">0</span><span class="sp">3</span>, is the same. Taking +the three other sums analogous to S<span class="su">0</span> we thus reach the result that +the series</p> + +<p class="center">φ(z) = −2Σ (z − Ω)<span class="sp">−3</span>,</p> + +<p class="noind">where Ω is mω + m′ω′, and m, m′ are to take all positive and negative +integer values, and z is any point outside small circles described with +the points Ω as centres, is <i>absolutely convergent</i>. 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 <i>uniformly convergent</i> 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 ω, ω′; for φ(z + ω) is the same sum as φ(z) with the terms +in a slightly different order. Thus φ(z + ω) = φ(z) and φ(z + ω′) = φ(z).</p> + +<p>Consider now the function</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(z) =</td> <td>1</td> +<td rowspan="2">+ <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">0</span> <span class="f150">{</span> φ(z) +</td> <td>2</td> +<td rowspan="2"><span class="f150">}</span> dz,</td></tr> +<tr><td class="denom">z<span class="sp">2</span></td> <td class="denom">z<span class="sp">3</span></td></tr></table> + +<p class="noind">where, for the subject of integration, the area of uniform convergence +clearly includes the point z = 0; this gives</p> + +<table class="math0" summary="math"> +<tr><td>dƒ(z)</td> +<td rowspan="2">= φ(z)</td></tr> +<tr><td class="denom">dz</td></tr></table> + +<p class="noind">and</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(z) =</td> <td>1</td> +<td rowspan="2">+ <span class="f150">Σ</span><span class="sp">′</span> <span class="f150">{</span></td> <td>1</td> +<td rowspan="2">−</td> <td>1</td> +<td rowspan="2"><span class="f150">}</span> ,</td></tr> +<tr><td class="denom">z<span class="sp">2</span></td> <td class="denom">(z − Ω)<span class="sp">2</span></td> +<td class="denom">Ω<span class="sp">2</span></td></tr></table> + +<p class="noind">wherein Σ′ is a sum excluding the term for which m = 0 and m′ = 0. +Hence ƒ(z + ω) − ƒ(z) and ƒ(z + ω′) − ƒ(z) are both independent of z. +Noticing, however, that, by its form, ƒ(z) is an even function of z, +and putting z = −½ω, z = −½ω′ respectively, we infer that also ƒ(z) +has the two periods ω and ω′. In the primary parallelogram Π<span class="su">0</span>, +however, ƒ(z) is only infinite at z = 0 in the neighbourhood of which +its expansion is of the form z<span class="sp">−2</span> + (power series in z). Thus ƒ(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.</p> +</div> + +<p>It can be shown that any single valued meromorphic function +of z with ω and ω′ as periods can be expressed rationally in terms +of ƒ(z) and φ(z), and that [φ(z)]<span class="sp">2</span> is of the form 4[ƒ(z)]<span class="sp">3</span> + Aƒ(z) + B, +where A, B are constants.</p> + +<p><span class="pagenum"><a name="page319" id="page319"></a>319</span></p> + +<div class="condensed"> +<p>To prove the last of these results, we write, for |z| < |Ω|,</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2">−</td> <td>1</td> +<td rowspan="2">=</td> <td>2z</td> +<td rowspan="2">+</td> <td>3z²</td> +<td rowspan="2">+ ...,</td></tr> +<tr><td class="denom">(z − Ω)²</td> <td class="denom">Ω²</td> +<td class="denom">Ω³</td> <td class="denom">Ω<span class="sp">4</span></td></tr></table> + +<p class="noind">and hence, if Σ′Ω<span class="sp">−2n</span> = σ<span class="su">n</span>, since Σ′Ω<span class="sp">−(2n−1)</span> = 0, we have, for sufficiently +small z greater than zero,</p> + +<p class="center">ƒ(z) = z<span class="sp">−2</span> + 3σ<span class="su">2</span>·z<span class="sp">2</span> + 5σ<span class="su">3</span>·z<span class="sp">4</span> + ...</p> + +<p class="noind">and</p> + +<p class="center">φ(z) = −2z<span class="sp">−3</span> + 6σ<span class="su">2</span>·z + 20σ<span class="su">3</span>·z<span class="sp">3</span> + ...;</p> + +<p class="noind">using these series we find that the function</p> + +<p class="center">F(z) = [φ(z)]² − 4[ƒ(z)]³ + 60σ<span class="su">2</span>ƒ(z) + 140σ<span class="su">3</span></p> + +<p class="noind">contains no negative powers of z, being equal to a power series in z² +beginning with a term in z². The function F(z) is, however, doubly +periodic, with periods ω, ω′, and can only be infinite when either +ƒ(z) or φ(z) is infinite; this follows from its form in ƒ(z) and φ(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 ƒ(z) = ζ and φ(z) = dζ/dz we see that</p> + +<table class="math0" summary="math"> +<tr><td>dz</td> +<td rowspan="2">= (4ζ³ − 60σ<span class="su">2</span>ζ − 140σ<span class="su">3</span>)<span class="sp">−1/2</span>.</td></tr> +<tr><td class="denom">dζ</td></tr></table> + +<p class="noind">Historically it was in the discussion of integrals such as</p> + +<p class="center">∫ dζ (4ζ³ − 60σ<span class="su">2</span>·ζ − 140σ<span class="su">3</span>)<span class="sp">−1/2</span>,</p> + +<p class="noind">regarded as a branch of Integral Calculus, that the doubly periodic +functions arose. As in the familiar case</p> + +<p class="center">z = <span class="f150">∫</span> <span class="sp1">ζ</span><span class="su1">0</span> (1 − ζ²)<span class="sp">−1/2</span> dζ,</p> + +<p class="noind">where ζ = sin z, it has proved finally to be simpler to regard ζ as a +function of z. We shall come to the other point of view below, +under § 20, <i>Elliptic Integrals</i>.</p> +</div> + +<p>To prove that any doubly periodic function F(z) with periods +ω, ω′, having poles at the points z = a<span class="su">1</span>, ... z = a<span class="su">m</span> of a parallelogram, +these being, for simplicity of explanation, supposed to be +all of the first order, is rationally expressible in terms of φ(z) +and ƒ(z), and we proceed as follows:—</p> + +<div class="condensed"> +<p>Consider the expression</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">Φ(z) =</td> <td>(ζ, 1)<span class="su">m</span> + η(ζ, 1)<span class="su">m−2</span></td></tr> +<tr><td class="denom">(ζ − A<span class="su">1</span>) (ζ − A<span class="su">2</span>)...(ζ − A<span class="su">m</span>)</td></tr></table> + +<p class="noind">where A<span class="su">s</span> = ƒ(a<span class="su">s</span>), ζ is an abbreviation for ƒ(z) and η for φ(z), and +(ζ, 1)<span class="su">m</span>, (ζ, 1)<span class="su">m−2</span>, denote integral polynomials in ζ, 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 a<span class="su">1</span>, ... a<span class="su">m</span> is one of the points mω + m′ω′ where f(z) = ∞. +The function Φ(z) is a monogenic function of z with the periods ω, ω′, +becoming infinite (and having singularities) only when (1) ζ = ∞ or +(2) one of the factors ζ-A<span class="su">s</span> is zero. In a period parallelogram +including z = 0 the first arises only for z = 0; since for ζ = ∞, η is in +a finite ratio to ζ<span class="sp">3/2</span>; the function Φ(z) for ζ = ∞ is not infinite +provided the coefficient of ζ<span class="sp">m</span> in (ζ, 1)<span class="su">m</span> is not zero; thus Φ(z) is +regular about z = 0. When ζ − A<span class="su">s</span> = 0, that is ƒ(z) = f(a<span class="su">s</span>), we have +z = ±a<span class="su">s</span> + mω + m′ω′, 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 −a<span class="su">1</span>, −a<span class="su">2</span>, ... −a<span class="su">m</span>; that is, if φ(a<span class="su">s</span>) = B<span class="su">s</span>, and therefore +φ(−a<span class="su">s</span>) = −B<span class="su">s</span>, so that we have the m relations</p> + +<p class="center">(A<span class="su">s</span>, 1)<span class="su">m</span> − B<span class="su">s</span>(A<span class="su">s</span>, 1)<span class="su">m−2</span> = 0;</p> + +<p class="noind">then the function Φ(z) will only have the m poles a<span class="su">1</span>, ... a<span class="su">m</span>. Denoting +further the m zeros of F(z) by a<span class="su">1</span>′, ... a<span class="su">m</span>′, putting ƒ(a<span class="su">s</span>′) = A<span class="su">s</span>′, +φ(a<span class="su">s</span>′) = B<span class="su">s</span>′, suppose the coefficients of the numerator of Φ(z) to +satisfy the further m − 1 conditions</p> + +<p class="center">(A<span class="su">s</span>′, 1)<span class="su">m</span> + B<span class="su">s</span>′ (A<span class="su">s</span>′, 1)<span class="su">m−2</span> = 0</p> + +<p class="noind">for s = 1, 2, ... (m − 1). The ratios of the 2m coefficients in the +numerator of Φ(z) can always be chosen so that the m + (m − 1) linear +conditions are all satisfied. Consider then the ratio</p> + +<p class="center">F(z) / Φ(z);</p> + +<p class="noind">it is a doubly periodic function with no singularity other than the +one pole a<span class="su">m</span>′. It is therefore a constant, the numerator of Φ(z) +vanishing spontaneously in a<span class="su">m</span>′. We have</p> + +<p class="center">F(z) = AΦ(z),</p> + +<p class="noind">where A is a constant; by which F(z) is expressed rationally in +terms of ƒ(z) and φ(z), as was desired.</p> + +<p>When z = 0 is a pole of F(z), say of order r, the other poles, each of +the first order, being a<span class="su">1</span>, ... a<span class="su">m</span>, similar reasoning can be applied to +a function</p> + +<table class="math0" summary="math"> +<tr><td>(ζ, 1)<span class="su">h</span> + η(ζ, 1)<span class="su">k</span></td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">(ζ − A<span class="su">1</span>) ... (ζ − A<span class="su">m</span>)</td></tr></table> + +<p class="noind">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 a<span class="su">1</span>, ... a<span class="su">m</span> 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.</p> + +<p>One important application of the result is the theorem that the +functions ƒ(z + t), φ(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 ƒ(z) and φ(z), and therefore, so far as they +depend on z and t, rationally in terms of ƒ(z), ƒ(t), φ(z) and φ(t). +It can in fact be shown, by reasoning analogous to that given above, +that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(z + t) + ƒ(z) + ƒ(t) = ¼ <span class="f150">[</span></td> <td>φ(z) − φ(t)</td> +<td rowspan="2"><span class="f150">]</span> <span class="sp1">2</span>.</td></tr> +<tr><td class="denom">ƒ(z) − ƒ(t)</td></tr></table> + +<p>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).</p> + +<p>The functions ƒ(z), φ(z) above are usually denoted by ℜ(z), ℜ′(z); +further the fundamental differential equation is usually written</p> + +<p class="center">(ℜ′z)² = 4(ℜz)³ − g<span class="su">2</span>ℜz − g<span class="su">3</span>,</p> + +<p class="noind">and the roots of the cubic on the right are denoted by e<span class="su">1</span>, e<span class="su">2</span>, e<span class="su">3</span>; +for the odd function, ℜ′z, we have, for the congruent arguments +−½ωand ½ω, ℜ′ (½ω) = −ℜ′ (−½ω) = −ℜ′ (½ω), and hence ℜ′ (½ω) = 0; +hence we can take e<span class="su">1</span> = ℜ (½ω), e<span class="su">2</span> = ℜ (½ω + ½ω′), e<span class="su">3</span> = ℜ (½ω). It can +then be proved that [ℜ(z) − e<span class="su">1</span>] [ℜ (z + ½ω) − e<span class="su">1</span>] = (e<span class="su">1</span> − e<span class="su">2</span>) (e<span class="su">1</span> − e<span class="su">3</span>), with +similar equations for the other half periods. Consider more particularly +the function ℜ(z) − e<span class="su">1</span>; like ℜ(z) it has a pole of the second +order at z = 0, its expansion in its neighbourhood being of the form +z<span class="sp">−2</span> (1 − e<span class="su">1</span>z<span class="sp">2</span> + Az<span class="sp">4</span> + ...); having no other pole, it has therefore either +two zeros, or a double zero in a period parallelogram (ω, ω′). In fact +near its zero ½ω its expansion is (x − ½ω) ℜ′ (½ω) + ½(z − ½ω)² ℜ″ (½ω) + +...; we have seen that ℜ′ (½ω) = 0; thus it has a zero of the second +order wherever it vanishes. Thus it appears that the square root +[ℜ(z) − e<span class="su">1</span>]<span class="sp">1/2</span>, 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 ƒ<span class="su">1</span>(z), we have ƒ<span class="su">1</span>(z + ω) = ±ƒ<span class="su">1</span>(z), ƒ<span class="su">1</span>(z + ω′) = ±ƒ<span class="su">1</span>(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 (ω, ω′); from the expansion of ƒ<span class="su">1</span>(z) about +z = 0, namely z<span class="sp">− 1</span> (1 − ½e<span class="su">1</span>z² + ...), it follows that ƒ<span class="su">1</span>(z) is an odd +function, and hence ƒ<span class="su">1</span> (−½ω′) = −ƒ<span class="su">1</span> (½ω′), which is not zero since +[ƒ<span class="su">1</span> (½ω′)]² = e<span class="su">3</span> − e<span class="su">1</span>, so that we have ƒ<span class="su">1</span> (z + ω′) = −ƒ<span class="su">1</span>(z); an equation +f<span class="su">1</span>(z + ω) = −ƒ<span class="su">1</span>(z) would then give ƒ<span class="su">1</span>(z + ω + ω′) = ƒ<span class="su">1</span>(z), and hence +ƒ<span class="su">1</span>(½ω + ½ω′) = ƒ<span class="su">1</span>(−½ω − ½ω′), of which the latter is −ƒ<span class="su">1</span>(½ω + ½ω′); this +would give ƒ<span class="su">1</span>(½ω + ½ω′) = 0, while [ƒ<span class="su">1</span>(½ω + ½ω′)]² = e<span class="su">2</span> − e<span class="su">1</span>. We thus +infer that ƒ<span class="su">1</span>(z + ω) = ƒ<span class="su">1</span>(z), ƒ<span class="su">1</span>(z + ω′) = −ƒ<span class="su">1</span>(z), ƒ<span class="su">1</span>(z + ω + ω′) = −ƒ<span class="su">1</span>(z). +The function ƒ<span class="su">1</span>(z) is thus doubly periodic with the periods ω and +2ω′; in a parallelogram of which two sides are ω and 2ω′ it has +poles at z = 0, z = ω′ each of the first order, and zeros of the first +order at z = ½ω, z = ½ω + ω′; it is thus a doubly periodic function +of the second order with two different poles of the first order in its +parallelogram (ω, 2ω′). We may similarly consider the functions +ƒ<span class="su">2</span>(z) = [ℜ(z) − e<span class="su">2</span>]<span class="sp">1/2</span>, ƒ<span class="su">3</span>(z) = [ℜ(z) − e<span class="su">3</span>]<span class="sp">1/2</span>; they give</p> + +<table class="math0" summary="math"> +<tr><td>ƒ<span class="su">2</span>(z + ω + ω′) = ƒ<span class="su">2</span>(z), ƒ<span class="su">2</span>(z + ω) = −ƒ<span class="su">2</span>(z), ƒ<span class="su">2</span>(z + ω′) = −ƒ<span class="su">2</span>(z),</td> +<td>ƒ<span class="su">3</span>(z + ω′) = ƒ<span class="su">3</span>z, ƒ<span class="su">3</span>(z + ω) = −ƒ<span class="su">3</span>(z), ƒ<span class="su">3</span>(z + ω + ω′) = −ƒ<span class="su">3</span>(z).</td></tr> +</table> + +<p class="noind">Taking u = z (e<span class="su">1</span> − e<span class="su">3</span>)<span class="sp">1/2</span>, with a definite determination of the constant +(e<span class="su">1</span> − e<span class="su">3</span>)<span class="sp">1/2</span>, it is usual, taking the preliminary signs so that for z = 0 +each of zƒ<span class="su">1</span>(z), zƒ<span class="su">2</span>(z), zƒ<span class="su">3</span>(z) is equal to +1, to put</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">sn(u) =</td> <td>(e<span class="su">1</span> − e<span class="su">3</span>)<span class="sp">1/2</span></td> +<td rowspan="2">,  cn(u) =</td> <td>ƒ<span class="su">1</span>(z)</td> +<td rowspan="2">,  dn(u) =</td> <td>f<span class="su">2</span>(z)</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">ƒ<span class="su">3</span>(z)</td> <td class="denom">ƒ<span class="su">3</span>(z)</td> +<td class="denom">ƒ<span class="su">3</span>(z)</td></tr></table> + +<table class="math0" summary="math"> +<tr><td>k² = (e<span class="su">2</span> − e<span class="su">3</span>) / (e<span class="su">1</span> − e<span class="su">3</span>),  K = ½ω (e<span class="su">1</span> − e<span class="su">3</span>)<span class="sp">1/2</span>,   iK′ = ½ω′ (e<span class="su">1</span> − e<span class="su">3</span>)<span class="sp">1/2</span>;</td></tr> +</table> + +<p class="noind">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 sn²(u) + cn²(u) = 1, k²sn²(u) + dn²(u) = 1; if x = sn(u) +we at once find, from the relations given here, that</p> + +<table class="math0" summary="math"> +<tr><td>du</td> +<td rowspan="2">= [(1 − x²) (1 − k²x²)]<span class="sp">−1/2</span>;</td></tr> +<tr><td class="denom">dx</td></tr></table> + +<p class="noind">if we put x = sinφ we have</p> + +<table class="math0" summary="math"> +<tr><td>du</td> +<td rowspan="2">= [1 − k²sin²φ]<span class="sp">−1/2</span>,</td></tr> +<tr><td class="denom">dφ</td></tr></table> + +<p class="noind">and if we call φ the amplitude of u, we may write φ = 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 Δam(u), where +Δ(φ) meant (1 − k²sin²φ)<span class="sp">1/2</span>. The addition equation for each of the +functions ƒ<span class="su">1</span>(z), ƒ<span class="su">2</span>(z), ƒ<span class="su">3</span>(z) is very simple, being</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ(z + t) = ½ <span class="f150">(</span></td> <td>∂</td> +<td rowspan="2">+</td> <td>∂</td> +<td rowspan="2"><span class="f150">)</span> log</td> <td>ƒ(z) + ƒ(t)</td> +<td rowspan="2">=</td> <td>ƒ(z)ƒ′(t) − ƒ(t)ƒ′(z)</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">∂z</td> <td class="denom">∂i</td> +<td class="denom">ƒ(z) − ƒ(t)</td> <td class="denom">ƒ²(z) − ƒ²(t)</td></tr></table> + +<p class="noind">where f<span class="su">1</span>′(z) means dƒ<span class="su">1</span>(z)/dz, which is equal to −ƒ<span class="su">2</span>(z)·ƒ<span class="su">3</span>(z), and ƒ²(z) +<span class="pagenum"><a name="page320" id="page320"></a>320</span> +means [ƒ(z)]<span class="sp">2</span>. This may be verified directly by showing, if R denote +the right side of the equation, that ∂R/∂z = ∂R/∂t; this will require +the use of the differential equation</p> + +<p class="center">[ƒ<span class="su">1</span>′<span class="sp">(z)</span>]<span class="sp">2</span> = [ƒ<span class="su">1</span><span class="sp">2</span>(z) + e<span class="su">1</span> − e<span class="su">2</span>] [ƒ<span class="su">1</span><span class="sp">2</span>(z) + e<span class="su">1</span> − e<span class="su">3</span>],</p> + +<p class="noind">and in fact we find</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">(</span></td> <td>∂<span class="sp">2</span></td> +<td rowspan="2">−</td> <td>∂<span class="sp">2</span></td> +<td rowspan="2"><span class="f150">)</span> log [ƒ(z) + ƒ(t)] = ƒ<span class="sp">2</span>(z) − ƒ<span class="sp">2</span>(t) = <span class="f150">(</span></td> <td>∂<span class="sp">2</span></td> +<td rowspan="2">−</td> <td>∂<span class="sp">2</span></td> +<td rowspan="2"><span class="f150">)</span> log [ƒ(z) − ƒ(t)];</td></tr> +<tr><td class="denom">∂z<span class="sp">2</span></td> <td class="denom">dt<span class="sp">2</span></td> +<td class="denom">∂z<span class="sp">2</span></td> <td class="denom">dt<span class="sp">2</span></td></tr></table> + +<p class="noind">hence it will follow that R is a function of z + t, and R is at once seen +to reduce to ƒ(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 s<span class="su">1</span>, c<span class="su">1</span>, d<span class="su">1</span>, s<span class="su">2</span>, c<span class="su">2</span>, d<span class="su">2</span> denote respectively sn(u<span class="su">1</span>), cn(u<span class="su">1</span>), dn(u<span class="su">1</span>), sn(u<span class="su">2</span>), +cn(u<span class="su">2</span>), dn(u<span class="su">2</span>), they can be put into the forms</p> + +<table class="math0" summary="math"> +<tr><td>sn(u<span class="su">1</span> + u<span class="su">2</span>) = (s<span class="su">1</span>c<span class="su">2</span>d<span class="su">2</span> + s<span class="su">2</span>c<span class="su">1</span>d<span class="su">1</span>) / D,</td> +<td>cn(u<span class="su">1</span> + u<span class="su">2</span>) = (c<span class="su">1</span>c<span class="su">2</span> − s<span class="su">1</span>s<span class="su">2</span>d<span class="su">1</span>d<span class="su">2</span>) / D,</td> +<td>dn(u<span class="su">1</span> + u<span class="su">2</span>) = (d<span class="su">1</span>d<span class="su">2</span> − k<span class="sp">2</span>s<span class="su">1</span>s<span class="su">2</span>c<span class="su">1</span>c<span class="su">2</span>) / D,</td></tr> +</table> + +<p class="noind">where</p> + +<p class="center">D = 1 − k<span class="sp">2</span>s<span class="su">1</span><span class="sp">2</span>s<span class="su">2</span><span class="sp">2</span>.</p> + +<p>The introduction of the function ƒ<span class="su">1</span>(z) is equivalent to the introduction +of the function ℜ(z; ω, 2ω′) constructed from the periods +ω, 2ω′ as was ℜ(z) from ω and ω′; denoting this function by ℜ<span class="su">1</span>(z) +and its differential coefficient by ℜ′<span class="su">1</span>(z), we have in fact</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ƒ<span class="su">1</span>(z) = ½</td> <td>ℜ′<span class="su">1</span>(z)</td></tr> +<tr><td class="denom">ℜ<span class="su">1</span>(ω′) − ℜ<span class="su">1</span>(z)</td></tr></table> + +<p class="noind">as we see at once by considering the zeros and poles and the limit of +zƒ<span class="su">1</span>(z) when z = 0. In terms of the function ℜ<span class="su">1</span>(z) the original function +ℜ(z) is expressed by</p> + +<p class="center">ℜ(z) = ℜ<span class="su">1</span>(z) + ℜ<span class="su">1</span>(z + ω′) − ℜ<span class="su">1</span>(ω′),</p> + +<p class="noind">as a consideration of the poles and expansion near z = 0 will show.</p> + +<p>A function having ω, ω′ for periods, with poles at two arbitrary +points a, b and zeros at a′, b′, where a′ + b′ = a + b save for an expression +mω + m′ω′, in which m, m′ are integers, is a constant multiple of</p> + +<table class="math0" summary="math"> +<tr><td>{ℜ [z − ½(a′ + b′)] − ℜ [a′ − ½(a′ + b′)]} / {ℜ [z − ½(a + b)] − ℜ [a − ½(a + b)]};</td></tr> +</table> + +<p class="noind">if the expansion of this function near z = a be</p> + +<p class="center">λ(z − a)<span class="sp">−1</span> + μ + <span class="f150">Σ</span> <span class="su">n=1</span> μ<span class="su">n</span> (z − a)<span class="sp">n</span>,</p> + +<p class="noind">the expansion near z = b is</p> + +<p class="center">−λ (z − b)<span class="sp">− 1</span> + μ + <span class="f150">Σ</span> <span class="su">n=1</span> (−1)<span class="sp">n</span> μ<span class="su">n</span> (z − b)<span class="sp">n</span>,</p> + +<p class="noind">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.</p> + +<p>From the function ℜ(z) we can obtain another function, termed the +Zeta-function; it is usually denoted by ζ(z), and defined by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ζ(z) −</td> <td>1</td> +<td rowspan="2">= <span class="f150">∫</span> <span class="sp1">π</span><span class="su1">0</span> <span class="f150">[</span></td> <td>1</td> +<td rowspan="2">− ℜ(z) <span class="f150">]</span> dz = <span class="f150">Σ</span><span class="sp1">′</span> <span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2">+</td> <td>z</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">z</td> <td class="denom">z<span class="sp">2</span></td> +<td class="denom">z − Ω</td> <td class="denom">Ω</td> +<td class="denom">Ω<span class="sp">2</span></td></tr></table> + +<p class="noind">for which as before we have equations</p> + +<table class="math0" summary="math"> +<tr><td>ζ(z + ω) = ζ(z) + 2πiη,   ζ(z + ω′) = ζ(z) + 2πiη′,</td></tr> +</table> + +<p class="noind">where 2η, 2η′ are certain constants, which in this case do not both +vanish, since else ζ(z) would be a doubly periodic function with only +one pole of the first order. By considering the integral</p> + +<p class="center">∫ ζ(z)dz</p> + +<p class="noind">round the perimeter of a parallelogram of sides ω, ω′ containing +z = 0 in its interior, we find ηω′ − η′ω = 1, so that neither of η, η′ +is zero. We have ζ′(z) =−ℜ(z). From ζ(z) by means of the equation</p> + +<table class="math0" summary="math"> +<tr><td>σ(z)</td> +<td rowspan="2">= exp <span class="f150">{</span> <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">0</span> <span class="f150">[</span> ζ(x) −</td> <td>1</td> +<td rowspan="2"><span class="f150">]</span> dz <span class="f150">}</span> = Π′ <span class="f150">[ (</span> 1 −</td> <td>z</td> +<td rowspan="2"><span class="f150">)</span> exp <span class="f150">(</span></td> <td>z</td> +<td rowspan="2">+</td> <td>z<span class="sp">2</span></td> +<td rowspan="2"><span class="f150">) ]</span>,</td></tr> +<tr><td class="denom">z</td> <td class="denom">z</td> +<td class="denom">Ω</td> <td class="denom">Ω</td> +<td class="denom">2Ω<span class="sp">2</span></td></tr></table> + +<p class="noind">we determine an integral function σ(z), termed the Sigma-function, +having a zero of the first order at each of the points z = Ω; it can be +seen to satisfy the equations</p> + +<table class="math0" summary="math"> +<tr> <td>σ(z + ω)</td> +<td rowspan="2">= −exp [2πiη(z + ½ω)],   </td> <td>σ(z + ω′)</td> +<td rowspan="2">= −exp [2πiη′ (z + ½ω′)].</td></tr> +<tr><td class="denom">σ(z)</td> <td class="denom">σ(z)</td></tr></table> + +<p class="noind">By means of these equations, if a<span class="su">1</span> + a<span class="su">2</span> + ... + a<span class="su">m</span> = a′<span class="su">1</span> + a′<span class="su">2</span> + ... ++ a′<span class="su">m</span>, it is readily shown that</p> + +<table class="math0" summary="math"> +<tr><td>σ(z − a′<span class="su">1</span>) σ(z − a′<span class="su">2</span>) ... σ(z − a′<span class="su">m</span>)</td></tr> +<tr><td class="denom">σ(z − a<span class="su">1</span>) σ(z − a<span class="su">2</span>) ... σ(z − a<span class="su">m</span>)</td></tr></table> + +<p class="noind">is a doubly periodic function having a<span class="su">1</span>, ... a<span class="su">m</span> as its simple poles, +and a′<span class="su">1</span>, ... a′<span class="su">m</span> as its simple zeros. Thus the function σ(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 ζ(z) = σ′(z)/σ(z).</p> + +<p>The functions ζ(z), ℜ(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</p> + +<p class="center">A<span class="su">1</span>(z − a)<span class="sp">−1</span> + A<span class="su">2</span>(z − a)<span class="su">−2</span> + ... + A<span class="su">m+1</span>(z − a)<span class="sp">−(m+1)</span>,</p> + +<p class="noind">then the difference</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">F(z) − A<span class="su">1</span>ζ (z − a) − A<span class="su">2</span>ℜ (z − a) − ... +</td> <td>A<span class="su">m+1</span></td> +<td rowspan="2">(−1)<span class="sp">m</span> ℜ<span class="sp">m−1</span> (z − a)</td></tr> +<tr><td class="denom">m!</td></tr></table> + +<p class="noind">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 ∫F(z)dz can +then be expressed in terms of z, functions ℜ(z − a) and their differential +coefficients, functions ζ(z − a) and functions logσ(z − a).</p> +</div> + +<p>§ 15. <i>Potential Functions.</i> <i>Conformal Representation in +General.</i>—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(iφ) = r(cosφ + i sinφ) be an internal point +of this circle we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u + iv =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>(U + iV)</td> +<td rowspan="2">dt,</td></tr> +<tr><td class="denom">2πi</td> <td class="denom">t − z</td></tr></table> + +<p class="noind">where U + iV is the value of the function at a point of the circumference +and t = aE(iθ); this is the same as</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u + iv =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>(U + iV) [1 − (r/a) E (iθ − iφ)]</td> +<td rowspan="2">dθ.</td></tr> +<tr><td class="denom">2π</td> <td class="denom">1 + (r/a)² − 2(r/a) cos (θ − φ)</td></tr></table> + +<p class="noind">If in the above formula we replace z by the external point +(a²/r) E(iφ) the corresponding contour integral will vanish, so that +also</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">0 =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>(U + iV) [(r/a)² − (r/a) E (iθ − iφ)]</td> +<td rowspan="2">dθ;</td></tr> +<tr><td class="denom">2π</td> <td class="denom">1 + (r/a)² − 2(r/a) cos (θ − φ)</td></tr></table> + +<p class="noind">hence by subtraction we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>U(a² − r²)</td> +<td rowspan="2">dθ,</td></tr> +<tr><td class="denom">2π</td> <td class="denom">a² + r² − 2ar cos (θ − φ)</td></tr></table> + +<p class="noind">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(iθ) +of the circumference, and ω the angle which QP makes with OQ +produced, this integral is at once found to be the same as</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> Udω −</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> Udθ</td></tr> +<tr><td class="denom">π</td> <td class="denom">2π</td></tr></table> + +<p class="noind">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.</p> + +<div class="condensed"> +<p>Conversely, let U be any continuous real function on the circumference, +U<span class="su">0</span> being the value of it at a point P<span class="su">0</span> of the circumference, +and describe a small circle with centre at P<span class="su">0</span> cutting the given circle in +A and B, so that for all points P of the arc AP<span class="su">0</span>B we have |U − U<span class="su">0</span>| < ε, +where ε is a given small real quantity. Describe a further circle, +centre P<span class="su">0</span> 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′P<span class="su">0</span>B′ and this second circle; then for all positions of P upon the +greater arc AB of the original circle QP² is greater than a definite +finite quantity which is not zero, say QP² > D². Consider now the +integral</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u′ =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> U</td> <td>(a² − r²)</td> +<td rowspan="2">dθ =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> Udω −</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> Udθ,</td></tr> +<tr><td class="denom">2π</td> <td class="denom">a² + r² − 2ar cos (θ − φ)</td> +<td class="denom">π</td> <td class="denom">2π</td></tr></table> + +<p class="noind">which we evaluate as the sum of two, respectively along the small arc +AP<span class="su">0</span>B and the greater arc AB. It is easy to verify that, for the +whole circumference,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">U<span class="su">0</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> U<span class="su">0</span></td> <td>a² − r²</td> +<td rowspan="2">dθ =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> U<span class="su">0</span> dω −</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> U<span class="su">0</span> dθ.</td></tr> +<tr><td class="denom">2π</td> <td class="denom">a² + r² − 2ar cos (θ − φ)</td> +<td class="denom">π</td> <td class="denom">2π</td></tr></table> + +<p class="noind">Hence we can write</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u′ − U<span class="su">0</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> <span class="su">AP<span class="su">0</span>B</span> (U − U<span class="su">0</span>)dω −</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> <span class="su">AP<span class="su">0</span>B</span> (U − U<span class="su">0</span>)dθ +</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> <span class="su">AB</span> (U − U<span class="su">0</span>)</td> <td>(a² − r²)</td> +<td rowspan="2">dθ.</td></tr> +<tr><td class="denom">2π</td> <td class="denom">2π</td> +<td class="denom">2π</td> <td class="denom">QP²</td></tr></table> + +<p class="noind">If the finite angle between QA and QB be called Φ and the finite +angle AOB be called Θ, the sum of the first two components is +numerically less than</p> + +<table class="math0" summary="math"> +<tr><td>ε</td> +<td rowspan="2">(Φ + Θ).</td></tr> +<tr><td class="denom">2π</td></tr></table> + +<p class="noind">If the greatest value of |(U − U<span class="su">0</span>)| on the greater arc AB be called H, +the last component is numerically less than</p> + +<table class="math0" summary="math"> +<tr><td>H</td> +<td rowspan="2">(a² − r²)</td></tr> +<tr><td class="denom">D²</td></tr></table> + +<p class="noind">of which, when the circle, of centre P<span class="su">0</span>, passing through A′B′ is +sufficiently small, the factor a² − r² 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 P<span class="su">0</span>, is +U<span class="su">0</span>. From the form</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u′ =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> Udω −</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> Udθ,</td></tr> +<tr><td class="denom">π</td> <td class="denom">2π</td></tr></table> + +<p class="noind">since the inclination of QP to a fixed direction is, when Q varies, P +remaining fixed, a solution of the differential equation</p> + +<table class="math0" summary="math"> +<tr><td>∂²ψ</td> +<td rowspan="2">+</td> <td>∂²</td> +<td rowspan="2">= 0,</td></tr> +<tr><td class="denom">∂x²</td> <td class="denom">∂y²</td></tr></table> + +<p class="noind">where z, = x + iy, is the point Q, we infer that u′ is a differentiable +<span class="pagenum"><a name="page321" id="page321"></a>321</span> +function satisfying this equation; indeed, when r < a, we can write</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"><span class="f150">∫</span> U</td> <td>(a² − r²)</td> +<td rowspan="2">dθ =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> U <span class="f150">[</span> 1 + 2</td> <td>r</td> +<td rowspan="2">cos (θ − φ) + 2</td> <td>r²</td> +<td rowspan="2">cos 2(θ − φ) + ... <span class="f150">]</span> dθ</td></tr> +<tr><td class="denom">2π</td> <td class="denom">a² + r² − 2ar cos (θ − φ)</td> +<td class="denom">2π</td> <td class="denom">a</td> +<td class="denom">a²</td></tr></table> + +<p class="center">= a<span class="su">0</span> + a<span class="su">1</span>x + b<span class="su">1</span>y + a<span class="su">2</span> (x² − y²) + 2b<span class="su">2</span>xy + ...,</p> + +<p class="noind">where</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">a<span class="su">0</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span> Udθ,   a<span class="su">1</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>U cosθ</td> +<td rowspan="2">dθ,   b<span class="su">1</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>U sinθ</td> +<td rowspan="2">dθ,</td></tr> +<tr><td class="denom">2π</td> <td class="denom">π</td> +<td class="denom">a</td> <td class="denom">π</td> +<td class="denom">a</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">a<span class="su">2</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>U cos 2θ</td> +<td rowspan="2">dθ,   b<span class="su">2</span> =</td> <td>1</td> +<td rowspan="2"><span class="f150">∫</span></td> <td>U sin 2θ</td> +<td rowspan="2">dθ.</td></tr> +<tr><td class="denom">π</td> <td class="denom">a²</td> +<td class="denom">π</td> <td class="denom">a²</td></tr></table> + +<p>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 ∂²ψ/∂x² + ∂²ψ/∂y²; 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.</p> + +<p>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 x<span class="su">0</span>, y<span class="su">0</span> of this region by a power +series in x − x<span class="su">0</span>, y − y<span class="su">0</span>, with real coefficients, these various series being +obtainable from one of them by continuation. For any region R<span class="su">0</span> +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 R<span class="su">0</span>. Each of these series, and +therefore the function, will be differentiable; suppose that at all +points of R<span class="su">0</span> the function satisfies the equation</p> + +<table class="math0" summary="math"> +<tr><td>∂²P</td> +<td rowspan="2">+</td> <td>∂P²</td> +<td rowspan="2">= 0,</td></tr> +<tr><td class="denom">∂x²</td> <td class="denom">∂y²</td></tr></table> + +<p class="noind">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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">Q = <span class="f150">∫</span> <span class="sp1">(x, y)</span> <span class="f150">(</span></td> <td>∂P</td> +<td rowspan="2">dy −</td> <td>∂P</td> +<td rowspan="2">dx <span class="f150">)</span>.</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td></tr></table> + +<p>The functions P, Q, being given by a finite number of power series, +will be single valued in R<span class="su">0</span>, and P + iQ will be a monogenic function of +z within R<span class="su">0</span>· In drawing this inference it is supposed that the region +R<span class="su">0</span> is such that every closed path drawn in it is capable of being +deformed continuously to a point lying within R<span class="su">0</span>, that is, is <i>simply +connected</i>.</p> + +<p>Suppose in particular, c being any point interior to R<span class="su">0</span>, 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</p> + +<p class="center">ζ = (z − c) exp (−P − iQ)</p> + +<p class="noind">will be developable by a power series in (z − z<span class="su">0</span>) about every point z<span class="su">0</span> +interior to R<span class="su">0</span>, 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 ζ the boundary of R will become a circle of radius unity +with centre at ζ=0, this latter point corresponding to z=c. A +closed path within R<span class="su">0</span>, passing once round z=c, will lead to a closed +path passing once about ζ = 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 dζ/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.</p> + +<p>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/π <span class="f150">∫</span> Udω − 1/π <span class="f150">∫</span> Udθ 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.</p> + +<p>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 α, β, γ, ... be n +real numbers, whose sum is n − 2, the integral</p> + +<p class="center">z = ∫ (t − a)<span class="sp">α−1</span> (t − b)<span class="sp">β−1</span> ... dt,</p> + +<p class="noind">as t describes the real axis, describes in the plane of z a polygon of n +sides with internal angles equal to απ, βπ, ..., 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 +Σα = n − 2 ensures merely that the point t = ∞ 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 = ∞.</p> + +<p>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 +ζ = [(z² + h²)/(1 + h²z²)]<span class="sp">1/2</span>, with ζ = 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 ζ +upon another plane, as z describes the continuous curve ABCDE, +ζ will describe the real axis from ζ = 1 to ζ = − 1, the point C giving +ζ = 0, and the points B, D giving the points ζ = ±h. Near z = 0 +the expansion of ζ is ζ − h = z² (1 − h<span class="sp">4</span> / 2h) + ..., or ζ + h = −z² (1 − h<span class="sp">4</span> / 2h) + ...; +in either case an increase of ½π in the phase of z gives an increase +of π in the phase of ζ − h or ζ + h. Near z = ih the expansion of ζ is +ζ = (z − ih)<span class="sp">1/2</span> [2ih/(1 − h<span class="sp">4</span>)]<span class="sp">1/2</span> + ..., and an increase of 2π in the phase of +z − ih also leads to an increase of π in the phase of ζ. Then as z +describes the semicircle EFA, ζ also describes a semicircle of radius +unity, the point z = i becoming ζ = 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 ζ-plane, the function</p> + +<p class="center">z = [(ζ² − h²) / (1 − h²ζ²)]<span class="sp">1/2</span></p> + +<p class="noind">being single valued in the latter semicircle. By means of a transformation +t = (ζ + 1)² / (ζ − 1)², the semicircle in the plane of ζ can +further be conformably represented upon the upper half of the whole +plane of t.</p> + +<p>As another illustration we may take the conformal representation +of an equilateral triangle upon a half plane. Taking the elliptic +function ℜ(u) for which ℜ′²(u) = 4ℜ³(u) − 4, so that, with ε = exp (<span class="spp">2</span>⁄<span class="suu">3</span>πi), +we have e<span class="su">1</span> = 1, e<span class="su">2</span> = ε², e<span class="su">3</span> = ε, the half periods may be taken to be</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">½ω = <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">1</span></td> <td>dt</td> +<td rowspan="2">,   ½ω′ = <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">e<span class="su">3</span></span></td> <td>dt</td> +<td rowspan="2">= ½εω;</td></tr> +<tr><td class="denom">2(t³ − 1)<span class="sp">1/2</span></td> <td class="denom">2(t³ − 1)<span class="sp">1/2</span></td></tr></table> + +<p class="noind">drawing the equilateral triangle whose vertices are O, of argument O, +A of argument ω, and B of argument ω + ω′ = −ε²ω, and the equilateral +triangle whose angular points are O, B and C, of argument ω′, +let E, of argument <span class="spp">1</span>⁄<span class="suu">3</span>(2ω + ω′), and D, of argument <span class="spp">1</span>⁄<span class="suu">3</span>(ω + 2ω′), 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 = ξ + iη be any point of the +interior of the triangle OEH and v = εu<span class="su">0</span> = ε(ξ − iη) 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 −εv, ω + εu, ω − η²v, ω + ω′ + ε²u, +ω + ω′ − v, ω + ω′ − u, ω + ω′ + εv, ω′ − εu, ω′ + ε²v, −ε²u. Further, when +u is real, since the term − 2(u + mω + m′ε²ω)<span class="sp">−3</span>, which is the conjugate +complex of −2(u + mω + m′ε²ω)<span class="sp">3</span>, arises in the infinite sum +which expresses ℜ′(u), namely as −2(u + μω + μ′εω)<span class="sp">−3</span>, where +μ = m − m′, μ′ = −m′, it follows that ℜ′(u) is real; in a similar +way we prove that ℜ′(u) is pure imaginary when u is pure imaginary, +and that ℜ′(u) = ℜ′(εu) = ℜ′(ε²u), as also that for v = εu<span class="su">0</span>, ℜ′(v) is the +conjugate complex of ℜ′(u). Hence it follows that the variable</p> + +<p class="center">t = ½ iℜ′(u)</p> + +<p><span class="pagenum"><a name="page322" id="page322"></a>322</span></p> + +<p class="noind">takes each real value once as u passes along the perimeter of the +triangle ODE, being as can be shown respectively ∞, 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</p> + +<p class="center">u = <span class="spp">1</span>⁄<span class="suu">3</span> i <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">t</span> (t<span class="sp">2</span> − 1)<span class="sp">−2/3</span>dt</p> + +<p class="noind">in accordance with the general theory.</p> + +<p>It can be deduced that τ = t<span class="sp">2</span> represents the triangle ODH on the +upper half plane of τ, and ζ = (i − τ<span class="sp">−1</span>)<span class="sp">1/2</span> represents similarly the +triangle OBD.</p> +</div> + +<p>§ 16. <i>Multiple valued Functions. Algebraic Functions.</i>—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<span class="sp">1/2</span>, for which two values +exist for any value of z; another is the generalized logarithm +λ(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 = ∞ 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<span class="sp">−1</span>) [1 − (1 − cz<span class="sp">−1</span>)]<span class="sp">−1</span>, and a +series in 1/z to a series in z − c, when z is near enough to c, by +means of 1/z = 1/c [1 + (z − c / c)]<span class="sp">−1</span>.</p> + +<div class="condensed"> +<p>The commonest case of the occurrence of multiple valued functions +is that in which the function s satisfies an algebraic equation ƒ(s, z) = +p<span class="su">0</span>s<span class="sp">n</span> + p<span class="su">1</span>s<span class="sp">n−1</span> + ... + p<span class="su">n</span> = 0, wherein p<span class="su">0</span>, p<span class="su">1</span>, ... p<span class="su">n</span> are integral polynomials +in z. Assuming ƒ(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<span class="sp">n</span> vanishes, as also the values +of z for which beside ƒ(s, z) = 0 we have also ∂f(s, z)/∂s = 0, and also +in general the point z = ∞, 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 ∂f(s, z)/∂s = 0 is satisfied by one or more of the +roots s of ƒ(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)<span class="sp">1/r</span>, these series of the cycle being obtainable from +one another by replacing (z − c)<span class="sp">1/r</span> by ω(z − r)<span class="sp">1/r</span>, where ω, equal to +exp (2πih/r), is one of the rth roots of unity. Putting then z − c = t<span class="sp">r</span> +we may say that the r roots of a cycle are given by a single power +series in t, an increase of 2π in the phase of t giving an increase of +2πr 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 2πr, is to be looked upon as defining a single +<i>place</i> among the aggregate of values of z and s which satisfy ƒ(s, z) = 0; +two such places may be at the same <i>point</i> (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 ∂f(s, z)/∂s = 0 there +are less than n places. Similar remarks hold for the neighbourhood +of z = ∞; there may be n places whose neighbourhood is given by n +power series in z<span class="sp">− 1</span> or fewer, one of these being associated with a +series in t, where t = (z<span class="sp">−1</span>)<span class="sp">1/r</span>; 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)<span class="sp">1/r</span>, z<span class="sp">−1</span>, (z<span class="sp">−1</span>)<span class="sp">1/r</span>. that the neighbourhood of +any place (c, d) for which ƒ(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<span class="sp">−1</span> when c is infinite, and similarly the expression s − d by s<span class="sp">−1</span> when +d is infinite. The last case arises when we consider the finite values +of z for which the polynomial coefficient of s<span class="sp">n</span> vanishes. Of such a +pair of expressions we may obtain a continuation by writing t = t<span class="su">0</span> + +λ<span class="su">1</span>τ + λ<span class="su">2</span>τ² + ..., where τ is a new variable and λ<span class="su">1</span> is not zero; +in particular for an ordinary finite place this equation simply becomes +t = t<span class="su">0</span> + τ. 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 ƒ(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 ƒ(s, z) = 0 defines a <i>monogenic algebraic +construct</i>. With less accuracy we may say that an irreducible +algebraic equation ƒ(s, z) = 0 determines a single monogenic function +s of z.</p> + +<p>Any rational function of z and s, where ƒ(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<span class="sp">m</span>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, = −μ, the function +is infinite to order μ at the place. More generally, if A be an +arbitrary constant, and, near (c, d), R(s, z) −A is of the form t<span class="sp">m</span>H(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 μ 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 = ∞, 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 μ 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 ∞) the same number of times on +the algebraic construct; this number is called the <i>order</i> of the +rational function.</p> + +<p>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 +ƒ(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 ƒ(s, z) = 0; about any place (c, d) for which +z = c + P(t), s = d + Q(t), expand the product</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">R(s, z)</td> <td>dz</td></tr> +<tr><td class="denom">dt</td></tr></table> + +<p class="noind">in powers of t and pick out the coefficient of t<span class="sp">−1</span>. There is only a +finite number of places of this kind. The theorem is that the sum +of these coefficients of t<span class="sp">−1</span> is zero. This we express by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">[</span> R(s, z)</td> <td>dz</td> +<td rowspan="2"><span class="f150">]</span><span class="su">t<span class="sp">−1</span></span> = 0.</td></tr> +<tr><td class="denom">dt</td></tr></table> + +<p class="noind">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 = ∞ we have z<span class="sp">−1</span> = t, and therefore dz/dt = −t<span class="sp">−2</span>; +in that case, then, the theorem is that in any rational function of z,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">Σ (</span></td> <td>A<span class="su">1</span></td> +<td rowspan="2">+</td> <td>A<span class="su">2</span></td> +<td rowspan="2">+ ... +</td> <td>A<span class="su">m</span></td> +<td rowspan="2"><span class="f150">)</span> + Pz<span class="sp">h</span> + Qz<span class="sp">h−1</span> + ... + R,</td></tr> +<tr><td class="denom">z − a </td> <td class="denom">(z − a)²</td> +<td class="denom">(z − a)<span class="sp">m</span></td></tr></table> + +<p class="noind">the sum ΣA<span class="su">1</span> 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 = ∞; an obvious result. In general, if for a finite place +of the algebraic construct associated with ƒ(s, z) = 0, whose neighbourhood +is given by z = c + t<span class="sp">r</span>, s = d + Q(t), there be a coefficient of t<span class="sp">−1</span> in +R(s, z) dz/dt, this will be r times the coefficient of t<span class="sp">−r</span> in R(s, z) or +R[d + Q(t), c + t<span class="sp">r</span>], namely will be the coefficient of t<span class="sp">−r</span> in the sum of +the r series obtainable from R [d + Q(t), c + t<span class="sp">r</span>] by replacing t by ωt, +where ω is an rth root of unity; thus the sum of the coefficients of +t<span class="sp">−1</span> 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)<span class="sp">−1</span> in R(s<span class="su">1</span>, z) + +R(s<span class="su">2</span>z) + ... + R(s<span class="su">n</span>, z), where s<span class="su">1</span>, ... s<span class="su">n</span> are the n values of s for a +value of z near to z = c; this latter sum Σ R(s<span class="su">i</span>, z) is, however, a +rational function of z only. Similarly, near z = ∞, for a place given +by z<span class="sp">−1</span>=t<span class="sp">r</span>, s = d + Q(t), or s<span class="sp">−1</span> = Q(t), the coefficient of t<span class="sp">−1</span> in R(s, z) dz/dt +is equal to −r times the coefficient of t<span class="sp">r</span> in R[d + Q(t), t<span class="sp">−r</span>], that is +equal to the negative coefficient of z<span class="sp">−l</span> in the sum of the r series +R[d + Q(ωt), t<span class="sp">−r</span>], so that, as before, the sum of the coefficients of +t<span class="sp">−1</span> in R(s, z) dz/dt at the various places which arise for z = ∞ is equal +to the negative coefficient of z<span class="sp">− 1</span> in the same rational function of z, +Σ R(s<span class="su">i</span>, z). Thus, from the corresponding theorem for rational functions +of one variable, the general theorem now being proved is seen to +follow.</p> + +<p>Apply this theorem now to the rational function of s and z,</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"> </td> <td>dR(s, z)</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">R(s, z)</td> <td class="denom">dz</td></tr></table> + +<p class="noind">at a zero of R(s, z) near which R(s, z) = t<span class="sp">m</span>H(t), we have</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2"> </td> <td>dR(s, z)</td> +<td rowspan="2"> </td> <td>dz</td> +<td rowspan="2">=</td> <td>d</td> +<td rowspan="2">{λ [R(s, z)] },</td></tr> +<tr><td class="denom">R(s, z)</td> <td class="denom">dz</td> +<td class="denom">dt</td> <td class="denom">dt</td></tr></table> + +<p class="noind">where λ denotes the generalized logarithmic function, that is equal +to</p> + +<p class="center">mt<span class="sp">−1</span> + power series in t;</p> + +<p class="noind">similarly at a place for which R(s, z) = t<span class="sp">−μ</span>K(t); the theorem</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">[</span></td> <td>1</td> +<td rowspan="2"> </td> <td>dR(s, z)</td> +<td rowspan="2"> </td> <td>dz</td> +<td rowspan="2"><span class="f150">]</span><span class="su">t<span class="sp">−1</span></span> = 0</td></tr> +<tr><td class="denom">R(s, z)</td> <td class="denom">dz</td> +<td class="denom">dt</td></tr></table> + +<p class="noind">thus gives Σm = Σμ, 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.</p> + +<p><span class="pagenum"><a name="page323" id="page323"></a>323</span></p> + +<p>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 s² = 4z³ + +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 <i>order</i>. 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 § 17 below.</p> +</div> + +<p>§ 17. <i>Integrals of Algebraic Functions.</i>—In treatises on Integral +Calculus it is proved that if R(z) denote any rational function, +an indefinite integral ∫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 s² = az² + bz + c and R(s, z) be any rational +function of s, z any integral ∫R(s, z)dz can be evaluated in terms +of rational functions of s, z and logarithms of such functions; +the simplest case is ∫s<span class="sp">− 1</span>dz or ∫(az² + bz + c)<span class="sp"> −1/2</span>dz. More generally +if f(s, z) = 0 be such a relation connecting s, z that when θ is an +appropriate rational function of s and z both s and z are rationally +expressible, in virtue of ƒ(s, z) = 0 in terms of θ, the integral +∫R(s, z)dz is reducible to a form ∫H(θ)dθ, where H(θ) is rational +in θ, 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 +∫R(s, z)dz wherein s² is a cubic polynomial in z. The answer is +in the negative. For instance, no one of the three integrals</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span></td> <td>dz</td> +<td rowspan="2">, <span class="f150">∫</span></td> <td>zdz</td> +<td rowspan="2">, <span class="f150">∫</span></td> <td>dz</td></tr> +<tr><td class="denom">s</td> <td class="denom">s</td> +<td class="denom">(z − c)s</td></tr></table> + +<p class="noind">can be expressed by rational and logarithms of rational functions +of s and z; but it can be shown that every integral ∫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 § 20, <i>Elliptic Integrals</i>). A similar +theorem is true when s² = quartic polynomial in z; in fact when +s² = A(z − a) (z − b) (z − c) (z − d), putting y = s(z − a)<span class="sp">−2</span>, x = (z − a)<span class="sp">−1</span>, +we obtain y<span class="sp">2</span> = cubic polynomial in x. Much less is the theorem +true when the fundamental relation ƒ(s, z) = 0 is of more general +type. There exists then, however, a very general theorem, +known as <i>Abel’s Theorem</i>, which may be enunciated as follows: +Beside the rational function R(s, z) occurring in the integral +∫R(s, z)dz, consider another rational function H(s, z); let +(a<span class="su">1</span>), ... (a<span class="su">m</span>) denote the places of the construct associated +with the fundamental equation ƒ(s, z) = 0, for which H(s, z) is +equal to one value A, each taken with its proper multiplicity, +and let (b<span class="su">1</span>), ... (b<span class="su">m</span>) denote the places for which H(s, z) = B, +where B is another value; then the sum of the m integrals +<span class="f150">∫</span> <span class="sp1">(b<span class="su">i</span>)</span><span class="su1">(a<span class="su">i</span>)</span> R(s, z)dz is equal to the sum of the coefficients of t<span class="sp">−1</span> in the +expansions of the function</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">R(s, z)</td> <td>dz</td> +<td rowspan="2">λ <span class="f150">(</span></td> <td>H(s, z) − B</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">dt</td> <td class="denom">H(s, z) − A</td></tr></table> + +<p class="noind">where λ 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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">[</span></td> <td>1</td> +<td rowspan="2">R(s, z)</td> <td>dz</td> +<td rowspan="2"><span class="f150">]</span><span class="su">t<span class="sp">−1</span></span> = 0,</td></tr> +<tr><td class="denom">H(s, z) − μ</td> <td class="denom">dt</td></tr></table> + +<p class="noind">wherein μ is a constant. (For illustrations see below, under +§ 20, <i>Elliptic Integrals</i>.)</p> + +<p>§ 18. <i>Indeterminateness of Algebraic Integrals.</i>—The theorem +that the integral <span class="f150">∫</span> <span class="sp1">x</span><span class="su1">a</span> ƒ(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 ƒ(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 ƒ(z) is not a single valued function, so that the final +value of ƒ(z) at a is equal to its initial value. It is necessary for +the statement that this condition may be capable of being +satisfied.</p> + +<div class="condensed"> +<p>For instance, the integral <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">1</span> z<span class="sp">−1</span>dz is liable to an additive indeterminateness +equal to the value obtained by a closed path about z = 0, +which is equal to 2πi; if we put u = <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">1</span> z<span class="sp">−1</span>dz and consider z as a +function of u, then we must regard this function as unaffected by +the addition of 2πi to its argument u; we know in fact that +z = exp (u) and is a single valued function of u, with the period 2πi. +Or again the integral <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">0</span> (1 + z²)<span class="sp">−1</span>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 = <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">0</span> (1 + z²)<span class="sp">−1</span>dz, the +function z of u is periodic with period π, this being the function +tan (u). Next we take the integral u = <span class="f150">∫</span> <span class="sp1">(z)</span><span class="su1">(0)</span> (1 − z²)<span class="sp">−1/2</span>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 − z²)<span class="sp">−1/2</span>. 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 − z²)<span class="sp">1/2</span> = −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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫</span> <span class="sp1">−1</span><span class="su1">0</span></td> <td>dz</td> +<td rowspan="2">+ <span class="f150">∫</span> <span class="sp1">0</span><span class="su1">−1</span></td> <td>dz</td> +<td rowspan="2">+ <span class="f150">∫</span> <span class="sp1">1</span><span class="su1">0</span></td> <td>dz</td> +<td rowspan="2">+ <span class="f150">∫</span> <span class="sp1">0</span><span class="su1">1</span></td> <td>dz</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">(1 − z²)<span class="sp">1/2</span></td> <td class="denom">−(1 − z²)<span class="sp">1/2</span></td> +<td class="denom">−(1 − z²)<span class="sp">1/2</span></td> <td class="denom">(1 − z²)<span class="sp">1/2</span></td></tr></table> + +<p class="noind">where, in each case, (1 − z²)<span class="sp">1/2</span> is real and positive; that is, it gives</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−4 <span class="f150">∫</span> <span class="sp1">1</span><span class="su1">0</span></td> <td>dz</td></tr> +<tr><td class="denom">(1 − z²)<span class="sp">1/2</span></td></tr></table> + +<p class="noind">or 2π. Thus the additive indeterminateness of the integral is of the +form 2kπ, where k is an integer, and the function z of u, which is +sin (u), has 2π for period. Take now the case</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u = <span class="f150">∫</span> <span class="sp1">(z)</span><span class="su1">(z<span class="su">0</span>)</span></td> <td>dz</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">√{ (z − a) (z − b) (z − c) (z − d) }</td></tr></table> + +<p class="noind">adopting a definite determination for the phase of each of the +factors z − a, z − b, z − c, z − d at the arbitrary point z<span class="su">0</span>, 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 z<span class="su">0</span> describe a closed loop about the point z = a, consisting, +suppose, of a straight path from z<span class="su">0</span> to a, followed by a vanishing +circle whose centre is at a, completed by the straight path from a +to z<span class="su">0</span>. 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 z<span class="su">0</span> 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 z<span class="su">0</span> 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 z<span class="su">0</span> enclosing both a and b but +not either c or d, the value obtained will be A − B, and on returning +to z<span class="su">0</span> 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/ζ be reduced to a +circuit about ζ = 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 +u<span class="su">0</span> + m(A − B) + n(A − C), where m and n are integers. The value of +A − B is independent of the position of z<span class="su">0</span>, 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 ∫ 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 +<span class="pagenum"><a name="page324" id="page324"></a>324</span> +equation ƒ(s, z) = 0. Such an integral ƒK(z, s)dz is called an Abelian +Integral.</p> +</div> + +<p>§ 19. <i>Reversion of an Algebraic Integral</i>.—In a limited number of +cases the equation u = ∫ [z<span class="su">0</span> 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 § 14, <i>Doubly Periodic Functions</i>, 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</p> + +<p class="center">u = <span class="f150">∫</span> <span class="sp1">z</span><span class="su1">z<span class="su">0</span></span> [(z − a) (z − b) (z − c) (z − d) (z − e) (z − f) ]<span class="sp">−1/2</span>dz,</p> + +<p class="noind">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.</p> + +<p>§ 20. <i>Elliptic Integrals</i>.—An integral of the form ∫ 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 <i>elliptic integral</i>.</p> + +<div class="condensed"> +<p>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 z<span class="su">0</span> for which the radical s is not zero, +if we put z − z<span class="su">0</span> = t, we can find s − s<span class="su">0</span> = a power series in t, say +s=s<span class="su">0</span> + Q(t); when z is in the neighbourhood of a value, a, for which +s vanishes, if we put z = a + t², we shall obtain s = tQ(t), where Q(t) is a +power series in t; when z is very large and s² is a quartic polynomial +in z, if we put z<span class="sp">−1</span> = t, we shall find s<span class="sp">−1</span> = t²Q(t); when z is very large +and s² is a cubic polynomial in z, if we put z<span class="sp">−1</span> = t², we shall find +s<span class="sp">−l</span> = t³Q(t). By means of substitutions of these forms the character +of the integral ∫ R(z, s)dz may be investigated for any position of z; +in any case it takes a form ∫ [Ht<span class="sp">−m</span> + Kt<span class="sp">−m+1</span> + ... + Pt<span class="sp">−1</span> + R + St + ... ]dt +involving only a finite number of negative powers of t in the subject +of integration. Consider first the particular case ∫ s<span class="sp">−1</span>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 <i>everywhere finite</i>, and is said to be +of <i>the first kind</i>; it can, moreover, be shown without difficulty that +no integral ∫ R(z, s)dz, save a constant multiple of ∫ s<span class="sp">−1</span>dz, has this +property. Consider next, s² being of the form a<span class="su">0</span>z<span class="sp">4</span> + 4a<span class="su">1</span>z³ + ..., +wherein a<span class="su">0</span> may be zero, the integral ∫ (a<span class="su">0</span>z² + 2a<span class="su">1</span>z) s<span class="sp">−1</span>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<span class="sp">−1</span> + Q(t), where +Q(t) is a power series; denoting by √a<span class="su">0</span> a particular square root of a<span class="su">0</span> +when a<span class="su">0</span> is not zero, the integral becomes infinite for z = ∞ for both +signs of s, the value of A being + √a<span class="su">0</span> or − √a<span class="su">0</span> according as s is +√a<span class="su">0</span>·z² (1 + [2a<span class="su">1</span>/a<span class="su">0</span>] z<span class="sp">−1</span> + ... ) or is the negative of this; hence the integral +J<span class="su">1</span> = <span class="f150">∫</span> ( [a<span class="su">0</span>z² + 2a<span class="su">1</span>z]/s + √a<span class="su">0</span>) dz becomes infinite when z is infinite, for +the former sign of s, its infinite term being 2√a<span class="su">0</span>·t<span class="sp">−1</span> or 2a<span class="su">0</span>·z, +but does not become infinite for z infinite for the other sign of s. +When a<span class="su">0</span> = 0 the signs of s for z = ∞ 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 = ∞, its infinite part being 2√a<span class="su">1</span>·t<span class="sp">−1</span> or 2√a<span class="su">1</span>·√z. Similarly +if z<span class="su">0</span> be any finite value of z which is not a root of the polynomial +ƒ(z) to which s² is equal, and s<span class="su">0</span> denotes a particular one of the determinations +of s for z=z<span class="su">0</span>, the integral</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">J<span class="su">2</span> = <span class="f150">∫ {</span></td> <td>s²<span class="su">0</span> + ½(z − z<span class="su">0</span>) ƒ′(z<span class="su">0</span>)</td> +<td rowspan="2">+</td> <td>s<span class="su">0</span></td> +<td rowspan="2"><span class="f150">}</span> dz,</td></tr> +<tr><td class="denom">(z − z<span class="su">0</span>)² s</td> <td class="denom">(z − z<span class="su">0</span>)²</td></tr></table> + +<p class="noind">wherein ƒ′(z) = dƒ(z)/dz, becomes infinite for z = z<span class="su">0</span>, s = s<span class="su">0</span>, but not for +z = z<span class="su">0</span>, s = −s<span class="su">0</span>. its infinite term in the former case being the negative of +2s<span class="su">0</span>(z − z<span class="su">0</span>). For no other finite or infinite value of z is the integral +infinite. If z = θ be a root of ƒ(z), in which case the corresponding +value of s is zero, the integral</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">J<span class="su">3</span> = ½ƒ′(θ) <span class="f150">∫</span></td> <td>dz</td></tr> +<tr><td class="denom">(z − θ) s</td></tr></table> + +<p class="noind">becomes infinite for z=0, its infinite part being, if z − θ = t², equal to +−[ƒ′(θ)]½ t<span class="sp">−1</span>: and this integral is not elsewhere infinite. In each +of these cases, of the integrals J<span class="su">1</span>, J<span class="su">2</span>, J<span class="su">3</span>, the subject of integration +has been chosen so that when the integral is written near its point of +infinity in the form ∫[At<span class="sp">−2</span> + Bt<span class="sp">−1</span> + 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 <i>second kind</i>, 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 s² = ƒ(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 = ∞, +(2) z = z<span class="su">0</span>, (3) z = a, is given respectively by (s d/dz)<span class="sp">m−1</span> J<span class="su">1</span>, (s d/dz)<span class="sp">m−1</span> J<span class="su">2</span>, +(s d/dz)<span class="sp">m−1</span> J<span class="su">3</span>, 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 J<span class="su">1</span>, J<span class="su">2</span>, J<span class="su">3</span> and their differential +coefficients just written down, obtain, as the result, an integral +without algebraic infinities. But, in fact, if J, J<span class="sp">1</span> denote any two +of the three integrals J<span class="su">1</span>, J<span class="su">2</span>, J<span class="su">3</span>, there exists an equation AJ + BJ′ + +Cƒs<span class="sp">−1</span>dz = rational function of s, z, where A, B, C are properly chosen +constants. For the rational function</p> + +<table class="math0" summary="math"> +<tr><td>s + s<span class="su">0</span></td> +<td rowspan="2">+ z √a<span class="su">0</span></td></tr> +<tr><td class="denom">z − z<span class="su">0</span></td></tr></table> + +<p>is at once found to become infinite for (z<span class="su">0</span>, s<span class="su">0</span>), not for (z<span class="su">0</span>, −s<span class="su">0</span>), its +infinite part for the first point being 2s/(z − z<span class="su">0</span>), and to become +infinite for z infinitely large, and one sign of s only when these are +separable, its infinite part there being 2z √a<span class="su">0</span> or 2 √a<span class="su">1</span> √z when a<span class="su">0</span> = 0. +It does not become infinite for any other pair (z, s) satisfying the +relation s<span class="sp">2</span> = ƒ(z); this is in accordance with the easily verified +equation</p> + +<table class="math0" summary="math"> +<tr><td>s + s<span class="su">0</span></td> +<td rowspan="2">+ z √a<span class="su">0</span> − J<span class="su">1</span> + J<span class="su">2</span> + (a<span class="su">0</span>z<span class="su">0</span><span class="sp">2</span> + 2a<span class="su">1</span>z<span class="su">0</span>) <span class="f150">∫</span></td> <td>dz</td> +<td rowspan="2">= 0;</td></tr> +<tr><td class="denom">z − z<span class="sp">0</span></td> <td class="denom">s</td></tr></table> + +<p class="noind">and there exists the analogous equation</p> + +<table class="math0" summary="math"> +<tr><td>s</td> +<td rowspan="2">+ z √a<span class="su">0</span> − J<span class="su">1</span> + J<span class="su">3</span> + (a<span class="su">0</span>θ<span class="sp">2</span> + 2a<span class="su">1</span>θ) <span class="f150">∫</span></td> <td>dz</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">z − θ</td> <td class="denom">s</td></tr></table> + +<p class="noind">Consider now the integral</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">P = <span class="f150">∫ (</span></td> <td>s + s<span class="su">0</span></td> +<td rowspan="2">+ z √a<span class="su">0</span> <span class="f150">)</span></td> <td>dz</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">z − z<span class="su">0</span></td> <td class="denom">2s</td></tr></table> + +<p class="noind">this is at once found to be infinite, for finite values of z, only for +(z<span class="su">0</span>, s<span class="su">0</span>), its infinite part being log (z − z<span class="su">0</span>), and for z = ∞, for one sign +of s only when these are separable, its infinite part being −log t, +that is −log z when a<span class="su">0</span> ≠ 0, and −log (z<span class="sp">1/2</span>) when a<span class="su">0</span> = 0. And, if +ƒ(θ) = 0, the integral</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">P<span class="su">1</span> = <span class="f150">∫ (</span></td> <td>s</td> +<td rowspan="2">+ z √a<span class="su">0</span> <span class="f150">)</span></td> <td>dz</td></tr> +<tr><td class="denom">z − θ</td> <td class="denom">2s</td></tr></table> + +<p class="noind">is infinite at z = θ, s = 0 with an infinite part log t, that is log (z − θ)<span class="sp">1/2</span>, +is not infinite for any other finite value of z, and is infinite like P for +z = ∞. An integral possessing such logarithmic infinities is said +to be of the third kind.</p> + +<p>Hence it appears that any elliptic integral, by subtraction from +it of an appropriate sum formed with constant multiples of the +integral J<span class="su">3</span> and the rational functions of the form (s d/dz)<span class="sp">m−1</span> J<span class="su">1</span> +with constant multiples of integrals such as P or P<span class="su">1</span>, with constant +multiples of the integral u = ∫s<span class="sp">−1</span>dz, and with rational functions, +can be reduced to an integral H becoming infinite only for z = ∞, +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<span class="sp">1/2</span>). Such an integral +H = ∫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 = ∞ +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</p> + +<table class="math0" summary="math"> +<tr><td>s + s<span class="su">0</span></td> +<td rowspan="2">+ z √a<span class="su">0</span> or</td> <td>s</td> +<td rowspan="2">+ z √a<span class="su">0</span></td></tr> +<tr><td class="denom">z − z<span class="su">0</span></td> <td class="denom">z − θ</td></tr></table> + +<p class="noind">and their differential coefficients.</p> + +<p>The consideration of elliptic integrals is therefore reducible to +that of the three</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u = <span class="f150">∫</span></td> <td>dz</td> +<td rowspan="2">,   J = <span class="f150">∫ (</span></td> <td>a<span class="su">0</span>z<span class="sp">2</span> + 2a<span class="su">1</span>z</td> +<td rowspan="2">+ z √a<span class="su">0</span> <span class="f150">)</span> dz,   P = <span class="f150">∫ (</span></td> <td>s + s<span class="su">0</span></td> +<td rowspan="2">+ z √a<span class="su">0</span> <span class="f150">)</span></td> <td>dz</td></tr> +<tr><td class="denom">s</td> <td class="denom">s</td> +<td class="denom">z − z<span class="su">0</span></td> <td class="denom">2s</td></tr></table> + +<p class="noind">respectively of the first, second and third kind. Now the equation +s<span class="sp">2</span> = a<span class="su">0</span>z<span class="sp">4</span> + ... = a<span class="su">0</span> (z − θ) (z − φ) (z − ψ) (z − χ), by putting</p> + +<p class="center">y = 2s (z − θ)<span class="sp">−2</span> [a<span class="su">0</span> (θ − φ) (θ − ψ) (θ − χ) ]<span class="sp">−1/2</span></p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x =</td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">)</span></td></tr> +<tr><td class="denom">z − θ</td> <td class="denom">3</td> +<td class="denom">θ − φ</td> <td class="denom">θ − ψ</td> +<td class="denom">θ − χ</td></tr></table> + +<p class="noind">is at once reduced to the form y<span class="sp">2</span> = 4x<span class="sp">3</span> − g<span class="su">2</span>x − g<span class="su">3</span> = 4(x − e<span class="su">1</span>) (x − e<span class="su">2</span>) (x − e<span class="su">3</span>), +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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u = <span class="f150">∫</span></td> <td>dx</td> +<td rowspan="2">,   J = <span class="f150">∫</span></td> <td>xdx</td> +<td rowspan="2">,   P = <span class="f150">∫</span></td> <td>y + y<span class="su">0</span></td> +<td rowspan="2"> </td> <td>dx</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">y</td> <td class="denom">y</td> +<td class="denom">x − x<span class="su">0</span></td> <td class="denom">2y</td></tr></table> + +<p class="noind">Of these consider the first, putting</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u = <span class="f150">∫</span> <span class="sp1">(∞)</span><span class="su1">(x)</span></td> <td>dx</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">y</td></tr></table> + +<p class="noind">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<span class="sp">−1</span> = t<span class="sp">2</span>, y<span class="sp">−1</span> = +2t<span class="sp">3</span> (1 − ¼ g<span class="su">2</span>t<span class="sp">4</span> − ¼ g<span class="su">3</span>t<span class="sp">6</span>)<span class="sp">−1/2</span>, we have</p> + +<p class="center">u = <span class="f150">∫</span> <span class="sp1">t</span><span class="su1">0</span> (1 + <span class="spp">1</span>⁄<span class="suu">8</span> g<span class="su">2</span>t<span class="sp">4</span> + ... ) dt = t + <span class="spp">1</span>⁄<span class="suu">40</span> g<span class="su">2</span>t<span class="sp">5</span> + ...,</p> + +<p><span class="pagenum"><a name="page325" id="page325"></a>325</span></p> + +<p class="noind">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</p> + +<p class="center">x = u<span class="sp">−2</span> + <span class="spp">1</span>⁄<span class="suu">20</span> g<span class="su">2</span>u<span class="sp">2</span> + ...</p> + +<p>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 φ(u). In the range in question it is single valued and satisfies the +differential equation</p> + +<p class="center">[φ′(u)]<span class="sp">2</span> = 4[φ(u)]<span class="sp">3</span> − g<span class="su">2</span>φ(u) − g<span class="su">3</span>;</p> + +<p class="noind">in terms of it we can write x = φ(u), y = − φ′(u), and, φ′(u) being an +odd function, the sign attached to y in the original integral for x = ∞ +is immaterial. Now for any two values u, v in the range in question +consider the function</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">F(u, v) = ¼ <span class="f150">[</span></td> <td>φ′(u) − φ′(v)</td> +<td rowspan="2"><span class="f150">]</span><span class="sp1">2</span> − φ(u) − φ(v);</td></tr> +<tr><td class="denom">φ(u) − φ(v)</td></tr></table> + +<p class="noind">it is at once seen, from the differential equation, to be such that +∂F/∂u = ∂F/∂v; it is therefore a function of u + v; supposing +|u + v| < R we infer therefore, by putting v = 0, that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ(u + v) = ¼ <span class="f150">[</span></td> <td>φ′(u) − φ′(v)</td> +<td rowspan="2"><span class="f150">]</span><span class="sp1">2</span> − φ(u) − φ(v).</td></tr> +<tr><td class="denom">φ(u) − φ(v)</td></tr></table> + +<p class="noind">By repetition of this equation we infer that if u<span class="su">1</span>, ... u<span class="su">n</span> be any arguments +each of which is in absolute value less than R, whose sum is also +in absolute value less than R, then φ(u<span class="su">1</span> + ... + u<span class="su">n</span>) is a rational +function of the 2n functions φ(u<span class="su">s</span>), φ′(u<span class="su">s</span>); and hence, if |u| < R, +that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ(u) = H <span class="f150">[</span> φ <span class="f150">(</span></td> <td>u</td> +<td rowspan="2"><span class="f150">)</span>,   φ′ <span class="f150">(</span></td> <td>u</td> +<td rowspan="2"><span class="f150">) ]</span>,</td></tr> +<tr><td class="denom">n</td> <td class="denom">n</td></tr></table> + +<p class="noind">where H is some rational function of the arguments φ(u/n), φ′(u/n). +In fact, however, so long as |u/n| < R, each of the functions φ(u/n), +φ′(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 [φ(u/n), φ′(u/n)] is single valued and without singularities other +than poles so long as |u| < nR; it agrees with φ(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 [φ′(u)]<span class="sp">2</span> = 4[φ(u)]<span class="sp">3</span> − g<span class="su">2</span>φ(u) − g<span class="su">3</span>. 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 φ(u + α), wherein α is arbitrary, +being such. Taking now any range of values of u, from u = 0, +and putting for any value of u, x = φ(u), y = −φ′(u), so that +y<span class="sp">2</span>=4x<span class="sp">3</span>-g<span class="su">2</span>x-g<span class="su">3</span>, we clearly have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u = <span class="f150">∫</span> <span class="sp1">(∞)</span><span class="su1">(x, y)</span></td> <td>dx</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">y</td></tr></table> + +<p class="noind">conversely if x<span class="su">0</span> = φ(u<span class="su">0</span>), y<span class="su">0</span> = −φ′(u<span class="su">0</span>) and ξ, η be any values satisfying +η<span class="su">2</span> = 4ξ<span class="sp">2</span> − g<span class="su">2</span>ξ − g<span class="su">3</span>, which are sufficiently near respectively to x<span class="su">0</span>, y<span class="su">0</span>, +while v is defined by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">v − u<span class="su">0</span> = − <span class="f150">∫</span> <span class="sp1">(ξ, η)</span><span class="su1">(x<span class="su">0</span>, y<span class="su">0</span>)</span></td> <td>dξ</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">η</td></tr></table> + +<p class="noind">then ξ, η are respectively φ(v) and −φ′(v); for this equation leads +to an expansion for ξ − x<span class="su">0</span> in terms of v = u<span class="su">0</span> and only one such expansion, +and this is obtained by the same work as would be necessary +to expand φ(v) when v is near to u<span class="su">0</span>; the function φ(u) can therefore +be continued by the help of this equation, from v = u<span class="su">0</span>, provided +the lower limit of |ξ − x<span class="su">0</span>| necessary for the expansions is not zero +in the neighbourhood of any value (x<span class="su">0</span>, y<span class="su">0</span>). In fact the function φ(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 φ(u) is greater than zero; the same will therefore be the case +for the lower limit of the radii |ξ − x<span class="su">0</span>| necessary for the continuations +spoken of above provided that the values of (ξ, η) considered do not +lead to infinitely increasing values of v; there does not exist, however, +any definite point (ξ<span class="su">0</span>, η<span class="su">0</span>) in the neighbourhood of which the +integral <span class="f150">∫</span> <span class="sp1">(ξ, η)</span><span class="su1">(x<span class="su">0</span>, y<span class="su">0</span>)</span> dξ/η increases indefinitely, it is only by a path of infinite +length that the integral can so increase. We infer therefore that +if (ξ, η) be any point, where η<span class="su">2</span> = 4ξ<span class="sp">3</span> − g<span class="su">2</span>ξ − g<span class="su">3</span>, and v be defined by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">v = <span class="f150">∫</span> <span class="sp1">(∞)</span><span class="su1">(ξ, η)</span></td> <td>dx</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">y</td></tr></table> + +<p class="noind">then ξ = φ(v) and η = −φ′(v). Thus this equation determines (ξ, η) +without ambiguity. In particular the additive indeterminatenesses +of the integral obtained by closed circuits of the point of integration +are periods of the function φ(u); by considerations advanced above +it appears that these periods are sums of integral multiples of two +which may be taken to be</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ω = 2 <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">e<span class="su">1</span></span></td> <td>dx</td> +<td rowspan="2">,   ω′ = 2 <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">e<span class="su">3</span></span></td> <td>dx</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">y</td> <td class="denom">y</td></tr></table> + +<p class="noind">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 +(x<span class="su">0</span>, y<span class="su">0</span>), other than one enclosing two of the points (e<span class="su">1</span>, 0), (e<span class="su">2</span>, 0), +(e<span class="su">3</span>, 0), (∞, ∞), which leads back to the initial point (x<span class="su">0</span>, y<span class="su">0</span>), which is +impossible. On the whole, therefore, it appears that the function +φ(u) agrees with the function ℜ(u) previously discussed, and the +discussion of the elliptic integrals can be continued in the manner +given under § 14, <i>Doubly Periodic Functions</i>.</p> +</div> + +<p>§ 21. <i>Modular Functions.</i>—One result of the previous theory +is the remarkable fact that if</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ω = 2 <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">e<span class="su">1</span></span></td> <td>dx</td> +<td rowspan="2">,   ω′ = 2 <span class="f150">∫</span> <span class="sp1">∞</span><span class="su1">e<span class="su">3</span></span></td> <td>dx</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">y</td> <td class="denom">y</td></tr></table> + +<p class="noind">where y<span class="sp">2</span> = 4(x − e<span class="su">1</span>) (x − e<span class="su">2</span>) (x − e<span class="su">3</span>), then we have</p> + +<p class="center">e<span class="su">1</span> = (½ω)<span class="sp">−2</span> + Σ′ {[(m + ½) ω + m′ω′]<span class="sp">−2</span> − [mω + m′ω′]<span class="sp">−2</span>},</p> + +<p class="noind">and a similar equation for e<span class="su">3</span>, 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 ω′/ω.</p> + +<div class="condensed"> +<p>It is easy to see that the series for ℜ(u), u<span class="sp">−2</span> + Σ′[(u + mω + m′ω′)<span class="sp">2</span> − +(mω + m′ω′)<span class="sp">2</span>], is unaffected by replacing ω, ω′ by two quantities Ω, Ω′ +equal respectively to pω + qω′, p′ω′ + q′ω′, 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 Ω = pω + qω′, Ω′ = p′ω + q′ω′, +wherein pq′ − p′q = 1, can be built up by repetitions of the two particular +substitutions (Ω = −ω′, Ω′ = ω), (Ω = ω, Ω′ = ω + ω′). Consider +the function of the ratio ω′/ω expressed by</p> + +<p class="center">h = −ℜ (½ω′) / ℜ(½ω);</p> + +<p class="noind">it is at once seen from the properties of the function ℜ(u) that by +the two particular substitutions referred to we obtain the corresponding +substitutions for h expressed by</p> + +<p class="center">h′ = 1/h,   h′ = 1 − h;</p> + +<p class="noind">thus, by all the integer substitutions Ω = pω + qω′, Ω′ = p′ω + q′ω′, 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 θ,</p> + +<table class="math0" summary="math"> +<tr><td>(1 − θ + θ<span class="sp">2</span>)<span class="sp">3</span></td> +<td rowspan="2">=</td> <td>(1 − h + h<span class="sp">2</span>)<span class="sp">3</span></td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">θ<span class="sp">2</span>(1 − θ)<span class="sp">2</span></td> <td class="denom">h<span class="sp">2</span>(1 − h)<span class="sp">2</span></td></tr></table> + +<p class="noind">the function of τ, = ω′/ω, expressed by the right side, is thus +unaltered by every one of the substitutions τ′ = (p′ + q′τ / p + qτ), wherein +p, q, p′, q′ are integers having pq′ − p′q = 1. If the imaginary part +σ, of τ, which we may write τ = ρ + iσ, is positive, the imaginary part +of τ′, which is equal to σ(pq′ − p′q)/[(p + qρ)<span class="sp">2</span> + q<span class="sp">2</span>σ<span class="sp">2</span>], is also positive; +suppose σ to be positive; it can be shown that the upper half of the +infinite plane of the complex variable τ 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, τ′ = (p′ + q′τ)/(p + qτ) leads from an arbitrary point τ, +of one of these regions, to a point τ′ of another; taking τ = ρ + iσ, +one of these regions may be taken to be that for which −½ < ρ < ½, +ρ<span class="sp">2</span> + σ<span class="sp">2</span> > 1, together with the points for which ρ 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 τ = (p′ + q′τ)/(p + qτ), and hence by a definite combination +of the substitutions τ′ = −1/τ, τ′ = 1 + τ. Upon the infinite half +plane of τ, the function considered above,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">z(τ) = <span class="spp">4</span>⁄<span class="suu">27</span></td> <td>[ℜ<span class="sp">2</span> (½ω) + ℜ (z(½ω) ℜ (½ω′) + ℜ<span class="sp">2</span> (½ω′)]<span class="sp">3</span></td> +<td rowspan="2"></td></tr> +<tr><td class="denom">ℜ<span class="sp">2</span> (½ω) ℜ<span class="sp">2</span> (½ω′) [ℜ (½ω) + ℜ (½ω′]<span class="sp">2</span></td></tr></table> + +<p class="noind">is a single valued monogenic function, whose only essential singularities +are the points τ′ = (p′ + q′τ)/(p + qτ) for which τ = ∞, namely +those for which τ′ is any real rational value; the real axis is thus a +line over which the function z(τ) cannot be continued, having an +essential singularity in every arc of it, however short; in the fundamental +region, z(τ) has thus only the single essential singularity, +r = ρ + iσ, where σ = ∞; in this fundamental region z(τ) takes any +assigned complex value just once, the relation z(τ′) = z(τ) requiring, +as can be shown, that τ′ is of the form (p′ + q′τ)/(p + qτ), in which +p, q, p′, q′ are integers with pq′ − p′q = 1; the function z(τ) 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 τ, J(τ) = z(τ) [z(τ) − 1], +it can be shown that J(τ) = 0 for τ = exp (<span class="spp">2</span>⁄<span class="suu">3</span>πi), that J(τ) = 1 for τ = i, +these being values of τ on the boundary of the fundamental region; +like z(τ) it has an essential singularity for τ = ρ + iσ, σ = + ∞. In the +<span class="pagenum"><a name="page326" id="page326"></a>326</span> +theory of linear differential equations it is important to consider the +inverse function τ(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 τ of period 3, such as +τ′ = −(1 + τ)<span class="sp">−1</span>), having a cycle of two values about J = 1 (the circuit +leading to a linear substitution for τ of period 2, such as τ′ = −τ<span class="sp">−1</span>), +and having a cycle of infinitely many values about J = ∞ (the circuit +leading to a linear substitution for τ which is not periodic, such as +τ′ = 1 + τ). These are the only singularities for the function τ(J). +Each of the functions</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">[J(τ)]<span class="sp">1/3</span>,   [J(τ) − 1]<span class="sp">1/2</span>,   <span class="f150">[</span> −</td> <td>ℜ (½ω) + 2ℜ (½ω′)</td> +<td rowspan="2"><span class="f150">]</span><span class="sp1">1/8</span>,</td></tr> +<tr><td class="denom">ℜ (½ω) − ℜ (½)ω′)</td></tr></table> + +<p class="noind">beside many others (see below), is a single valued function of τ, +and is expressible without ambiguity in terms of the single valued +function of τ,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">η(τ) = exp <span class="f150">(</span></td> <td>iπτ</td> +<td rowspan="2"><span class="f150">) Π</span> <span class="sp1">∞</span><span class="su1">n=1</span> [1 − exp (2iπnτ)] = exp <span class="f150">(</span></td> <td>iπτ</td> +<td rowspan="2"><span class="f150">) Σ</span> <span class="sp1">∞</span><span class="su1">m=−∞</span> (−1)<span class="sp">m</span> exp [(3m<span class="sp">2</span> + m) iπτ].</td></tr> +<tr><td class="denom">12</td> <td class="denom">12</td></tr></table> + +<p>It should be remarked, however, that η(τ) is not unaltered by all +the substitutions we have considered; in fact</p> + +<p class="center">η(−τ<span class="sp">−1</span>) = (−iτ) ½η (τ),   η(1 + τ) = exp (<span class="spp">1</span>⁄<span class="suu">12</span> iπ) η(τ).</p> + +<p>The aggregate of the substitutions τ′ = (p′ + q′τ)/(p + qτ), wherein +p, q, p′, q′ are integers with pq′ − p′q = 1, represents a <i>Group</i>; the +function J(τ), unaltered by all these substitutions, is called a <i>Modular +Function</i>. More generally any function unaltered by all the substitutions +of a group of linear substitutions of its variable is called an +<i>Automorphic Function</i>. A rational function, of its variable h, of this +character, is the function (1 − h + h<span class="sp">2</span>)<span class="sp">3</span> h<span class="sp">−2</span>(1 − h)<span class="sp">−2</span> 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<span class="sp">2</span> = 4x<span class="sp">3</span> − g<span class="su">2</span>x − g<span class="su">3</span> (and in fact many other algebraic equations, +see below, under § 23, <i>Geometrical Applications of Elliptic +Functions</i>) in terms of single valued functions x = ℜ(u), y = −ℜ′(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.</p> + +<p>The modular function J(τ) considered above, unaltered by the +group of linear substitutions τ′ = (p′ + q′τ) / (p + qτ), 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</p> + +<table class="math0" summary="math"> +<tr><td>s″′</td> +<td rowspan="2">−</td> <td>3</td> +<td rowspan="2"><span class="f150">(</span></td> <td>s″</td> +<td rowspan="2"><span class="f150">)</span><span class="sp1">2</span> =</td> <td>1 − α<span class="sp">2</span></td> +<td rowspan="2">+</td> <td>1 − β<span class="sp">2</span></td> +<td rowspan="2">+</td> <td>α<span class="sp">2</span> + β<span class="sp">2</span> − γ<span class="sp">2</span> − 1</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">s′</td> <td class="denom">2</td> +<td class="denom">s′</td> <td class="denom">2(x − 1)<span class="sp">2</span></td> +<td class="denom">2x<span class="sp">2</span></td> <td class="denom">2x (x − 1)</td></tr></table> + +<p class="noind">where s′ = ds/dx, &c., of which the dependent variable s is equal to τ. +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(α,β,γ, x), +we have in fact τ = s(½, <span class="spp">1</span>⁄<span class="suu">3</span>, 0, J). If we introduce also the function +of τ given by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">λ =</td> <td>2ℜ (½ω′) + ℜ (½ω)</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">ℜ (½ω′) − ℜ (½ω)</td></tr></table> + +<p class="noind">we similarly have τ = s(0, 0, 0, λ); this function λ is a single valued +function of τ, which is also a modular function, being unaltered by a +group of integral substitutions also of the form τ′ = (p′ + q′τ)/(p + qτ), +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 τ = ρ + iσ, may be +taken to be that given by −1 < ρ < 1, (ρ + ½)<span class="sp">2</span> + σ<span class="sp">2</span> > ¼, (ρ − ½)<span class="sp">2</span> + σ<span class="sp">2</span> > ¼, +and is built up of six of the regions which arose for the general +modular group associated with J(τ). Within this fundamental +region, λ takes every complex value just once, except the values +λ = 0, 1, ∞, which arise only at the angular points τ = 0, τ = ∞, τ = − 1 +and the equivalent point τ = 1; these angular points are essential +singularities for the function λ(τ). For λ(τ) as for J(τ), the region of +existence is the upper half plane of τ, there being an essential singularity +in every length of the real axis, however short.</p> + +<p>If, beside the plane of τ, we take a plane to represent the values of +λ, the function τ = s(0, 0, 0, λ) being considered thereon, the values of +τ belonging to the interior of the fundamental region of the τ-plane +considered above, will require the consideration of the whole of the +λ-plane taken once with the exception of the portions of the real +axis lying between −∞ and 0 and between 1 and +∞, the two +sides of the first portion corresponding to the circumferences of the +τ-plane expressed by (ρ + ½)<span class="sp">2</span> + σ<span class="sp">2</span> = ¼, (ρ − ½)<span class="sp">2</span> + σ<span class="sp">2</span> = ¼, while the two +sides of the latter portion, for which λ is real and > 1, correspond +to the lines of the τ-plane expressed by ρ = ±1. The line for +which λ is real, positive and less than unity corresponds to the +imaginary axis of the τ-plane, lying in the interior of the fundamental +region. All the values of τ = s(0, 0, 0, λ) may then be derived +from those belonging to the fundamental region of the τ-plane by +making λ describe a proper succession of circuits about the points +λ = 0, λ = 1; any such circuit subjects τ to a linear substitution +of the subgroup of τ considered, and corresponds to a change of τ +from a point of the fundamental region to a corresponding point +of one of the other regions.</p> +</div> + +<p>§ 22. <i>A Property of Integral Functions deduced from the Theory +of Modular Functions</i>.—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 λ = exp(z) +pass through a closed succession of values in the plane of λ +having λ = 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, λ) by a circuit of λ about λ = 0; more generally, if +φ(z) be an integral function of z, never becoming either zero or +unity for finite values of z, the function λ = φ(z), however z vary +in the finite part of the plane, will never make, in the plane of λ, +a circuit about either λ = 0 or λ = 1, and s(0, 0, 0, λ), that is +s[0, 0, 0, φ(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, φ(z)] is also an integral function—whose imaginary +part, moreover, by the property of s(0, 0, 0, λ), remains positive +for all finite values of z. In that case, however, exp {is[0, 0, 0, φ(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 <i>an integral function φ(z) which does not vanish for any finite +value of z, takes the value unity and hence</i> (by considering the +function A<span class="sp">−1</span>φ(z)) <i>takes every other value for some definite value +of z</i>; or, an integral function for which both the equations +φ(z) = A, φ(z) = B are unsatisfied by definite values of z, does not +exist, A and B being arbitrary constants.</p> + +<div class="condensed"> +<p>A similar theorem can be proved in regard to the values assumed +by the function φ(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.</p> + +<p>Another application of modular functions is to prove that the +function s(α, β, γ, λ) is a single valued function of τ = s(0, 0, 0, λ); +for, putting τ′ = (τ − i)/(τ + i), the values of τ′ which correspond to the +singular points λ = 0, 1, ∞ of s(α, β, γ, λ), though infinite in number, +all lie on the circumference of the circle |τ′| = 1, within which therefore +s(α, β, γ, x) is expressible in a form <span class="f150">Σ</span> <span class="sp1">∞</span><span class="su1">n=0</span> a<span class="su">n</span>τ′<span class="sp">n</span>. More generally any +monogenic function of λ which is single valued save for circuits of +the points λ = 0, 1, ∞, is a single valued function of τ = s(0, 0, 0, λ). +Identifying λ with the square of the modulus in Legendre’s form of +the elliptical integral, we have τ = iK′/K, where</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">K = <span class="f150">∫</span> <span class="sp1">1</span><span class="su1">0</span></td> <td>dt</td> +<td rowspan="2">,   K′ = <span class="f150">∫</span> <span class="sp1">1</span><span class="su1">0</span></td> <td>dt</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">√[1 − t<span class="sp">2</span>] [1 − λt<span class="sp">2</span>]</td> <td class="denom">√[1 − t<span class="sp">2</span>] [1 − (1 − λ) t<span class="sp">2</span>]</td></tr></table> + +<p class="noind">functions such as λ<span class="sp">1/4</span>, (1 − λ)<span class="sp">1/4</span>, [λ(1 − λ)]<span class="sp">1/4</span>, which have only λ = 0, 1, ∞ +as singular points, were expressed by Jacobi as power series in q = e<span class="sp">iπτ</span>, +and therefore, at least for a limited range of values of τ, as single +valued functions of τ; it follows by the theorem given that any +product of a root of λ and a root of 1 − λ is a single valued function +of τ. More generally the differential equation</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x(1 − x)</td> <td>d<span class="sp">2</span>y </td> +<td rowspan="2">+ [γ − (α + β + 1)x]</td> <td>dy</td> +<td rowspan="2">− αβγ = 0</td></tr> +<tr><td class="denom">dx<span class="sp">2</span></td> <td class="denom">dx</td></tr></table> + +<p class="noind">may be solved by expressing both the independent and dependent +variables as single valued functions of a single variable τ, the expression +for the independent variable being x = λ(τ).</p> +</div> + +<p>§ 23. <i>Geometrical Applications of Elliptic Functions.</i>—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 ½n(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 +<span class="pagenum"><a name="page327" id="page327"></a>327</span> +ƒ in n(n − 2) − n(n − 3) − (n − 3) = 3 other points, and will contain +homogeneously at least ½(n − 1)n − ½n(n − 3) −(n − 3) = 3 arbitrary +constants, and so will be of the form λφ + λ<span class="su">1</span>φ<span class="su">1</span> + λ<span class="su">2</span>φ<span class="su">2</span> + +... = 0, wherein λ<span class="su">3</span>, λ<span class="su">4</span>, ... are in general zero. Put now +ξ = φ<span class="su">1</span>/φ, η = φ<span class="su">2</span>/φ and eliminate x, y between these equations and +ƒ(x, y) = 0, so obtaining a rational irreducible equation F(ξ, η) = 0, +representing a further plane curve. To any point (x, y) of ƒ will +then correspond a definite point (ξ, η) of F.</p> + +<div class="condensed"> +<p>For a general position of (x, y) upon ƒ the equations +φ<span class="su">1</span>(x′, x′)/φ(x′, x′) = φ<span class="su">1</span>(x, y)/φ(x, y), φ<span class="su">2</span>(x′, x′)/φ(x′, x′) = φ<span class="su">2</span>(x, y)/φ(x, y), +subject to ƒ(x′, x′) = 0, will have the same number of solutions (x′, x′); +if their only solution is x′ = x, x′ = y, then to any position (ξ, η) of F +will conversely correspond only one position (x, y) of ƒ. If these +equations have another solution beside (x, y), then any curve +λφ + λ<span class="su">1</span>φ<span class="su">1</span> + λ<span class="su">2</span>φ<span class="su">2</span> = 0 which passes (through the double points of ƒ +and) through the n − 2 points of ƒ constituted by the fixed n− 3 +points and a point (x<span class="su">0</span>, y<span class="su">0</span>), will necessarily pass through a further +point, say (x<span class="su">0</span>′, y<span class="su">0</span>′), and will have only one further intersection with +ƒ; such a curve, with the n − 2 assigned points, beside the double +points, of ƒ, will be of the form μψ + μ<span class="su">1</span>ψ<span class="su">1</span> + ... = 0, where μ<span class="su">2</span>, μ<span class="su">3</span>, ... +are generally zero; considering the curves ψ + tψ<span class="su">1</span> = 0, for variable t, +one of these passes through a further arbitrary point of ƒ, by choosing +t properly, and conversely an arbitrary value of t determines a single +further point of ƒ; the co-ordinates of the points of ƒ 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 ½(n − 3)n but ½(n − 3)n + 1 double points. +We may therefore assume that to every point of F corresponds +only one point of ƒ, 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 ƒ, +that is, at the least, by taking these to be consecutive points, will +involve the co-ordinates of one point of ƒ, and will not be rational +in the coefficients of ƒ unless we can specify a point of ƒ whose co-ordinates +are rational in these. The curve F is intersected by a +straight line aξ + bη + c = 0 in as many points as the number of +unspecified intersections of ƒ with aφ + bφ<span class="su">1</span> + cφ<span class="su">2</span> = 0, that is, 3; or F +will be a cubic curve, without double points.</p> + +<p>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</p> + +<p class="center">ZY² = aX³ + bX²Z + cXZ² + dZ³;</p> + +<p class="noind">by putting X equal to λX + μZ, that is, choosing a suitable line +through Y to be X = 0, and choosing λ properly, this is reduced to +the form</p> + +<p class="center">ZY² = 4X³ − g<span class="su">2</span>XZ² − g<span class="su">3</span>Z³,</p> + +<p class="noind">of which a representation is given, valid for every point, in terms of +the elliptic functions ℜ(u), ℜ′(u), by taking X = Zℜ(u), Y = Zℜ′(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</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">u = <span class="f150">∫</span> <span class="sp1">(x)</span><span class="su1">(∞)</span></td> <td>ZdX − XdZ</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">ZY</td></tr></table> + +<p class="noind">where (∞) denotes the point of inflection.</p> + +<p>It thus appears that the co-ordinates of any point of a plane curve, +ƒ, of order n with ½(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, φ, of order m be drawn, passing through +the double points of the curve, the values of the argument u at the +remaining intersections of φ with ƒ, have a sum which is unaffected +by variation of the coefficients of φ, 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 y² = 4x³ − g<span class="su">2</span>x − g<span class="su">3</span>, in points (u<span class="su">1</span>), (u<span class="su">2</span>), (u<span class="su">3</span>), the sum +u<span class="su">1</span> + u<span class="su">2</span> + u<span class="su">3</span> is zero, or a period; or the general theorem is a corollary +from Abel’s theorem proved under § 17, <i>Integrals of Algebraic +Functions</i>. 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 δm, δn in m +and n, is obtained by differentiation of the equation for the three +abscissae, namely the equation</p> + +<p class="center">F(x) = 4x³ − g<span class="su">2</span>x − g<span class="su">3</span> − (mx + n)² = 0,</p> + +<p class="noind">and is thus given by</p> + +<table class="math0" summary="math"> +<tr><td>dx</td> +<td rowspan="2">=</td> <td>xδm + δn</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">y</td> <td class="denom">F′(x)</td></tr></table> + +<p class="noind">and the sum of three such fractions as that on the right for the three +roots of F(x) = 0 is zero; hence u<span class="su">1</span> + u<span class="su">2</span> + u<span class="su">3</span> is independent of the +straight line considered; if in particular this become the inflexional +tangent each of u<span class="su">1</span>, u<span class="su">2</span>, u<span class="su">3</span> vanishes. It may be remarked in passing +that x<span class="su">1</span> + x<span class="su">2</span> + x<span class="su">3</span> = ¼m², and hence is ¼ {(y<span class="su">1</span> − y<span class="su">2</span>)/(x<span class="su">1</span> − x<span class="su">2</span>)}²; so that we +have another proof of the addition equation for the function ℜ(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 ½(n − 3)n double points.</p> +</div> + +<p>§ 24. <i>Integrals of Algebraic Functions in Connexion with the +Theory of Plane Curves.</i>—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, ƒ(x, y) = o.</p> + +<div class="condensed"> +<p>The algebraical integrals ∫ R(x, y)dx associated with this may as +before be divided into those of the <i>first kind</i>, which have no infinities, +those of the <i>second kind</i>, possessing only algebraical infinities, +and those of the <i>third kind</i>, 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 +ƒ(x, y) = 0; a rational function can be constructed with poles of the +first order at p + 1 arbitrary positions (x, y), satisfying ƒ(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 ƒ(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.</p> + +<p>There is, however, a difference of essential kind in regard to the +inversion of integrals of the first kind; if u = ∫R(x, y)dx be such an +integral, it can be shown, in common with all algebraic integrals +associated with ƒ(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 ∫R<span class="su">i</span>(x, y)dx +denote one of the integrals of the first kind, is that the p equations</p> + +<p class="center">∫ R<span class="su">i</span> (x<span class="su">1</span>, y<span class="su">1</span>)dx<span class="su">1</span> + ... + ∫ R<span class="su">i</span> (x<span class="su">p</span>, y<span class="su">p</span>)dx<span class="su">p</span> = u<span class="su">i</span>,</p> + +<p class="noind">determine the rational symmetric functions of the p positions (x<span class="su">1</span>, y<span class="su">1</span>), +... (x<span class="su">p</span>, y<span class="su">p</span>) as single valued functions of the p variables, u<span class="su">1</span>, ... u<span class="su">p</span>. +It is thus necessary to enter into the theory of functions of several +independent variables; and the equation ƒ(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.</p> +</div> + +<p>§ 25. <i>Monogenic Functions of Several Independent Variables.</i>—A +monogenic function of several independent complex variables +u<span class="su">i</span>, ... u<span class="su">p</span> 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 <i>poles</i>, or <i>polar points of indetermination</i>, +or <i>essential singularities</i>.</p> + +<div class="condensed"> +<p>A pole is a point (u<span class="sp">(0)</span><span class="su">1</span>, ... u<span class="sp">(0)</span><span class="su">p</span>) in the neighbourhood of which the +function is expressible as a quotient of converging power series in +u<span class="su">1</span> − u<span class="sp">(0)</span><span class="su">1</span> ... u<span class="su">p</span> − u<span class="sp">(0)</span><span class="su">p</span>; of these the denominator series D must +vanish at (u<span class="sp">(0)</span><span class="su">1</span>, ... u<span class="sp">(0)</span><span class="su">p</span>), 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 (u<span class="sp">(0)</span><span class="su">1</span>, ... u<span class="sp">(0)</span><span class="su">p</span>), or if it does, it must +be possible to write D = MD<span class="su">0</span>, N = MN<span class="su">0</span>, where M is a converging +power series vanishing at (u<span class="sp">(0)</span><span class="su">1</span>, ...u<span class="sp">(0)</span><span class="su">p</span>), and N<span class="su">0</span> is a converging power +series, in (u<span class="su">1</span> − u<span class="sp">(0)</span><span class="su">1</span> ... u<span class="su">p</span> − u<span class="sp">(0)</span><span class="su">p</span>), 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 u<span class="su">1</span>, ... u<span class="su">p</span> approach to u<span class="sp">(0)</span><span class="su">1</span>, ... u<span class="sp">(0)</span><span class="su">p</span> 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 u<span class="su">1</span>, ... u<span class="su">p</span> is indefinitely great; +such points are brought under the preceding definitions by means +<span class="pagenum"><a name="page328" id="page328"></a>328</span> +of the convention that for u<span class="sp">(0)</span><span class="su">i</span> = ∞, the difference u<span class="su">i</span> − u<span class="sp">(0)</span><span class="su">i</span> is to be +understood to stand for u<span class="sp">−1</span><span class="su">i</span>. This being so, a single valued function +of u<span class="su">1</span>, ... u<span class="su">p</span> 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.</p> + +<p>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 u<span class="su">1</span>, ... u<span class="su">p</span> are <i>simultaneously</i> increased each by a +proper constant, these being the additive constants of indeterminateness +for the p integrals ∫ R<span class="su">i</span>(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 u<span class="su">1</span>, ... u<span class="su">p</span>, called the <i>Theta Function</i>. This is +expressible as a series of positive and negative integral powers of +quantities exp (c<span class="su">1</span>u<span class="su">1</span>), exp (c<span class="su">2</span>u<span class="su">2</span>), ... exp (c<span class="su">p</span>u<span class="su">p</span>), wherein c<span class="su">1</span>, ... c<span class="su">p</span> are +proper constants; for p = 1 this theta function is essentially the +same as that above given under a different form (see § 14, <i>Doubly +Periodic Functions</i>), the function σ(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 x<span class="su">1</span>, ... x<span class="su">p</span>, y denote p + 1 such functions; then each of the partial +derivatives dx<span class="su">i</span>/∂u<span class="su">i</span> will equally be a meromorphic function of the +same periods, and so expressible rationally in terms of x<span class="su">1</span>, ... x<span class="su">p</span>, y; +thus there will exist p equations of the form</p> + +<p class="center">dx<span class="su">i</span> = R<span class="su">1</span>du<span class="su">1</span> + ... + R<span class="su">p</span>du<span class="su">p</span>,</p> + +<p class="noind">and hence p equations of the form</p> + +<p class="center">du<span class="su">i</span> = H<span class="su">i, 1</span>dx<span class="su">1</span> + ... + H<span class="su">i, p</span>dx<span class="su">p</span>,</p> + +<p class="noind">wherein H<span class="su">i, j</span> are rational functions of x<span class="su">1</span>, ... x<span class="su">p</span>, y, these being connected +by a fundamental algebraic (rational) equation, say ƒ(x<span class="su">1</span>, ... x<span class="su">p</span>, y) += 0. This then is the generalized form of the corresponding equation +for p = 1.</p> +</div> + +<p>§ 26. <i>Multiply-Periodic Functions and the Theory of Surfaces.</i>—The +theory of algebraic integrals ∫ R(x, y)dx, wherein x, y are +connected by a rational equation ƒ(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.</p> + +<div class="condensed"> +<p>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 ∫(Rds + Sdy), belonging to a surface ƒ(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 ƒ(x, y, z) = 0 have a double +curve with triple points triple also for the surface, and φ(x, y, z) = 0 +be a surface of order n − 4 passing through the double curve, the +double integral</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">∫ ∫</span></td> <td>φ dx dy</td></tr> +<tr><td class="denom">∂f/∂z</td></tr></table> + +<p class="noind">is everywhere finite; and, the most general everywhere finite +integral of this form remains invariant in a birational transformation +of the surface ƒ, 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<span class="su">a</span>, 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<span class="su">g</span>. The +difference p<span class="su">g</span> − p<span class="su">a</span>, 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.</p> + +<p>On a plane algebraical curve we may consider linear series of sets +of points, obtained by the intersection with it of curves λφ + λ<span class="su">1</span>φ<span class="su">1</span> + +... = 0, wherein λ, λ<span class="su">1</span>, ... are variable coefficients; such a series +consists of the sets of points where a rational function of given poles, +belonging to the construct ƒ(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 λφ + λ<span class="su">1</span>φ<span class="su">1</span> + ... = 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 <i>order</i>, which is the number of points forming a +set of the series, and the <i>dimension</i>, which is the number of parameters +λ<span class="su">1</span>/λ, λ<span class="su">2</span>/λ, ... entering linearly in the equation of the series. +The series is <i>complete</i> 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 <i>order</i>, being in the number of variable intersections of two +curves of the system, the <i>dimension</i>, being the number of linear +parameters λ<span class="su">1</span>/λ, λ<span class="su">2</span>/λ, ... in the equation for the system, and the +<i>deficiency</i> 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 <i>characteristic</i> 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<span class="su">g</span> − p<span class="su">a</span> 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<span class="su">g</span> − p<span class="su">a</span> than the +dimension of the linear system. The extra p = p<span class="su">g</span> − p<span class="su">a</span> variable parameters +so entering may be regarded as the independent co-ordinates +of an algebraic construct ƒ(y, x<span class="su">1</span>, ... x<span class="su">p</span>) = 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</p> + +<p class="center">u<span class="su">i</span> = ∫ R<span class="su">1</span>(x, y) dx<span class="su">1</span> + ... + R<span class="su">p</span>(x, y) dx<span class="su">p</span>,</p> + +<p class="noind">wherein R<span class="su">i</span>(x, y) denotes a rational function of x<span class="su">1</span>, ... x<span class="su">p</span> and y. +The original surface has correspondingly p integrals of the form +∫(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<span class="su">g</span> − p<span class="su">a</span> 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<span class="su">g</span> − p<span class="su">a</span> = 0 there exists no algebraic system of curves not +contained in a linear system.</p> + +<p>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<span class="su">g</span> − p<span class="su">a</span> = 0 the +co-ordinates of the points are rational functions of two parameters; +it is necessary that p<span class="su">g</span> − p<span class="su">a</span> = 0, but this is not sufficient. For surfaces, +beside the p<span class="su">g</span> 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<span class="su">k</span>. It can then be stated that a sufficient +condition for a surface to be rational consists of the two conditions +p<span class="su">a</span> = 0, P<span class="su">2</span> = 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, +<span class="pagenum"><a name="page329" id="page329"></a>329</span> +of which the co-ordinates are meromorphic functions of two variables +of the simplest kind, with four sets of periods, is characterized by +p<span class="su">g</span> = 1, p<span class="su">a</span> = −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.</p> + +<p><span class="sc">Bibliography.</span>—The learner will find a lucid introduction to the +theory in E. Goursat, <i>Cours d’analyse mathématique</i>, t. ii. (Paris, +1905), or, with much greater detail, in A.R. Forsyth, <i>Theory of +Functions of a Complex Variable</i> (2nd ed., Cambridge, 1900); for +logical rigour in the more difficult theorems, he should consult +W.F. Osgood, <i>Lehrbuch der Functionentheorie</i>, 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, <i>The Theory of Sets of Points</i> (Cambridge, +1906), chs. viii.-xiii., and C. Jordan, <i>Cours d’analyse</i>, t. i. (Paris, +1893), chs. i., ii.; a comprehensive account of the <i>Theory of Functions +of Real Variables</i> 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, <i>Introduction to the Theory +of Analytic Functions</i> (London, 1898), an elementary treatise; +for the theory of the convergence of series there is also T.J. I’A. +Bromwich, <i>An Introduction to the Theory of Infinite Series</i> (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 <i>Acta mathematica</i>; earlier expositions of the theory +of functions on the basis of power series are in C. Méray, <i>Leçons +nouvelles sur l’analyse infinitésimale</i> (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, <i>Traité d’analyse</i>, t. ii. +(Paris, 1893). For elliptic functions there is an introductory book, +P. Appell and E. Lacour, <i>Principes de la théorie des fonctions elliptiques +et applications</i> (Paris, 1897), beside the treatises of G.H. Halphen, +<i>Traité des fonctions elliptiques et de leurs applications</i> (three parts, +Paris, 1886 ff.), and J. Tannery et J. Molk, <i>Éléments de la théorie +des fonctions elliptiques</i> (Paris, 1893 ff.); a book, A.G. Greenhill, +<i>The Applications of Elliptic Functions</i> (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, +<i>Theorie der elliptischen Modulfunctionen</i> (Leipzig, 1890); see also +the most interesting smaller volume, F. Klein, <i>Über das Ikosaeder</i> +(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, <i>Théorie des fonctions algébriques +et de leurs intégrales</i>; for Abelian functions see also H. Stahl, <i>Theorie +der Abel’schen Functionen</i> (Leipzig, 1896), and H.F. Baker, <i>An +Introduction to the Theory of Multiply Periodic Functions</i> (Cambridge, +1907), and H.F. Baker, <i>Abel’s Theorem and the Allied Theory, including +the Theory of the Theta Functions</i> (Cambridge, 1897); for +theta functions of one variable a standard work is C.G. Jacobi, +<i>Fundamenta nova, &c.</i> (Königsberg, 1828); for the general theory +of theta functions, consult W. Wirtinger, <i>Untersuchungen über Theta-Functionen</i> +(Leipzig, 1895). For a history of the theory of algebraic +functions consult A. Brill and M. Noether, <i>Die Entwicklung der +Theorie der algebraischen Functionen in älterer und neuerer Zeit, +Bericht der deutschen Mathematiker-Vereinigung</i> (1894); and for a +special theory of algebraic functions, K. Hensel and G. Landsberg, +<i>Theorie der algebraischen Function u.s.w.</i> (Leipzig, 1902). The +student will, of course, consult also Riemann’s and Weierstrass’s +<i>Ges. Werke</i>. For the applications to geometry in general an important +contribution, of permanent value, is E. Picard and G. Simart, +<i>Théorie des fonctions algébriques de deux variables indépendantes</i> +(Paris, 1897-1906). This work contains, as Note v. t. ii. p. 485, a +valuable summary by MM. Castelnuovo and Enriques, <i>Sur quelques +résultats nouveaux dans la théorie des surfaces algébriques</i>, 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.</p> + +<p>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 <i>Encyclopaedia of Mathematics</i>, and the Royal Society’s +<i>Reference Catalogue of Current Scientific Literature, Pure Mathematics</i>, +published yearly, should also be consulted.</p> +</div> +<div class="author">(H. F. Ba.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1ga" id="ft1ga" href="#fa1ga"><span class="fn">1</span></a> The word “function” (from Lat. <i>fungi</i>, 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 <i>funzione</i>, 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.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUNDY, BAY OF,<a name="ar71" id="ar71"></a></span> 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. <i>c.</i> 1628) in 1604 and +named by him La Baye Française.</p> + + +<hr class="art" /> +<p><span class="bold">FUNERAL RITES,<a name="ar72" id="ar72"></a></span> the ceremonies associated with different +methods of disposing of the dead. (See also <span class="sc"><a href="#artlinks">Burial and Burial +Acts</a></span>; <span class="sc"><a href="#artlinks">Cemetery</a></span>; and <span class="sc"><a href="#artlinks">Cremation</a></span>.) 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.</p> + +<div class="condensed"> +<p>The main ideas, then, underlying funeral rites may roughly be +enumerated as follows:</p> + +<p>1. The pollution or taboo attaching to a corpse.</p> + +<p>2. Mourning.</p> + +<p>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.</p> + +<p>4. Communion with the dead in a funeral feast and otherwise.</p> + +<p>5. Sacrifice for the dead and expiation of their sins.</p> + +<p>6. Death witchery.</p> + +<p>7. Protection of the dead from ghouls.</p> + +<p>8. Fear of ghosts.</p> +</div> + +<p>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” (<i>Zend Avesta</i> in <i>Sacred +Books of the East</i>, pt. i. p. 120), and the paralysis is inflicted by +the innumerable <i>drugs</i> or evil spirits which invest a corpse. +Fire and earth, being alike creations of the good and pure god +<span class="pagenum"><a name="page330" id="page330"></a>330</span> +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 <i>aspergillum</i>; 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 (<i>The Melanesians</i>, 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 (<i>Religion des Veda</i>, 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, <i>De +opinionibus quorundam Graecorum</i>). 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.</p> + +<p>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 +<i>lugubria</i> 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 (<i>Golden Bough</i>, 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.</p> + +<p>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 <i>Alhuemapu</i>, 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.</p> + +<p>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.</p> + +<p>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 <i>ka</i> 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, +<i>Cities and Cemeteries of Etruria</i>, vol i. p. lxx.).</p> + +<p>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 <i>manes</i> of objects is +liberated by such fracture or inversion and so passes into the +<span class="pagenum"><a name="page331" id="page331"></a>331</span> +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 <i>simulacra</i> +for consumption at funerals is still an important industry +in Chinese cities. The ancient Egyptians, assured that a man’s +<i>ka</i> 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 (<i>q.v.</i>), +and the custom was already of a hoary antiquity in 3200 <span class="scs">B.C.</span>, +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.</p> + +<p>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, <i>Prim. Cult.</i> +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 (<i>Hist. Eccl.</i> 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, <i>e.g.</i> 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.</p> + +<p>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 +(<i>Egyptian Ideas of the Future Life</i>, 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 <span class="scs">A.D.</span> 386 under the altar of his new basilica in Milan, +were <i>mirae magnitudinis ut prisca aetas ferebat</i>, and were also +coloured red. He imagined the red to be the remains of the +martyrs’ blood! <i>Hic sanguis clamat coloris indicio.</i> 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.</p> + +<p>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.”<a name="fa1g" id="fa1g" href="#ft1g"><span class="sp">1</span></a> +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 +(<i>Religion of the Semites</i>, 336) that the world-wide customs of +<span class="pagenum"><a name="page332" id="page332"></a>332</span> +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.</p> + +<p>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.”<a name="fa2g" id="fa2g" href="#ft2g"><span class="sp">2</span></a> 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.”<a name="fa3g" id="fa3g" href="#ft3g"><span class="sp">3</span></a></p> + +<p>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:</p> + +<div class="condensed"> +<p>“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 <i>let not any filthy +and unclean devil dare to approach him, such as assail the body and +souls of the heathen</i>, who possess not the birth of the holy font, and +have not the dread seal laid upon their graves.”</p> +</div> + +<p>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 <i>De opinionibus quorundam Graecorum</i> +(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 <i>Encaenia</i> 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.</p> + +<p>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.</p> + +<p>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 <i>quâ</i> 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, <i>The Melanesians</i>, +p. 264), or the skull and jaw bone are kept and “are +called <i>mangite</i>, which are <i>saka</i>, hot with spiritual power, and by +means of which the help of the <i>lio’a</i>, the powerful ghost of the +man whose relics these are, can be obtained” (<i>ibid.</i> 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.</p> + +<p>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 <i>Wah! Wah!</i> 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.” +<span class="pagenum"><a name="page333" id="page333"></a>333</span> +(Spencer and Gillen, <i>Native Tribes of Central Australia</i>, 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.</p> + +<div class="condensed"> +<p><span class="sc">Authorities.</span>—B. Spencer and F.J. Gillen, <i>The Native Tribes +of Central Australia</i> (London, 1899); F.B. Jevons, <i>Introduction to +History of Religion</i> (London, 1896); E.S. Hartland, <i>The Legend of +Perseus</i>, vol. ii.; J.G. Frazer, <i>The Golden Bough</i> (London, 1900); +L.W. Faraday, “Custom and Belief in the Icelandic Sagas,” in +<i>Folk-lore</i>, vol. xvii. No. 4; E.B. Tylor, <i>Primitive Culture</i> (London, +1903); E.A. W. Budge, <i>The Mummy</i> (Cambridge, 1893); C. Royer, +“Les Rites funéraires aux époques préhistoriques,” <i>Revue d’anthropologie</i> +(1876); Forrer, <i>Über die Totenbestattung bei den Pfahlbauern</i> +(Ausland, 1885); J. Lubbock, <i>Origin of Civilization</i> (London, 1875) +and <i>Prehistoric Times</i> (London, 1865); L.A. Muratori, “De antiquis +Christianorum sepulchris,” <i>Anecd. Graeca</i> (Padua, 1709); Onaphr. +Panvinius, <i>De ritu sepeliendi mortuos apua veteres Christianos</i>, reprinted +in Volbeding’s <i>Thesaurus</i> (Leipzig, 1841).</p> +</div> +<div class="author">(F. C. C.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1g" id="ft1g" href="#fa1g"><span class="fn">1</span></a> E.S. Hartland, <i>Legend of Perseus</i> (1895), ii. 278.</p> + +<p><a name="ft2g" id="ft2g" href="#fa2g"><span class="fn">2</span></a> Mary Kingsley, <i>West African Studies</i> (1901), p. 178.</p> + +<p><a name="ft3g" id="ft3g" href="#fa3g"><span class="fn">3</span></a> B. Spencer and F.J. Gillen, <i>The Native Tribes of Central Australia</i> +(1899), p. 48.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUNGI<a name="ar73" id="ar73"></a></span> (pl. of Lat. <i>fungus</i>, 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 (<i>chlamydospores</i>) 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 (<i>Penicillium</i>, <i>Aspergillus</i>) can +utilize almost any organic food-materials, other fungi are more +restricted in their choice—<i>e.g.</i> 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.</p> + +<p>The number of species in 1889 was estimated by Saccardo at +about 32,000, but of these 8500 were so-called <i>Fungi imperfecti</i>—<i>i.e.</i> +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 <i>Puccinia malvacearum</i>, <i>Peronospora +viticola</i>, <i>Hemileia vastatrix</i>, &c. Some fungi—<i>e.g.</i> 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, <i>Pezizae</i>, &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.</p> + +<p>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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:600px; height:489px" src="images/img333.jpg" alt="" /></td></tr> +<tr><td class="tcl f90"><span class="sc">Fig. 1.</span>—1, <i>Peronospora parasitica</i> (De Bary). Mycelium with +haustoria (<i>h</i>); 2, <i>Erysiphe</i>; A and B, mycelium (<i>m</i>), with haustoria +(<i>h</i>). (After De Bary.)</td></tr></table> + +<div class="condensed"> +<p class="pt2"><i>Organs.</i>—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 <i>clamp-connexion</i>. +Many parasitic hyphae put out minute lateral branches, +which pierce the cell-wall of the host and form a peg-like (<i>Trichosphaeria</i>), +sessile (<i>Cystopus</i>), or stalked (<i>Hemileia</i>), knot-like, or a +more or less branched (<i>Peronospora</i>) or coiled (<i>Protomyces</i>) haustorium. +In <i>Rhizopus</i> 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 (<i>appressorium</i>), the walls of which soften and gum +<span class="pagenum"><a name="page334" id="page334"></a>334</span> +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 (<i>Erysiphe</i>), or the swollen end of any hypha +which comes in contact with the surface of the host (<i>Piptocephalis</i>, +<i>Syncephalis</i>), haustoria piercing in each case the cell-wall below. +In <i>Botrytis</i> the appressoria assume the form of dense tassels of short +branches. In <i>Arthrobotrys</i> side-branches of the mycelium sling themselves +around the host (<i>Tylenchus</i>) much as tendrils round a support.</p> + +<p>Many fungi (<i>Phallus</i>, <i>Agaricus</i>, <i>Fumago</i>, &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 (<i>Armillaria mellea</i>).</p> + +<p><i>Sclerotia.</i>—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 (<i>Phallus</i>, mushroom, &c.), or flattened and anastomosing +ribbons and plates, often containing several kinds of hyphae +(<i>Merulius lacrymans</i>). 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 +(<i>Armillaria mellea</i>), 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 (<i>Peziza</i>, +<i>Coprinus</i>) to that of a man’s fist or head, and weighing 10 to 25 ℔ +or more (<i>Polyporus Mylittae</i>, <i>P. tumulosus</i>, <i>Lentinus Woermanni</i>, +<i>P. Sapurema</i>, &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 +(<i>Polyporus sacer</i>, <i>P. rhinoceros</i>, <i>Lentinus Woermanni</i>). 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.</p> + +<p><i>Tissue Differentiations.</i>—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 (<i>Phycomycetes</i>), 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.</p> + +<p>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, <i>Xylaria</i>, &c.). Fibre-like hyphae with +the lumen almost obliterated by the thick walls occur in mycelial +cords (<i>Merulius</i>). Latex-tubes abound in the tissues of <i>Lactarius</i>, +<i>Stereum</i>, <i>Mycena</i>, <i>Fistulina</i>, 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 <i>Merulius lacrymans</i> Hartig has observed +thin-walled hyphae with large lumina, the septa of which are perforated +like those of sieve-tubes.</p> + +<p>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 (<i>Trametes</i>, <i>Daedalea</i>), or swelling +to a gelatinous mucilage (<i>Tremella</i>, <i>Gymnosporangium</i>), while cutinization +of the outer layers is common. One of the most striking +alterations of cell-walls is that termed <i>carbonization</i>, in which the +substance gradually turns black, hard and brittle, as if charred—<i>e.g.</i> +<i>Xylaria</i>, <i>Ustulina</i>, 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 <i>pits</i> (<i>e.g.</i> <i>Uredospores</i>, septa of +Basidiomycetes), <i>spirals</i>, <i>reticulations</i>, <i>rings</i>, &c. (capillitium fibres +of <i>Podaxon</i>, <i>Calostoma</i>, <i>Battarrea</i>), 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.</p> + +<p><i>Cell-Contents.</i>—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 (<i>Uredineae</i>) +crystals of calcium oxalate (<i>Phallus</i>, <i>Russula</i>), proteid crystals +(<i>Mucor</i>, <i>Pilobolus</i>, &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, <i>Mucor</i>, +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 (<i>Amanita</i>), ergotin +(<i>Claviceps</i>), &c.</p> + +<p>Among the enzymes already extracted from fungi are <i>invertases</i> +(yeasts, moulds, &c.), which split cane-sugar and other complex +sugars with hydrolysis into simpler sugars such as dextrose and +levulose; <i>diastases</i>, which convert starches into sugars (<i>Aspergillus</i>, +&c.); <i>cytases</i>, which dissolve cellulose similarly (<i>Botrytis</i>, &c.); +<i>peptases</i>, using the term as a general one for all enzymes which +convert proteids into peptones and other bodies (<i>Penicillium</i>, &c.); +lipases, which break up fatty oils (<i>Empusa</i>, <i>Phycomyces</i>, &c.); +oxydases, which bring about the oxidations and changes of colour +observed in <i>Boletus</i>, and <i>zymase</i>, 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 <i>Aspergillus</i> 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 <i>Penicillium</i> +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 +(<i>Peziza aeruginosa</i>). 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 <i>Corticium +sanguineum</i>, blood-red; <i>Elaphomyces Leveillei</i>, yellow-green; +<i>Chlorosplenium aeruginosum</i>, verdigris green; and the <i>Dematei</i>, +brown or black.</p> + +<p><i>Nuclei.</i>—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 (<i>Aspergillus</i>) in each segment, or only two (<i>Exoascus</i>) +or one (<i>Erysiphe</i>) 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 μ in <i>Phycomyces</i>), 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 +<span class="pagenum"><a name="page335" id="page335"></a>335</span> +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.</p> + +<p><i>Spores.</i>—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 <i>Coniothecium</i>. 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.</p> + +<table class="flt" style="float: right; width: 330px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:275px; height:600px" src="images/img335a.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig.</span> 2.—<i>Peronospora parasitica</i> +(De Bary). Conidiophore +with conidia.</td></tr></table> + +<p>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, <i>e.g.</i> 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, <i>e.g.</i> 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, +<i>e.g.</i> 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.</p> + +<p><i>Sporophores.</i>—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.</p> + +<p>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 (<i>Erysiphe</i>); when the primary branch has +thus formed a basipetal series, branches may arise from below and +again repeat this process, thus forming a tuft (<i>Penicillium</i>). Or the +primary hypha may first swell at its apex, and put forth a series of +short peg-like branches (<i>sterigmata</i>) from the increased surface thus +provided, each of which develops a similar basipetal chain of conidia +(<i>Aspergillus</i>), and various combinations of these processes result in +the development of numerous varieties of exquisitely branched +sporophores of this type (<i>Botrytis</i>, <i>Botryosporium</i>, <i>Verticillium</i>, &c.).</p> + +<table class="pic" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:600px; height:566px" src="images/img335b.jpg" alt="" /></td></tr> +<tr><td class="caption" colspan="2"><span class="sc">Fig. 3.</span>—<i>Cystopus candidus</i>.</td></tr> + +<tr><td class="f90" style="width: 50%; vertical-align: top;"> +<p>A. <i>a</i>, Conidia.</p> +<p>  <i>b</i>, Conidiophores.</p> +<p>  <i>c</i>, Conidium emitting zoospores.</p> +<p>  <i>d</i>, Free zoospore.</p> +<p>B.<i>og</i>, Oogonium.</p></td> + +<td class="f90" style="width: 50%; vertical-align: top;"> +<p>  <i>os</i>, Oosphere.</p> +<p>  <i>an</i>, Antheridium.</p> +<p>C. Formation of zoospores by oospores.</p> +<p>  <i>z</i>, Free zoospores.</p> +<p>   (After De Bary.)</p></td></tr></table> + +<p class="pt2">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 <i>Penicillium</i>, +&c., branches put forth lower down may repeat the process (<i>Hormodendron</i>). +In all these cases we may speak of simple conidiophores. +The simple sporophore does not necessarily terminate in conidia, +however. In <i>Mucor</i>, 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 +(<i>Thamnidium</i>). Such may be termed sporangiophores. In <i>Sporodinia</i> +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.</p> + +<p>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 <i>Puccinia</i>, <i>Uromyces</i>, +&c., or of basidia in <i>Exobasidium</i>, <i>Corticium</i>, &c., or of asci in +<i>Exoascus</i>, <i>Ascocorticium</i>, &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 <i>Stilbum</i>, <i>Coremium</i>, &c. These lead +<span class="pagenum"><a name="page336" id="page336"></a>336</span> +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 +<i>Nectria</i> and many Pyrenomycetes, the clavate “receptacle” of +<i>Clavaria</i>, &c., passing into the complex forms met with in <i>Sparassis</i>, +<i>Xylaria</i>, <i>Polyporei</i>, and <i>Agaricini</i>, &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.</p> + +<p>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 (<i>e.g.</i> +<i>Onygena</i>). 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—<i>e.g.</i> <i>Solenia</i>, <i>Cyphella</i>—and even simpler +cases are met with in <i>Mortierella</i>, where the zygospore is invested by +the overgrowth of a dense mat of closely branching hyphae, and in +<i>Gymnoascus</i>, where a loose mat of similarly barren hyphae covers +in the tufts of asci as they develop.</p> + +<p>In such examples as the above we may regard the hymenium +(<i>Solenia</i>, <i>Cyphella</i>), 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 <i>Penicillium</i>, <i>Eurotium</i>, <i>Erysiphe</i>, +&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 (<i>Hypomyces</i>, <i>Nectria</i>) the +perithecia arise on an already mature stroma, while yet more numerous +examples can be given (<i>Poronia</i>, <i>Hypoxylon</i>, <i>Claviceps</i>, &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—<i>e.g.</i> <i>Fumago</i>—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 <i>Cicinnobolus</i>, <i>Pleospora</i>, +<i>Cucurbitaria</i>, <i>Leptosphaeria</i> and others. In other cases (<i>Diplodia</i>, +<i>Aecidium</i>, &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—<i>e.g.</i> puff-balls, earth-stars and various <i>Phalloideae</i>.</p> + +<p><i>Spore-Distribution.</i>—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 (<i>Pythium</i>, <i>Peronospora</i>, +<i>Saprolegnia</i>) we often find spores held under tension in sporangia +(<i>Pilobolus</i>) or in asci (<i>Peziza</i>) 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 (<i>e.g.</i> <i>Empusa</i>), and similarly with <i>basidiospores</i> (<i>Coprinus</i>, +<i>Agaricus</i>, &c.). A more complicated case is illustrated by <i>Sphaerobolus</i>, +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 <i>Mucor</i>, conidia of <i>Empusa</i>, &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 (<i>Sclerotinia</i>) +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 (<i>Uredineae</i>), conidia (<i>Claviceps</i>), basidiospores (<i>Phallus</i>, +<i>Coprinus</i>), &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 <i>Sclerotinia</i> to the stigma of <i>Vaccinium</i> and infect +it, and that flies carry away the foetid spores of <i>Phallus</i>, just as +pollen is dispersed by such insects. Whether the strong odour of +trimethylamine evolved by the spores of <i>Tilletia</i> attracts insects is +not known.</p> + +<p>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 <i>Boletus</i>, <i>Polyporus</i>—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.</p> +</div> + +<p><i>Classification.</i>—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.</p> + +<p>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.</p> + +<p>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:</p> + +<div class="condensed"> +<p><span class="sc">A. Phycomycetes.</span> Alga-like fungi with unicellular thallus +and well-marked sexual organs.</p> + +<div class="list"> +<p><span class="sc">Class I.</span>—Oomycetes. Mycelium usually well developed, but +sometimes poor or absent. Sexual reproduction by oogonia +and antheridia; asexual reproduction by zoospores or +conidia.</p> +</div> + +<div class="list1"> +<p>1. Monoblepharidineae. Mycelium present, antheridia with +antherozoids, oogonium with single oosphere: Monoblepharidaceae.</p> + +<p>2. Peronosporineae. Mycelium present; antheridia but no +antherozoids; oogonia with one or more oospheres: +Peronosporaceae, Saprolegniaceae.</p> + +<p>3. Chytridineae. Mycelium poorly developed or absent; +oogonia and antheridia (without antherozoids) known in +some cases; zoospores common: Chytridiaceae. Ancylistaceae.</p> +</div> + +<p><span class="pagenum"><a name="page337" id="page337"></a>337</span></p> + +<div class="list"> +<p><span class="sc">Class II.</span>—Zygomycetes. Mycelium well developed; sexual reproduction +by zygospores; asexual reproduction by sporangia +and conidia.</p> +</div> + +<div class="list1"> +<p>1. Mucorineae. Sexual reproduction as above, asexual by +sporangia or conidia or both: Mucoraceae. Mortierellaceae, +Chaetocladiaceae, Piptocephalidaceae.</p> + +<p>2. Entomophthorineae. Sexual reproduction typical but +with sometimes inequality of the fusing gametes (gametangia ?): +Entomophthoraceae.</p> +</div> + +<p>B. <span class="sc">Higher Fungi.</span> Fungi with segmental thallus; sexual +reproduction sometimes with typical antheridia and oogonia +(ascogonia) but usually much reduced.</p> + +<div class="list"> +<p><span class="sc">Class I.</span>—Ustilaginales. Forms with septate thallus, and reproduction +by chlamydospores which on germination produce +sporidia; sexuality doubtful.</p> + +<p><span class="sc">Class II.</span>—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.</p> +</div> + +<div class="list1"> +<p>Exoascineae, Saccharomycetineae, Perisporinea, Discomycetes, +Pyrenomycetes, Tuberineae, Laboulbeniineae.</p> +</div> + +<div class="list"> +<p><span class="sc">Class III.</span>—Basidiales. Thallus septate. Conidia (basidiospores) +borne in fours on a special conidiophore, the basidium. +Sexual reproduction always much reduced.</p> +</div> + +<div class="list1"> +<p>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).</p> + +<p>2. Basidiomycetes. Life-history always very simple, no well-marked +alternation of generations; basidium borne +directly on the mycelium.</p> +</div> + +<div class="list2"> +<p>(A) Protobasidiomycetes. Basidia septate. +Auriculariaceae, Pilacreaceae, Tremellinaceae.</p> + +<p>(B) Autobasidiomycetes. Basidia non-septate. +Hymenomycetes, Gasteromycetes.</p> +</div></div> + +<p>A. <span class="sc">Phycomycetes.</span>—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.</p> + +<div class="condensed"> +<p><i>Monoblepharidaceae</i> consists of a very small group of aquatic +forms living on fallen twigs in ponds and ditches. Only one genus, +<i>Monoblepharis</i>, can certainly be placed here, though a somewhat +similar genus, <i>Myrioblepharis</i>, with a peculiar multiciliate zoospore +like that of Vaucheria, is provisionally placed in the same group. +<i>Monoblepharis</i> 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. <i>Monoblepharis</i> +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 <i>Oedogonium</i> among the algae is very striking, as is also that of +<i>Myrioblepharis</i> and <i>Vaucheria</i>.</p> + +<p><i>Peronosporaceae</i> are a group of endophytic parasites—about 100 +species—of great importance as comprising the agents of “damping +off” disease (<i>Pythium</i>), vine-mildew (<i>Plasmopara</i>), potato disease +(Phytophthora), onion-mildew (<i>Peronospora</i>). <i>Pythium</i> 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. <i>Pythium</i> 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 (<i>Cystopus candidus</i>, <i>Phytophthora infestans</i>) as in <i>Pythium</i> +described above; in other cases the direct conidial germination is +characteristic of genera—<i>e.g.</i> <i>Peronospora</i>; while others emit +zoospores—<i>e.g.</i> <i>Plasmopara</i>, &c. In <i>Cystopus</i> (<i>Albugo</i>) 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 <i>Pythium</i>. +In <i>Cystopus Bliti</i> 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 <i>Cystopus Bliti</i> on the one hand and <i>Pythium de +Baryanum</i> 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.</p> + +<table class="pic" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:500px; height:482px" src="images/img337a.jpg" alt="" /></td></tr> +<tr><td class="tcl f80" colspan="2">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</td></tr> +<tr><td class="caption" colspan="2"><span class="sc">Fig. 4.</span>—Fertilization of the Peronosporeae. After Wager.</td></tr> + +<tr><td class="f90" style="width: 50%; vertical-align: top;"> +<p>1, <i>Peronospora parasitica</i>. Young +multinucleate oogonium (<i>og</i>) +and antheridium (<i>an</i>).</p> + +<p>2, <i>Albugo candida</i>. Oogonium +with the central uninucleate +oosphere and the fertilizing +tube (<i>a</i>) of the antheridium +which introduces the male +nucleus.</p></td> + +<td class="f90" style="width: 50%; vertical-align: top;"> +<p>3, The same. Fertilized egg-cell +(<i>o</i>) surrounded by the +periplasm (<i>p</i>).</p></td></tr></table> + +<table class="pic pt2" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:500px; height:615px" src="images/img337b.jpg" alt="" /></td></tr> +<tr><td class="caption" colspan="2"><span class="sc">Fig. 5.</span>—<i>Phytophthora infestans</i>. Fungus of Potato Disease.</td></tr> + +<tr><td class="f90" style="width: 50%; vertical-align: top;"> +<p>A, B, Section of Leaf of Potato +with sporangiophores of <i>Phytophthora +infestans</i> passing +through the stomata D, on +the under surface of the leaf.</p> + +<p>E, Sporangia.</p> + +<p>F, G, H, J, Further development +of the sporangia.</p></td> + +<td class="f90" style="width: 50%; vertical-align: top;"> +<p>K, Germination of the zoospores +formed in the sporangia.</p> + +<p>L, M, N, Fertilization of the +oogonium and development of +the oospore in <i>Peronospora</i>.</p></td></tr></table> + +<p class="pt2"><i>Saprolegniaceae</i> are aquatic forms found growing usually on dead +insects lying in water but occasionally on living fish (<i>e.g.</i> the salmon +disease associated with <i>Saprolegnia ferax</i>). The chief genera are +<span class="pagenum"><a name="page338" id="page338"></a>338</span> +<i>Saprolegnia, Achlya, Pythiopsis, Dictyuchus, Aplanes.</i> 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 <i>Saprolegnia</i> +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 <i>Saprolegnia</i>-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.</p> + +<p><i>Chytridineae.</i>—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 (<i>Olpidium</i>), or merely sends in +delicate protoplasmic filaments (<i>Rhizophydium</i>) 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 (<i>Chytridium</i>). In some cases resting spores are formed inside +the host (<i>Chytridium</i>), and give rise to zoosporangia on germination. +In a few species a sexual process is described, consisting in +the conjugation of similar cells (<i>Zygochytrium</i>) or the union of +two dissimilar ones (<i>Polyphagus</i>). In the development of distinct +antheridial and oogonial cells the allied Ancylistineae show +close alliances to <i>Pythium</i> and the Oomycetes. On the other hand, +the uniciliate zoospores of <i>Polyphagus</i> 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—<i>e.g.</i> <i>Olpidium Brassicae</i>; others discoloured spots +and even tumour-like swellings—<i>e.g.</i> <i>Synchytium Scabiosae</i>, <i>S. +Succisae</i>, <i>Urophlyctis</i>, &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.</p> + +<p>The <i>Zygomycetes</i>, 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 <i>Mucor</i> containing numerous +endospores, passing to cases where, as in <i>Thamnidium</i>, these are +accompanied with more numerous small sporangia (sporangioles) +containing few spores, and thence to <i>Chaetocladium</i> and <i>Piptocephalis</i>, +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>i.e.</i> 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.</p> + +<table class="flt" style="float: right; width: 400px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:350px; height:578px" src="images/img338.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Strasburger’s <i>Lehrbuch der Botanik</i>, by +permission of Gustav Fischer.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 6.</span>—<i>Mucor Mucedo.</i> Different +stages in the formation and germination +of the zygospore. (After Brefeld, +1-4. 5 from v. Tavel, <i>Pilze</i>.)</td></tr> + +<tr><td class="caption1"> +<p>1, Two conjugating branches in contact.</p> +<p>2, Septation of the conjugating cells (<i>a</i>) + from the suspensors (<i>b</i>).</p> +<p>3, More advanced stage, the conjugating + cells (<i>a</i>) are still distinct from + one another; the warty thickenings + of their walls have commenced to + form.</p> +<p>4, Ripe zygospore (<i>b</i>) between the suspensors +(<i>a</i>).</p> +<p>5, Germinating zygospore with a germ-tube + bearing a sporangium.</p></td></tr></table> + +<p>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 +<i>Pilobolus</i>, multicellular in <i>Sporodinia</i>. 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).</p> + +<p>Blakeslee has lately +made some very important +observations of the +Zygomycetes. It is well +known that while in some +forms, <i>e.g.</i> <i>Spordinia</i>, +zygospores are easily obtained, +in others, <i>e.g.</i> most +species of <i>Mucor</i>, 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 <i>same</i> mycelium and +so can be produced by the +growth from a single spore; +this group includes <i>Spordinia +grandis</i>, <i>Spinellus +fusiger</i>, some species of +<i>Mucor</i>, &c. The majority +of forms, however, fall +into the heterothallic +group, in which the association +of branches from +two mycelia <i>different in +nature</i> 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 <i>any</i> 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.</p> + +<p>The classification of the Mucorini depends on the prevalence and +characters of the conidia, and of the sporangia and zygospores—<i>e.g.</i> +the presence or absence of a columella in the former, the formation +of an investment round the latter. Most genera are saprophytes, +but some—<i>Chaetocladium</i>, <i>Piptocephalis</i>—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 <i>Mucor</i>, 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.</p> + +<p>The <i>Entomophthoraceae</i> contain three genera, <i>Empusa</i>, <i>Entomophthora</i> +and <i>Basidiobolus</i>. The two first genera consist of forms +which are parasitic on insects. <i>Empusa Muscae</i> 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. <i>B. ranarum</i> 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 <i>Pilobolus</i>.</p> +</div> + +<p><span class="pagenum"><a name="page339" id="page339"></a>339</span></p> + +<p>B. <span class="sc">Higher Fungi.</span>—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 <i>Eumycetes</i> 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 <i>Ustilaginales</i>, +of doubtful position, and the two very sharply marked +groups <i>Basidiales</i> and <i>Ascomycetes</i>.</p> + +<table class="flt" style="float: right; width: 350px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:300px; height:371px" src="images/img339a.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Vine’s <i>Students’ Text Book of +Botany</i>, by permission of Swan Sonnenschein +& Co.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 7.</span>—Germinating resting-gonidia. +A, of <i>Ustilago +receptaculorum</i>; B, of <i>Tilletia +Caries</i>.</td></tr> +<tr><td class="caption1"> +<p><i>sp</i>, The gonidium.</p> +<p><i>pm</i>, The promycelium.</p> +<p><i>d</i>, The sporidia: in B the + sporidia have coalesced + in pairs at <i>v</i>.</p></td></tr></table> + +<div class="condensed"> +<p>I. <i>Ustilaginales.</i>—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 +<i>Ustilago</i> and its allies this is transversely septate, and bears lateral +conidia (sporidia); in <i>Tilletia</i> and its allies non-septate, and bears +a terminal tuft of conidia (sporidia) (fig. 7). Brefeld regarded the +promycelium as a kind of <i>basidium</i>, bearing lateral or terminal +conidia (comparable to <i>basidiospores</i>), +but since the number of +basidiospores is not fixed, and the +basidium has not yet assumed very +definite morphological characters, +Brefeld termed the group <i>Hemibasidii</i>, +and regarded them as a half-way +stage in the evolution of the +true Basidiomycetes from Phycomycetes, +the <i>Tilletia</i> type leading +to the true basidium (Autobasidium), +the <i>Ustilago</i> 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 <i>Ustilaginales</i>. +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.</p> + +<p>II. <i>Ascomycetes.</i>—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 <i>ascus</i>. 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 <i>epiplasm</i> 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.</p> + +<table class="flt" style="float: right; width: 280px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:230px; height:313px" src="images/img339b.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Strasburger’s <i>Lehrbuch der +Botanik</i>, by permission of Gustav +Fischer.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 8.</span>—Development of the +Ascus.</td></tr> +<tr><td class="caption1"> +<p><i>A-C</i>, <i>Pyronema confluens</i>. +(After Harper.)</p> + +<p><i>D</i>, Young ascus of <i>Boudiera</i> +with eight spores. +(After Claussen.)</p></td></tr></table> + +<p>In some forms we find definite male and female sexual organs +(<i>Sphaerotheca</i>, <i>Pyronema</i>, &c.), in others the antheridium is abortive +or absent, but the ascogonium (oogonium) is still present and the +female nuclei fuse in pairs (<i>Lachnea +stercorea</i>, <i>Humaria granulata</i>, <i>Ascobolus +furfuraceus</i>); while in other +forms ascogonium and antheridium +are both absent and fusion occurs +between vegetative nuclei (<i>Humaria +rutilans</i>, and probably the majority +of other forms). In other cases the +sexual fusion is apparently absent +altogether, as in <i>Exoascus</i>. In the first +case (fig. 9) we have a true sexual +process, while in the second and third +cases we have a <i>reduced</i> 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.</p> + +<p>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 <i>Ascoidea</i>) or true Ascomycetes (like <i>Thelebolus</i>). <i>Eremascus</i> and +<i>Dipodascus</i>, which are often placed among the Hemiasci, possibly do +not belong to the Ascomycetes series at all.</p> + +<table class="pic" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:550px; height:422px" src="images/img339c.jpg" alt="" /></td></tr> +<tr><td class="tcl f80" colspan="2">From Strasburger’s <i>Lehrbuch der Botanik</i>, by permission of Gustav Fischer.</td></tr> +<tr><td class="caption" colspan="2">Fig. 9.—<i>Sphaerotheca Castagnei</i>. Fertilization and Development +of the Perithecium. (After Harper.)</td></tr> + +<tr><td class="f90" style="width: 50%; vertical-align: top;"> +<p>1, Oogonium (<i>og</i>) with the antheridial + branch (<i>az</i>) applied to its surface</p> + +<p>2, Separation of antheridium + (<i>an</i>).</p> + +<p>3, Passage of the antheridial + nucleus towards that of the + oogonium.</p></td> + +<td class="f90" style="width: 50%; vertical-align: top;"> +<p>4, Union of the nuclei.</p> + +<p>5, Fertilized oogonium surrounded + by two layers of + hyphae derived from the + stalk-cell (<i>st</i>).</p> + +<p>6, The multicellular ascogonium + derived by division from the + oogonium; the terminal cell + with the two nuclei (<i>as</i>) + gives rise to the ascus.</p></td></tr></table> + +<p class="pt2"><i>Exoascaceae</i> are a small group of doubtful extent here used to +include <i>Exoascus</i>, <i>Taphrina</i>, <i>Ascorticium</i> and <i>Endomyces</i>. 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 <i>Taphrineae</i>, +which include <i>Exoascus</i> and <i>Taphrina</i>, are important parasites—<i>e.g.</i> +pocket-plums and witches’ brooms on birches, &c., are due to +their action (fig. 10). <i>Exoascus</i> and <i>Ascorticium</i> present interesting +parallels to <i>Exobasidium</i> and <i>Corticium</i> among the Basidiomycetes.</p> + +<table class="flt" style="float: right; width: 380px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:330px; height:407px" src="images/img340.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Strasburger’s <i>Lehrbuch der +Botanik</i>, by permission of Gustav Fischer.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 10.</span>—<i>Taphrina Pruni.</i> +Transverse section through the +epidermis of an infected plum. +Four ripe asci, <i>a</i><span class="su">1</span>, <i>a</i><span class="su">2</span>, with eight +spores, <i>a</i><span class="su">3</span>, <i>a</i><span class="su">4</span>, with yeast-like +conidia abstricted from the spores. +After Sadebeck.</td></tr> +<tr><td class="caption1"><p>st, Stalk-cells of the asci.</p> +<p>m, Filaments of the mycelium cut transversely.</p> +<p>cut, Cuticle.</p> +<p>sp, Epidermis.</p></td></tr></table> + +<p><i>Saccharomycetaceae</i> include the well-known yeasts which belong +mainly to the genus <i>Saccharomyces</i>. They are characterized by +their unicellular nature, their power of rapid budding, their capacity +for fermenting various sugars, and their power of forming endogenous +<span class="pagenum"><a name="page340" id="page340"></a>340</span> +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 <i>Saccharomyces</i> +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>i.e.</i> +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.</p> + +<p>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:</p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb" rowspan="2">Species.</td> <td class="tccm allb" colspan="2">Optimum Temperature for</td> <td class="tccm allb" colspan="3">Characters of</td> <td class="tccm allb" rowspan="2">Sugars Fermented and<br />Products, &c.</td></tr> +<tr><td class="tccm allb">Spores.</td> <td class="tccm allb">Veils.</td> <td class="tccm allb">Fermentation.</td> <td class="tccm allb">Cells.</td> <td class="tccm allb">Spores.</td></tr> + +<tr><td class="tcl lb rb"><i>S. cereviseae I</i>.</td> <td class="tcc rb">30°</td> <td class="tcc rb">20°-28°</td> <td class="tcc rb">High</td> <td class="tcl rb">Rounded</td> <td class="tcl rb">Globoid</td> <td class="tclm rb cl" rowspan="3">Inverts maltose and saccharose<br />  and form alcohol 4-6 vol. %.</td></tr> +<tr><td class="tcl lb rb"><i>S. Pastorianus I</i></td> <td class="tcc rb">27°-5°</td> <td class="tcc rb">26°-28°</td> <td class="tcc rb">Low</td> <td class="tcl rb">Rounded</td> <td class="tcl rb">Globoid</td></tr> +<tr><td class="tcl lb rb"><i>S. ellipsoideus</i></td> <td class="tcc rb">25°</td> <td class="tcc rb">33°-34°</td> <td class="tcc rb">Low</td> <td class="tcl rb">Rounded</td> <td class="tcl rb">Globoid</td></tr> +<tr><td class="tcl lb rb"><i>S. anomalus</i></td> <td class="tcc rb">28°-31°</td> <td class="tcc rb">?</td> <td class="tcc rb">High</td> <td class="tcl rb">Elliptical</td> <td class="tcl rb">Hat-shaped</td> <td class="tcl rb">Ditto, and evolves a fragrant ether.</td></tr> +<tr><td class="tcl lb rb"><i>S. Ludwigii</i></td> <td class="tcc rb">30°-31°</td> <td class="tcc rb">?</td> <td class="tcc rb">?</td> <td class="tcl rb">Elongated</td> <td class="tcl rb">Globoid</td> <td class="tcl rb">Will not invert maltose.</td></tr> +<tr><td class="tcl lb rb bb"><i>S. membranaefaciens</i></td> <td class="tcc rb bb">30°</td> <td class="tcc rb bb">?</td> <td class="tcc rb bb">High</td> <td class="tcl rb bb">Elongated</td> <td class="tcl rb bb">Globoid</td> <td class="tcl rb bb">Inverts neither maltose nor saccharose.</td></tr> +</table> + +<p>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 (<i>e.g.</i> <i>Mucor</i>, 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 <i>Endomyces</i>, a hyphal fungus bearing +yeast-like asci, and such a form as <i>Saccharomyces anomalus</i>. 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.</p> + +<p><i>Schizosaccharomyces</i> 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 <i>Spirogyra</i>; 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 <i>Zygosaccharomyces</i> +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.</p> + +<p><i>Cytology.</i>—The study of the nucleus of yeast-cells is rendered +difficult by the presence of other deeply staining granules termed by +Guillermond <i>metachromatic granules</i>. 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 +<i>Schizosaccharomyces</i> and <i>Zygosaccharomyces</i>, 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 +<i>Schizosaccharomyces</i> and <i>Zygosaccharomyces</i> by a loss of sexuality, +the sporangium being formed parthenogenetically without any +nuclear fusion. This suggests a possible relationship to <i>Eremascus</i>, +which can only doubtfully be placed in the Ascomycetes (<i>vide supra</i>).</p> + +<p><i>Carpoascomycetes.</i>—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 <i>Humaria rutilans</i> 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 <i>double</i> reduction process in the ascus. +Miss Fraser in <i>Humaria rutilans</i> finds two reductions: a normal +synaptic reduction in the first nuclear division of the ascus, and a +peculiar reduction division termed <i>brachymeiosis</i> in the third ascus +division.</p> + +<p>Various types of ascocarp are characteristic of the different +divisions of the Carpoascomycetes: the cleistothecium, apothecium +and perithecium.</p> + +<p><span class="pagenum"><a name="page341" id="page341"></a>341</span></p> + +<p><i>Perisporineae.</i>—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 <i>cleistothecium</i> (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. +<i>Sphaerotheca Humuli</i> is the well known hop-mildew, <i>Sphaerotheca +Mors-Uvae</i> 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. +<i>Erysiphe</i>, <i>Uncinula</i> and <i>Phyllactinia</i> 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 <i>Sphaerotheca</i> +and <i>Erysiphe</i> 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 <i>Sphaerotheca Humuli</i> 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 <i>Erysiphe</i> and <i>Phyllactinia</i>.</p> + +<table class="pic" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:600px; height:640px" src="images/img341a.jpg" alt="" /></td></tr> +<tr><td class="caption" colspan="2"><span class="sc">Fig. 11.</span>—Development of <i>Eurotium repens</i>. (After De Bary.)</td></tr> + +<tr><td class="f90" style="width: 50%; vertical-align: top;"> +<p>A, Small portion of mycelium +with conidiophore (<i>c</i>), and +archicarp (<i>as</i>).</p> + +<p>B, The spiral archicarp (<i>as</i>), +with the antheridium (<i>p</i>).</p> + +<p>D, The same, beginning to be +surrounded by the hyphae +forming the perithecium wall.</p> + +<p>D, The perithecium.</p></td> + +<td class="f90" style="width: 50%; vertical-align: top;"> +<p>E, F, Sections of young perithecia.</p> + +<p><i>w</i>, Parietal cells.</p> + +<p><i>f</i>, Pseudo-parenchyma.</p> + +<p><i>as</i>, Ascogonium.</p> + +<p>G, An ascus.</p> + +<p>H, An ascospore.</p></td></tr></table> + +<p class="pt2">The Perisporiaceae are saprophytic forms, the two chief genera +being <i>Aspergillus</i> and <i>Penicillium</i>. The blue-green mould <i>P. +crustaceum</i> and the green mould <i>A. herbariorium</i> (= <i>Eurotium +herbariorum</i>) 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 <i>Eurotium</i> both +ascogonium and antheridium contain a number of nuclei (<i>i.e.</i> 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. <i>Aspergillus Oryzae</i> 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 saké. The process has now been +successfully introduced into European commerce.</p> + + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" style="vertical-align: bottom;"><img style="width:230px; height:214px" src="images/img341b.jpg" alt="" /></td> +<td class="figcenter"><img style="width:350px; height:395px" src="images/img341c.jpg" alt="" /></td></tr> +<tr><td class="tcl f80">From Strasburger’s <i>Lehrbuch +der Botanik</i>, by permission +of Gustav Fischer.</td> +<td class="caption"><span class="sc">Fig. 13.</span>—<i>Ascobolus furfuraceus.</i> +Diagrammatic section of the fructification. +(After Janczewski.)</td></tr> + +<tr><td class="caption"><span class="sc">Fig. 12.</span>—<i>Peziza aurantiaca.</i> +(After Krombholz, +nat. size.)</td> +<td class="tcl f90"> +<p>  <i>m</i>, Mycelium.</p> +<p>  <i>c</i>, Archicarp.</p> +<p>  <i>l</i>, Pollinodium.</p> +<p>  <i>s</i>, Ascogenous filaments.</p> +<p>  <i>a</i>, Asri.</p> +<p>  <i>r</i>, <i>p</i>, The sterile tissue from which the paraphyses <i>h</i> spring.</p></td></tr></table> + +<p class="pt2"><i>Discomycetes.</i>—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 +<i>apothecium</i>. 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 <i>Pyronema</i> and +<i>Boudiera</i>; in <i>Lachnea stercorea</i> +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 <i>Humaria +granulata</i> and <i>Ascobolus furfuraceus</i>, +where the antheridium is +entirely absent. In <i>H. rutilans</i>, +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 <i>Ascobolus</i> the ascogonium is +multicellular, the various cells +communicating by pores in +the transverse walls (fig. 13).</p> + +<table class="flt" style="float: right; width: 350px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:300px; height:490px" src="images/img341d.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Strasburger’s <i>Lehrbuch der Botanik</i>, +by permission of Gustav Fischer.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 14.</span>—Perithecium of Podospora +fimiseda in longitudinal section. After v. Tavel.</td></tr> +<tr><td class="caption1"> +<p><i>s</i>, Asci.</p> +<p><i>a</i>, Paraphyses.</p> +<p><i>e</i>, Periphyses.</p> +<p><i>m</i>, Mycelial hyphae.</p></td></tr></table> + +<p>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.</p> + +<p>The genus <i>Peziza</i> (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 <i>Sclerotinia</i> +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, <i>S. +heteroica</i>, is <i>heteroecious</i>; the +ascospores infecting the leaves of <i>Vaccinium uliginosum</i>, while the +conidia which then arise infect only <i>Ledum palustre</i>. This is the +only case of heteroecism known in the vegetable kingdom outside +the Uredineae.</p> + +<p><i>Pyrenomycetes.</i>—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 +<span class="pagenum"><a name="page342" id="page342"></a>342</span> +is characterized by a special type of ascocarp, the <i>perithecium</i>. +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.</p> + +<p><i>Tuberineae.</i>—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 (<i>Tuber brumale</i> and other species). The exact life-history +of the truffle is not known.</p> + +<table class="flt" style="float: left; width: 300px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:220px; height:266px" src="images/img342a.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Strasburger’s <i>Lehrbuch der Botanik</i>, +by permission of Gustav Fischer.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 15.</span>—<i>Armillaria mellea.</i> (After +Ruhland.)</td></tr> +<tr><td class="caption1"> +<p>A, Young basidium with the two +primary nuclei.</p> + +<p>B, After fusion of the two nuclei. +<i>Hypholoma appendiculatum</i>.</p> + +<p>C, A basidium before the four +nuclei derived from the secondary +nucleus of the basidium +have passed into the four +basidiospores.</p> + +<p>D, Passage of a nucleus through +the sterigma into the basidiospore.</p></td></tr></table> + +<p><i>Laboulbeniineae</i> 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.</p> + +<p><i>Basidiales.</i>—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 <i>Humaria +granulata</i> and of vegetative nuclei in <i>H. rutilans</i>, 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.</p> + +<table class="pic" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:550px; height:705px" src="images/img342b.jpg" alt="" /></td></tr> +<tr><td class="caption" colspan="2"><span class="sc">Fig. 16.</span>—<i>Puccinia graminis.</i></td></tr> + +<tr><td class="f90" style="width: 50%; vertical-align: top;"> +<p>A, Mass of teleutospores (<i>t</i>) on a + leaf of couch-grass.</p> + +<p><i>e</i>, Epidermis ruptured.</p> + +<p><i>b</i>, Sub-epidermal fibres. (After + De Bary.)</p></td> + +<td class="f90" style="width: 50%; vertical-align: top;"> +<p>B, Part of vertical section + through leaf of Berberis + vulgaris, with <i>a</i>, aecidium + fruits, <i>p</i>, peridium, and <i>sp</i>, + spermogonia. (After Sachs.)</p> + +<p>C, Mass of uredospores (<i>ur</i>), + with one teleutospore (<i>t</i>).</p> + +<p><i>sh</i>, Sub-hymenial hyphae. (After + De Bary.)</p></td></tr></table> + +<p class="pt2"><i>Uredineae.</i>—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 <i>eu</i> forms. +In the <i>opsis</i> forms the uredospores are absent, the mycelium from the +aecidiospores producing directly the teleutospores. In <i>brachy</i> and +<i>hemi</i> 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 <i>lepto</i> and <i>micro</i> forms both +aecidiospores and uredospores are absent, the sporidia producing a +mycelium which gives rise directly to teleutospores; in the <i>lepto</i> +forms the teleutospores can germinate directly, in the <i>micro</i> forms +only after a period of rest. We have thus a series showing a progressive +reduction in the complexity of the life-history, the <i>lepto</i> and +<i>micro</i> forms having a life-history like that of the Basidiomycetes. +The <i>eu</i> and <i>opsis</i> forms may exhibit the remarkable phenomenon +of heteroecism, <i>i.e.</i> 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 +<span class="pagenum"><a name="page343" id="page343"></a>343</span> +host on which the uredospores (if present) and the teleutospores +are developed. A few examples are appended:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">Species.</td> <td class="tcc allb">Teleutospores on</td> <td class="tcc allb">Aecidiospores on</td></tr> + +<tr><td class="tcl lb rb"><i>Coleosporium Senecionis</i></td> <td class="tcl rb"><i>Pinus</i></td> <td class="tcl rb"><i>Senecio</i></td></tr> +<tr><td class="tcl lb rb"><i>Melampsora Rostrupi</i></td> <td class="tcl rb"><i>Populus</i></td> <td class="tcl rb"><i>Mecurialis</i></td></tr> +<tr><td class="tcl lb rb"><i>Pucciniastrum Goeppertiana</i></td> <td class="tcl rb"><i>Vaccinium</i></td> <td class="tcl rb"><i>Abies</i></td></tr> +<tr><td class="tcl lb rb"><i>Gymnosporangium Sabinae</i></td> <td class="tcl rb"><i>Juniperus</i></td> <td class="tcl rb"><i>Pyrus</i></td></tr> +<tr><td class="tcl lb rb"><i>Uromyces Pisi</i></td> <td class="tcl rb"><i>Pisum, &c.</i></td> <td class="tcl rb"><i>Euphorbia</i></td></tr> +<tr><td class="tcl lb rb"><i>Puccinia graminis</i></td> <td class="tcl rb"><i>Triticum, &c.</i></td> <td class="tcl rb"><i>Berberis</i></td></tr> +<tr><td class="tcl lb rb"><i>P. dispersa</i></td> <td class="tcl rb"><i>Secale, &c.</i></td> <td class="tcl rb"><i>Anchusa</i></td></tr> +<tr><td class="tcl lb rb"><i>P. coronata</i></td> <td class="tcl rb"><i>Agrostis</i></td> <td class="tcl rb"><i>Rhamnus</i></td></tr> +<tr><td class="tcl lb rb"><i>P. Ari-Phalaridis</i></td> <td class="tcl rb"><i>Phalaris</i></td> <td class="tcl rb"><i>Arum</i></td></tr> +<tr><td class="tcl lb rb"><i>P. Caricis</i></td> <td class="tcl rb"><i>Carex</i></td> <td class="tcl rb"><i>Urtica</i></td></tr> +<tr><td class="tcl lb rb"><i>Cronartium Ribicola</i></td> <td class="tcl rb"><i>Ribes</i></td> <td class="tcl rb"><i>Pinus</i></td></tr> +<tr><td class="tcl lb rb bb"><i>Chrysomyxa Rhododendri</i></td> <td class="tcl rb bb"><i>Rhododendron</i></td> <td class="tcl rb bb"><i>Picea</i></td></tr> +</table> + +<p class="noind">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.</p> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:220px; height:426px" src="images/img343a.jpg" alt="" /></td></tr> +<tr><td class="caption80">From Strasburger’s <i>Lehrbuch der Botanik</i>, +by permission of Gustav Fischer.</td></tr> +<tr><td class="caption1"><span class="sc">Fig. 17.</span>—<i>Phragmidium Violaceum.</i> +(After Blackman.)</td></tr> +<tr><td class="caption1"> +<p>A, Portion of a young aecidium.</p> + +<p><i>st</i>, Sterile cell.</p> + +<p><i>a</i>, Fertile cells; at <i>a</i><span class="su">2</span> the + passage of a nucleus from + the adjoining cell is seen.</p> + +<p>B, Formation of the first spore-mother-cell + (<i>sm</i>), from the + basal cell (<i>a</i>) of one of the + rows of spores.</p> + +<p>C, A further stage in which + from sm<span class="su">1</span> the first aecidiospore + (<i>a</i>) and the intercalary + cell (<i>z</i>) have arisen.</p> + +<p><i>sm</i><span class="su">2</span>, The second spore-mother-cell.</p> + +<p>D, Ripe aecidiospore</p></td></tr></table> + +<p><i>Cytology of Uredineae.</i>—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 <i>eu</i> and <i>opsis</i> +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 <i>hemi</i>, +<i>brachy</i>, <i>micro</i> and <i>lepto</i> 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 <i>hemi</i> and +<i>brachy</i> forms, and before the +formation of teleutospores in +<i>micro</i> and <i>lepto</i> 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.</p> + +<p><i>Basidiomycetes.</i>—This group is characterized by its greatly reduced +life-history as compared with that of the <i>eu</i> forms among the Uredineae. +All the forms have the same life-history as the <i>lepto</i> 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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:500px; height:495px" src="images/img343b.jpg" alt="" /></td></tr> +<tr><td class="tcl f80">From <i>Annals of Botany</i>, by permission of the Clarendon Press.</td></tr> +<tr><td class="caption"><span class="sc">Fig. 18.</span></td></tr></table> + +<p>It must be +clearly borne in +mind that though +the Basidiomycetes +show no +traces of differentiated +sexual +organs yet, like +the <i>micro</i> and <i>lepto</i> +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 <i>Coprinus</i>, but +no fusion of cells or migration of nuclei was to be observed.</p> + +<table class="pic" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:600px; height:511px" src="images/img343c.jpg" alt="" /></td></tr> +<tr><td class="caption" colspan="2"><span class="sc">Fig. 19.</span>—Amanita muscaria.</td></tr> + +<tr><td class="f90" style="width: 50%; vertical-align: top;"> +<p>A, The young plant.</p> +<p>B, The mature plant.</p> +<p>C, Longitudinal section of mature plant.</p> +<p><i>p</i>, The <i>pileus</i>.</p></td> + +<td class="f90" style="width: 50%; vertical-align: top;"> +<p><i>g</i>, The gills.</p> +<p><i>a</i>, The <i>annulus</i>, or remnant of <i>velum partiale</i>,</p> +<p><i>v</i>, Remains of <i>volva</i> or <i>velum universale</i>.</p> +<p><i>s</i>, The stalk.</p></td></tr></table> + +<p class="pt2"><i>Protobasidiomycetes.</i>—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. +<i>Hirneola</i> (<i>Auricularia</i>) <i>Auricula-Judae</i> is the well-known +Jew’s Ear, so named from the resemblance of the sporophore to a +human ear.</p> + +<p>The Pilacreaceae are a family found by Brefeld to contain the genus +<i>Pilacre</i>. <i>P. Petersii</i> has a transversely divided basidium as in +<i>Auriculariaceae</i>, but the basidia are surrounded with a peridium-like +sheath. The <i>Tremellinaceae</i> are characterized by the possession of +basidia which are divided by two <i>vertical</i> walls at right angles to +one another. From each of the four segments in the case of <i>Tremella</i> +a long outgrowth arises which reaches to the surface of the hymenium +<span class="pagenum"><a name="page344" id="page344"></a>344</span> +and bears the basidiospores. In <i>Dacryomyces</i> only two outgrowths +and two spores are produced.</p> + +<p><i>Autobasidiomycetes</i>.—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, +<i>Hymenomycetes</i> and <i>Gasteromycetes</i>.</p> + +<table class="flt" style="float: right; width: 380px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:330px; height:269px" src="images/img344.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig.</span> 20.—<i>Agaricus mucidus</i>. Portion +of hymenium. <i>s</i>, Sporidia; <i>st</i>, sterigmata; +<i>g</i>, sterile cells; <i>c</i>, cystidium, with operculum +<i>o</i>.</td></tr></table> + +<p><i>Hymenomycetes</i> 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 (<i>e.g.</i> +<i>Exobasidium</i>) 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. <i>Armillaria +melleus</i> and +<i>Polyporus annosus</i> are +examples of parasitic +forms which attack and +destroy living trees, +while <i>Merulius lacrymans</i> is the well-known +“dry rot” fungus.</p> + +<p><i>Gasteromycetes</i> 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 <i>peridium</i> +and an inner spore-bearing mass, the <i>gleba</i>. 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 <i>Bovista, Lycoperdon</i> (puff-ball) <i>Scleroderma, Geaster</i> (earth-star, +<i>q.v.</i>). 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.</p> + +<p>The most complex members of the Gasteromycetes belong to the +<i>Phalloideae</i>, which is sometimes placed as a distinct division of the +Autobasidiomycetes. <i>Phallus impudicus</i>, 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.</p> +</div> + +<p><i>Physiology</i>.—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.</p> + +<div class="condensed"> +<p><i><span class="correction" title="amended from Parasatism">Parasitism</span>.</i> —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 (<i>e.g.</i> <i>Nectria</i>, +<i>Dasyscypha</i>, &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 (<i>Botrytis</i>). Many +fungi, however, cannot complete their life-history apart from the +host-plant. Such <i>obligate</i> parasites may be epiphytic (<i>Erysipheae</i>), +the mycelium remaining on the outside and at most merely sending +haustoria into the epidermal cells, or endophytic (<i>Uredineae</i>, +<i>Ustilagineae</i>, &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 (<i>Fumago</i>, &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 (<i>Olpidium</i>, +<i>Lagenidium</i>, <i>Sclerotinia</i>, &c.), but they are far more frequently +intercellular, at any rate while young, the mycelium growing in the +lacunae between the cells (<i>Peronospora</i>, <i>Uredineae</i>) into which it +may send short (<i>Cystopus</i>), or long and branched (<i>Peronospora +Calotheca</i>) haustoria, or it extends in the middle lamella (<i>Ustilago</i>), +or even in the solid substance of the cell-wall (<i>Botrytis</i>). No sharp +lines can be drawn, however, since many mycelia are intercellular at +first and subsequently become intracellular (<i>Ustilagineae</i>), 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 +(<i>Hemileia</i>) when the mycelium never extends far from the point of +infection, or general (<i>Phytophthora</i>) when it runs throughout the +plant. Destructive parasites rapidly ruin the whole plant-body +(<i>Pythium</i>), 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 (<i>Rhytisma</i>). A parasite may be restricted during +a long incubation-period, however, and rampant and destructive +later (<i>Ustilago</i>). 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 (<i>Botrytis cinerea</i>), others are restricted to one family +(<i>Cystopus Candidus</i>) or genus (<i>Phytophthora infestans</i>) or even +species (<i>Pucciniastrum Padi</i>), 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—<i>e.g.</i> <i>Dematophora necatrix</i> on +roots, <i>Calyptospora Goeppertiana</i> on stems, <i>Ustilago Scabiosae</i> in +anthers, <i>Claviceps purpurea</i> 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—<i>e.g.</i> <i>Pythium</i>, <i>Phytophthora +omnivora</i>.</p> + +<p><i>Chemotropism.</i>—Taken in conjunction with Pfeffer’s beautiful discovery +that certain chemicals exert a distinct attractive influence +on fungus hyphae (<i>chemotropism</i>), 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 <i>Botrytis</i> pierce the cell-walls of a lily by secreting a cytase +and dissolving a hole through the membrane. Miyoshi then demonstrated +that if <i>Botrytis</i> 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.</p> + +<p><i>Specialization of Parasitism.</i>—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 <i>Uredineae</i> and <i>Erysiphaceae</i> 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 +<span class="pagenum"><a name="page345" id="page345"></a>345</span> +sharply distinguishable by their infecting power. Eriksson found, +for example, that the well-known species <i>Puccinia graminis</i> 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 +<i>P. graminis Secalis</i>, <i>Tritici</i>, <i>Avenae</i>, <i>Airae</i>, <i>Agrostis</i>, <i>Poae</i>. The +first named will grow on rye and barley but not on wheat or oat. +The form <i>Tritici</i> is the least sharply marked and will grow on wheat, +barley, rye and oat but not on the other grasses. The form <i>Avenae</i> +will grow on oat and many grasses but not on the other three cereals +mentioned. The last three forms grow only on the genera <i>Aira</i>, +<i>Agrostis</i> and <i>Poa</i> 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 <i>Puccinia</i> parasitic on species of <i>Bromus</i>, +and by Neger, Marchal and especially Salmon in the Erysiphaceae. +In the last-named family the single morphological species <i>Erysiphe +graminis</i> is found growing on the cereals, barley, oat, wheat, rye +and a number of wild grasses (such as <i>Poa</i>, <i>Bromus</i>, <i>Dactylis</i>). 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 <i>Puccinia graminis</i> +so we find that both the conidia and ascospores of <i>E. graminis</i> show +this phenomenon. Salmon has further shown in investigating the +relation of <i>E. graminis</i> to various species of the genus, <i>Bromus</i>, 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 <i>B. racemosus</i> cannot infect <i>B. commutatus</i>. +If, however, conidia from <i>B. racemosus</i> are sown on <i>B. hordaceus</i>, +the conidia which develop on that plant are now able to infect +<i>B. commutatus</i>; thus <i>B. hordaceus</i> 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.</p> + +<p><i>Symbiosis.</i>—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 <i>symbiosis</i>. 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—<i>e.g.</i> 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 <span class="correction" title="added after">after</span> +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 rôle 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 <i>Aspergillus</i> and yeast in saké +manufacture, and between yeasts and bacteria in kephir and in the +ginger-beer plant are best worked out. For cases of symbiosis see +<span class="sc"><a href="#artlinks">Bacteriology</a></span>.</p> + +<p><span class="sc">Authorities.</span>—<i>General</i>: Engler and Prantl, <i>Die natürlichen +Pflanzenfamilien</i>, i. Teil (1892 onwards); Zopf, <i>Die Pilze</i> (Breslau, +1890); De Bary, <i>Comparative Morphology of Fungi</i>, &c. (Oxford, +1887); von Tafel, <i>Vergleichende Morphologie der Pilze</i> (Jena, 1892); +Brefeld, <i>Unters. aus dem Gesamtgebiete der Mykologie</i>, Heft i. 13 +(1872-1905); Lotsy, <i>Vorträge über botanische Stammesgeschichte</i> +(Jena, 1907). <i>Distribution</i>, &c.: Cooke, <i>Introduction to the Study +of Fungi</i> (London, 1895); Felix in <i>Zeitschr. d. deutsch. geologisch. +Gesellsch.</i> (1894-1896); Staub, <i>Sitzungsber. d. bot. Sec. d. Kgl. +ungarischen naturwiss. Gesellsch. zu Budapest</i> (1897). <i>Anatomy</i>, +&c.: Bommer, “Sclerotes et cordons mycéliens,” <i>Mém. de l’Acad. +Roy. de Belg.</i> (1894); Mangin, “Observ. sur la membrane des +mucorinées,” <i>Journ. de Bot.</i> (1899); Zimmermann, <i>Die Morph. +und Physiologie des Pflanzenzellkernes</i> (Jena, 1896); Wisselingh, +“Microchem. Unters. über die Zellwände d. Fungi,” <i>Pringsh. +Jahrb.</i> B. 31, p. 619 (1898); Istvanffvi, “Unters. über die phys. +Anat. der Pilze,” <i>Prings. Jahrb.</i> (1896). <i>Spore Distribution</i>: Fulton, +“Dispersal of the Spores of Fungi by Insects,” <i>Ann. Bot.</i> (1889); +Falck, “Die Sporenverbreitung bei den Basidiomyceten,” <i>Beitr. +zur Biol. d. Pflanzen</i>, ix. (1904). <i>Spores and Sporophores</i>: Zopf, +<i>Die Pilze</i>; also the works of von Tafel and Brefeld. <i>Classification</i>: +van Tieghem, <i>Journ. de bot.</i> p. 77 (1893), and the works of Brefeld, +Engler and Prantl, von Tafel, Saccardo and Lotsy already cited, +<i>Oomycetes</i>: Wager, “On the Fertilization of <i>Peronospora parasitica</i>,” +<i>Ann. Bot.</i> vol. xiv. (1900); Stevens, “The Compound +Oosphere of <i>Albugo Bliti</i>,” <i>Bot. Gaz.</i> vol. 28 (1899); “Gametogenesis +and Fertilization in <i>Albugo</i>,” <i>ibid</i>. vol. 32 (1901); +Miyake, “The Fertilization of <i>Pythium de Baryanum</i>,” <i>Ann. of Bot.</i> +vol. xv. (1901); Trow, “On Fertilization in the Saprolegnieae,” +<i>Ann. of Bot.</i> vol. xviii. (1904); Thaxter, “New and Peculiar Aquatic +Fungi,” <i>Bot. Gaz.</i> vol. 20 (1895); Lagerheim, “Unters. über die +Monoblepharideae,” <i>Bih. Svenska Vet. Acad. Handlingar</i>, 25. +Afd. iii. (1900); Woronin, “Beitrag zur Kenntnis der Monoblepharideen,” +<i>Mém. de l’Acad. Imp. d. Sc. de St-Pétersbourg</i>, 8 sér. +vol. 16 (1902). <i>Zygomycetes</i>: Harper, “Cell-division in Sporangia +and Asci,” <i>Ann. Bot.</i> vol. xiii. (1899); Klebs, <i>Die Bedingungen der +Fortpflanzung</i>, &c. (Jena, 1896), and “Zur Physiologie der Fortpflanzung” +<i>Prings. Jahr.</i> (1898 and 1899), “Über <i>Sporodinia +grandis</i>,” <i>Bot. Zeit.</i> (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 <i>Sporodinia grandis</i>,” <i>Ber. d. deutschen bot. Ges.</i> Bd. 19 (1901); +Blakeslee, “Sexual Reproduction in the Mucorineae,” <i>Proc. Am. +Acad.</i> (1904); “Zygospore germination in the Mucorineae,” <i>Annales +mycologici</i> (1906). <i>Ustilagineae</i>: Plowright, <i>British Uredineae and +Ustilagineae</i> (London, 1889); Massee, <i>British Fungi</i> (Phycomycetes +and Ustilagineae) (London, 1891); Brefeld, <i>Unters. aus dem +Gesamtgeb. der Mykol.</i> Hefte xi. and xii.; and Falck, “Die Bluteninfektion +bei den Brandpilzen,” ibid. Heft xiii. 1905; Dangeard, “La +Reproduction sexuelle des Ustilaginées,” C.R., Oct. 9, 1893; +Maire, “Recherches cytologiques et taxonomiques sur les Basidiomyceten,” +<i>Annexé au Bull. de la Soc. Mycol. de France</i> (1902). +<i>Saccharomycetaceae</i>: Jorgensen, <i>The Micro-organisms of Fermentation</i> +(1899); Barker, <i>Ann. of Bot.</i> vol. xiv. (1901); “On Spore-formation +among the Saccharomycetes,” <i>Journ. of the Fed. Institute +of Brewing</i>, vol. 8 (1902); Guillermond, <i>Recherches cytologiques +sur lés levures</i> (Paris, 1902); Hansen, <i>Centralbl. f. Bakt. u. Parasitenp.</i> +Abt. ii. Bd. 12 (1904). <i>Exoascaceae</i>: Giesenhagen, “<i>Taphrina, +Exoascus, Magnusiella</i>” (complete literature given), <i>Bot. +Zeit.</i> Bd. 7 (1901). <i>Erysiphaceae</i>: Harper, “Die Entwicklung des +Perithecium bei <i>Sphaerotheca castagnei</i>,” <i>Ber. d. deut bot Ges.</i> (1896); +“Sexual Reproduction and the Organization of the Nucleus in certain +Mildews,” <i>Publ. Carnegie Institution</i> (Washington, 1906); Blackman +& Fraser, “Fertilization in <i>Sphaerotheca</i>,” <i>Ann. of Bot.</i> (1905). +<i>Perisporiaceae</i>: Brefeld, <i>Untersuchungen aus dem Gesamtgeb. der +Mykol.</i> Heft 10 (1891); Fraser and Chamber, <i>Annales mycologici</i> +(1907). <i>Discomycetes</i>: Harper, “Über das Verhalten der Kerne bei +Ascomyceten,” <i>Jahr. f. wiss. Bot.</i> Bd. 29 (1890); “Sexual Reproduction +in <i>Pyronema confluens</i>,” <i>Ann. of Bot.</i> 14 (1900); Claussen, +“Zur Entw. der Ascomyceten,” Boudiera, Bot. Zeit. Bd. 63 (1905); +Dangeard, “Sur le <i>Pyronema confluens</i>,” <i>Le Botaniste</i>, 9 série (1903) +(and numerous papers in same journal earlier and later); Ramlow, +“Zur Entwick. von <i>Thelebolus stercoren</i>,” <i>Bot. Zeit.</i> (1906); Woronin, +“Über die Sclerotienkrankheit der Vaccineen Beeren,” <i>Mem. de +l’Acad. Imp. des Sciences de St-Pétersbourg</i>, 7 série, 36 (1888); +Dittrich, “Zur Entwickelungsgeschichte der Helvellineen,” Cohn’s +<i>Beitr. z. Biol. d. Pflanzen</i> (1892). <i>Pyrenomycetes</i>: Fisch, “Beitr. +z. Entwickelungsgeschichte einiger Ascomyceten,” <i>Bot. Zeit.</i> +(1882); Frank, “Über einige neue u. weniger bekannte Pflanzkrankh.,” +<i>Landw. Jahrb.</i> Bd. 12 (1883); Ward, “<i>Onygena +equina</i>, a horn-destroying fungus,” <i>Phil. Trans.</i>, vol. 191 +(1899); Dawson, “On the Biology of Poroniapunctata,” Ann. of +Bot. 14 (1900). <i>Tuberineae</i>: Buchholtz, “Zur Morphologie u. +Systematik der Fungi hypogaei,” <i>Ann. Mycol.</i> Bd. 1 (1903); +Fischer in Engler and Prantl, <i>Die natürlichen Pflanzenfamilien</i> +(1896). <i>Laboulbeniineae</i>: Thaxter, “Monograph of the Laboulbeniaceae,” +<i>Mem. Amer. Acad. of Arts and Sciences</i>, vol. 12 (1895). +<i>Uredineae</i>: Eriksson and Henning, <i>Die Getreideroste</i> (Stockholm, +1896); Eriksson, <i>Botan. Gaz.</i> vol. 25 (1896); “On the Vegetative +Life of some Uredineae,” Ann. of Bot. (1905); Klebahn, <i>Die wirtwechselnden +Rostpilze</i> (Berlin, 1904); Sapin-Trouffy, “Recherches +histologiques sur la famille des Urédinées,” <i>Le Botaniste</i> (1896-1897); +Blackman, “On the Fertilization, Alternation of Generations and +General Cytology of the Uredineae,” <i>Ann. of Bot.</i> vol. 18 (1904); +Blackman and Fraser, “Further Studies on the Sexuality of Uredineae,” +<i>Ann. of Bot.</i> vol. 20 (1906); Christman, “Sexual Reproduction +of Rusts,” <i>Ann. of Bot.</i> vol. 20 (1906); Ward, “The +Brooms and their Rust Fungus,” <i>Ann. of Bot.</i> vol. 15 (1901). +<i>Basidiomycetes</i>: Dangeard, “La Reprod. sexuelle des Basidiomycètes,” +<i>Le Botaniste</i> (1894 and 1900); Maire, “Recherches +cytologiques et taxonomiques sur les Basidiomycètes,” <i>Annexe du +Bull. de la Soc. Mycol. de France</i> (1902); Möller, “Protobasidiomyceten,” +<i>Schimper’s Mitt. aus den Tropen</i>, Heft 8 (Jena, 1895); +Nichols, “The Nature and Origin of the Binucleated Cells in certain +Basidiomycetes,” <i>Trans. Wisconsin Acad. of Sciences</i>, vol. 15 +(1905); Wager, “The Sexuality of the Fungi,” <i>Ann. of Bot.</i> 13 +(1899); Woronin, “<i>Exobasidium Vaccinii</i>,” <i>Verh. Naturf. Ges. zu +Freiburg</i>, Bd. 4 (1867). <i>Fermentation</i>: Buchner, “Gährung ohne Hefezellen,” +<i>Bot. Zeit.</i> Bd. 18 (1898); Albert, <i>Cent. f. Bakt.</i> Bd. 17 (1901); +<span class="pagenum"><a name="page346" id="page346"></a>346</span> +Green, <i>The Soluble Ferments and Fermentation</i> (Cambridge, 1899). +<i>Parasitism</i>: “On some Relations between Host and Parasite,” +<i>Proc. Roy. Soc</i>. vol. 47 (1890); “A Lily Disease,” <i>Ann. of Botany</i>, +vol. 2 (1888); Eriksson & Hennings, <i>Die Getreideroste (vide supra</i>); +Ward, “On the Question of Predisposition and Immunity in Plants,” +<i>Proc. Cambridge Phil. Soc</i>. vol. 11 (1902); also <i>Annals of Bot</i>. +vol. 16 (1902) and vol. 19 (1905); Neger, “Beitr. z. Biol. d. +Erysipheen” <i>Flora</i>, Bde. 88 and 90 (1901-1902); Salmon, “Cultural +Experiments with ‘Biologic Forms’ of the Erysiphaceae,” <i>Phil. +Trans</i>. (1904); “On Erysiphe graminis and its adaptative parasitism +within the genus, <i>Bromus</i>,” <i>Ann. Mycol</i>. vol. 11 (1904), also <i>Ann. +of Bot</i>. vol. 19 (1905). <i>Symbiosis</i>: Ward, “The Ginger-Beer +Plant,” <i>Phil. Trans. Roy. Soc</i>. (1892); “Symbiosis,” <i>Ann. of Bot</i>. 13 +(1899); Shalk, “Der Sinn der Mykorrhizenbildung,” <i>Jahrb. f. +wiss. Bot</i>. Bd. 34 (1900); Bernard, “On some Different Cases of +Germination,” <i>Gardener’s Chronicle</i> (1900); Pierce, <i>Publ. Univ. +California</i> (1900).</p> +</div> +<div class="author">(H. M. W.; V. H. B.)</div> + + +<hr class="art" /> +<p><span class="bold">FUNJ<a name="ar74" id="ar74"></a></span> (<span class="sc">Funniyeh, Fung, Fungha</span>), 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 +<span class="sc"><a href="#artlinks">Sennar</a></span> and <span class="sc"><a href="#artlinks">Sudan</a></span> (Anglo-Egyptian).</p> + + +<hr class="art" /> +<p><span class="bold">FUNKIA<a name="ar75" id="ar75"></a></span>, 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUNNEL<a name="ar76" id="ar76"></a></span> (through an O. Fr. <i>founil</i>, found in Breton, from +Lat. <i>infundibulum</i>, that through which anything is poured, +from <i>fundere</i>, 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 +<i>infundibulum</i> 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.</p> + + +<hr class="art" /> +<p><span class="bold">FUR<a name="ar77" id="ar77"></a></span> (connected with O. Fr. <i>forre</i>, 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.</p> + +<p>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.</p> + +<p><i>History.</i>—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.</p> + +<p>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 <i>Astoria</i>; 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.</p> + +<p><i>Natural Supplies.</i>—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, +<span class="pagenum"><a name="page347" id="page347"></a>347</span> +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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p><i>Quality and Colour.</i>—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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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, +<span class="pagenum"><a name="page348" id="page348"></a>348</span> +musquash, raccoon and wolverine are more valuable than the +paler skins.</p> + +<p><i>Collective Supplies and Sales.</i>—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:—</p> + +<table class="ws f90" summary="Contents"> +<tr><td class="tcl">January:</td> <td class="tcl">Frankfort-on-the-Oder</td> <td class="tcl">Small collection of provincial produce,<br />   such as otter, fox, fitch and marten.</td></tr> +<tr><td class="tcl">February:</td> <td class="tcl">Irbit, Siberia</td> <td class="tcl">General Russian furs.</td></tr> +<tr><td class="tcl">Easter:</td> <td class="tcl">Leipzig, Germany</td> <td class="tcl">General furs.</td></tr> +<tr><td class="tcl">August:</td> <td class="tcl">Nizhniy Novgorod, Russia</td> <td class="tcl">Persian lamb and general furs.</td></tr> +<tr><td class="tcl">August:</td> <td class="tcl">Kiakhta, Siberia</td> <td class="tcl">Chinese furs and ermine.</td></tr> +<tr><td class="tcl">December:</td> <td class="tcl">Ishim, Siberia</td> <td class="tcl">Chiefly squirrels.</td></tr> +</table> + +<p>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.</p> + +<p>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.</p> + +<table class="ws f90" summary="Contents"> +<tr><td class="tcc"><i>Year ending 31st of March 1906.</i></td> <td class="tcc">Total Number<br />of Skins.</td></tr> + +<tr><td class="tcl cl">Badger</td> <td class="tcr cl">28,634</td></tr> +<tr><td class="tcl">Badger, Japanese</td> <td class="tcr">6,026</td></tr> +<tr><td class="tcl cl">Bear</td> <td class="tcr cl">18,576</td></tr> +<tr><td class="tcl">Beaver</td> <td class="tcr">80,514</td></tr> +<tr><td class="tcl cl">Cat, Civet</td> <td class="tcr cl">157,915</td></tr> +<tr><td class="tcl">Cat, House</td> <td class="tcr">126,703</td></tr> +<tr><td class="tcl cl">Cat, Wild</td> <td class="tcr cl">32,253</td></tr> +<tr><td class="tcl">Chinchilla (La Plata), known also as Bastard</td> <td class="tcr">43,578</td></tr> +<tr><td class="tcl cl">Chinchilla Peruvian finest</td> <td class="tcr cl">5,603</td></tr> +<tr><td class="tcl">Deer, Chinese</td> <td class="tcr">124,355</td></tr> +<tr><td class="tcl cl">Ermine</td> <td class="tcr cl">40,641</td></tr> +<tr><td class="tcl">Fisher</td> <td class="tcr">5,949</td></tr> +<tr><td class="tcl cl">Fitch</td> <td class="tcr cl">77,578</td></tr> +<tr><td class="tcl">Fox, Blue</td> <td class="tcr">1,893</td></tr> +<tr><td class="tcl cl">Fox, Cross</td> <td class="tcr cl">10,276</td></tr> +<tr><td class="tcl">Fox, Grey</td> <td class="tcr">59,561</td></tr> +<tr><td class="tcl cl">Fox, Japanese</td> <td class="tcr cl">81,429</td></tr> +<tr><td class="tcl">Fox, Kit</td> <td class="tcr">4,023</td></tr> +<tr><td class="tcl cl">Fox, Red</td> <td class="tcr cl">158,961</td></tr> +<tr><td class="tcl">Fox, Silver</td> <td class="tcr">2,510</td></tr> +<tr><td class="tcl cl">Fox, White</td> <td class="tcr cl">27,463</td></tr> +<tr><td class="tcl">Goats, Chinese</td> <td class="tcr">261,190</td></tr> +<tr><td class="tcl cl">Hares</td> <td class="tcr cl">41,256</td></tr> +<tr><td class="tcl">Kangaroo</td> <td class="tcr">7,115</td></tr> +<tr><td class="tcl cl">Kid, Chinese linings and skins equal to</td> <td class="tcr cl">5,080,047</td></tr> +<tr><td class="tcl">Kolinsky</td> <td class="tcr">114,251</td></tr> +<tr><td class="tcl cl">Lamb, Mongolian linings and skins equal to</td> <td class="tcr cl">214,072</td></tr> +<tr><td class="tcl">Lamb, Slink linings and skins equal to</td> <td class="tcr">167,372</td></tr> +<tr><td class="tcl cl">Lamb, Tibet linings and skins equal to</td> <td class="tcr cl">794,130</td></tr> +<tr><td class="tcl">Leopard</td> <td class="tcr">3,574</td></tr> +<tr><td class="tcl cl">Lynx</td> <td class="tcr cl">88,822</td></tr> +<tr><td class="tcl">Marmot, linings and skins equal to</td> <td class="tcr">1,600,600</td></tr> +<tr><td class="tcl cl">Marten, Baum</td> <td class="tcr cl">4,573</td></tr> +<tr><td class="tcl">Marten, Japanese</td> <td class="tcr">16,461</td></tr> +<tr><td class="tcl cl">Marten, Stone</td> <td class="tcr cl">12,939</td></tr> +<tr><td class="tcl">Mink, Canadian and American</td> <td class="tcr">299,254</td></tr> +<tr><td class="tcl cl">Mink, Japanese</td> <td class="tcr cl">360,373</td></tr> +<tr><td class="tcl">Mouflon</td> <td class="tcr">23,594</td></tr> +<tr><td class="tcl cl">Musk-rat or Musquash, Brown</td> <td class="tcr cl">5,126,339</td></tr> +<tr><td class="tcl">Musk-rat or Musquash, Black</td> <td class="tcr">41,788</td></tr> +<tr><td class="tcl cl">Nutria</td> <td class="tcr cl">82,474</td></tr> +<tr><td class="tcl">Opossum, American</td> <td class="tcr">902,065</td></tr> +<tr><td class="tcl cl">Opossum, Australian</td> <td class="tcr cl">4,161,685</td></tr> +<tr><td class="tcl">Otter, River</td> <td class="tcr">21,235</td></tr> +<tr><td class="tcl cl">Otter, Sea</td> <td class="tcr cl">522</td></tr> +<tr><td class="tcl">Raccoon</td> <td class="tcr">310,712</td></tr> +<tr><td class="tcl cl">Sable, Canadian and American</td> <td class="tcr cl">97,282</td></tr> +<tr><td class="tcl">Sable, Japanese</td> <td class="tcr">556</td></tr> +<tr><td class="tcl cl">Sable, Russian</td> <td class="tcr cl">26,399</td></tr> +<tr><td class="tcl">Seals, Fur</td> <td class="tcr">77,000</td></tr> +<tr><td class="tcl cl">Seals, Hair</td> <td class="tcr cl">31,943</td></tr> +<tr><td class="tcl">Skunk</td> <td class="tcr">1,068,408</td></tr> +<tr><td class="tcl cl">Squirrel</td> <td class="tcr cl">194,596</td></tr> +<tr><td class="tcl">Squirrel Linings each averaging 126 skins</td> <td class="tcr">1,982,736</td></tr> +<tr><td class="tcl cl">Tiger</td> <td class="tcr cl">392</td></tr> +<tr><td class="tcl">Wallaby</td> <td class="tcr">60,956</td></tr> +<tr><td class="tcl cl">Wolf</td> <td class="tcr cl">56,642</td></tr> +<tr><td class="tcl">Wolverine</td> <td class="tcr">1,726</td></tr> +<tr><td class="tcl cl">Wombat</td> <td class="tcr cl">193,625</td></tr> +</table> + +<p>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 <i>first cost</i>, 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 <i>relative</i> value.</p> + +<p>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.</p> + +<div class="condensed"> +<p class="pt2 center"><i>Names, Qualities and Uses of Pelts.</i><a name="fa1h" id="fa1h" href="#ft1h"><span class="sp">1</span></a></p> + +<p><span class="sc">Astrachan.</span>—See <i>Lambs</i>, below.</p> + +<p><span class="sc">Badger.</span>—Size 2 × 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.</p> + +<p><span class="sc">Bear, Australian.</span>—See <i>Wombat</i>, below.</p> + +<p><span class="sc">Bear, Black.</span>—Size 6 × 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.</p> + +<p><span class="sc">Bear, Brown.</span>—Size 6 × 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 +<span class="pagenum"><a name="page349" id="page349"></a>349</span> +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.</p> + +<p><span class="sc">Bear, Grizzly.</span>—Size 8 × 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.</p> + +<p><span class="sc">Bear, Isabelline.</span>—See <i>Bear</i>, <i>Brown</i>, above.</p> + +<p><span class="sc">Bear, White.</span>—Size 10 × 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.</p> + +<p><span class="sc">Beaver.</span> Size 3 × 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.</p> + +<p><span class="sc">Broadtail.</span>—See <i>Lambs</i>, below.</p> + +<p><span class="sc">Caracal.</span>—A small lynx from India, the fur very poor, seldom +imported.</p> + +<p><span class="sc">Caracul.</span>—See <i>Goats</i> and <i>Lambs</i>, below.</p> + +<p><span class="sc">Cat, Civet.</span>—Size 9 × 4½ 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.</p> + +<p><span class="sc">Cat, House, &c.</span>—18 × 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 5½d. to 1s. 1d.</p> + +<p><span class="sc">Cheetah.</span>—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.</p> + +<p><span class="sc">Chinchilla, Peruvian</span> and <span class="sc">Bolivian.</span>—Size 12 × 7 in., fur 1 to +1¼ 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.</p> + +<p><span class="sc">Chinchilla, La Plata</span>, incorrectly named and known in the trade +as “bastard chinchilla,” size 9 × 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.</p> + +<p><span class="sc">Chinchillone.</span>—Size 13 × 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.</p> + +<p><span class="sc">Deer, Chinese</span> and <span class="sc">East Indian.</span>—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.</p> + +<p><span class="sc">Dog.</span>—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.</p> + +<p><span class="sc">Dog Wolf.</span>—See <i>Wolf</i>, below.</p> + +<p><span class="sc">Ermine.</span>—Size 12 × 2½ 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.</p> + +<p><span class="sc">Fisher.</span>—Size 30 × 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.</p> + +<p><span class="sc">Fitch.</span>—Size 12 × 3 in., of the marten species, also known as the +pole cat. Yellow underwool 1/3 in. deep, black top hair, 1½ to 1¾ 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.</p> + +<p><span class="sc">Fox, Blue.</span>—Size 24 × 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.</p> + +<p><span class="sc">Fox, Common.</span>—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.</p> + +<p><span class="sc">Fox, Cross.</span>—Size 20 × 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.</p> + +<p><span class="sc">Fox, grey.</span>—Size 27 × 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.</p> + +<p><span class="sc">Fox, Japanese.</span>—See <i>Fox, Red</i>, and <i>Raccoon</i>, below.</p> + +<p><span class="sc">Fox, Kit.</span>—Size 20 × 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.</p> + +<p><span class="sc">Fox, Red.</span>—Size 24 × 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.</p> + +<p><span class="sc">Fox, Silver.</span> Size 30 × 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 £1 to £320.</p> + +<p><span class="sc">Fox, White.</span>—Size 20 × 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 +<span class="pagenum"><a name="page350" id="page350"></a>350</span> +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.</p> + +<p><span class="sc">Genet.</span>—Size 10 × 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 <i>Cats</i>, above.) Value 1s. to 6s. 6d.</p> + +<p><span class="sc">Goats.</span>—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 <i>Lambs, caracul</i>. Value +of Chinese white 3s. 6d. to 6s. 6d.: grey, 4s. to 6s. 9d.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p><span class="sc">Guanaco.</span>—Size 30 × 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.</p> + +<p><span class="sc">Hamster.</span>—Size 8 × 3½ 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.</p> + +<p><span class="sc">Hare.</span>—Size 24 × 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.</p> + +<p><span class="sc">Jackal.</span>—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.</p> + +<p><span class="sc">Jaguar.</span>—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.</p> + +<p><span class="sc">Kaluga.</span>—See <i>Souslik</i>, below.</p> + +<p><span class="sc">Kangaroo.</span>—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 1½ d. to 5s. 3d., +wallaroo 1s. to 5s. 6d.</p> + +<p><span class="sc">Kids.</span>—See <i>Goats</i>, above.</p> + +<p><span class="sc">Kolinsky.</span>—Size 12 × 2½ 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.</p> + +<p><span class="sc">Lambs.</span>—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 × 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.</p> + +<p>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 × 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 moiré 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.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Persian Lamb</td> <td class="tcl">price 12s. 6d.</td> <td class="tcl">to 25s.</td></tr> +<tr><td class="tcl">Broadtail</td> <td class="tcl">price 10s.</td> <td class="tcl">to 35s.</td></tr> +</table> + +<p class="noind">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 × 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.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Astrachan</td> <td class="tcl">price 1s.</td> <td class="tcl">to 5s. 6d.</td></tr> +<tr><td class="tcl">Caracul Lamb</td> <td class="tcl">price 2s. 6d</td> <td class="tcl">to 10s. 6d.</td></tr> +<tr><td class="tcl">Shiraz</td> <td class="tcl">price 4s. 6d</td> <td class="tcl">to 10s.</td></tr> +<tr><td class="tcl">Bokharan</td> <td class="tcl">price 1s. 6d</td> <td class="tcl">to 3s. 6d.</td></tr> +</table> + +<p class="noind">Grey lambs, size 24 × 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 × 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.</p> + +<p><span class="sc">Leopard.</span>—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 +<span class="pagenum"><a name="page351" id="page351"></a>351</span> +from the softest and flattest of skins, but usually they are made into +settee covers, floor rugs and foot muffs. Value 2s. to 40s.</p> + +<p><span class="sc">Lion.</span>—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’ £10 +to £100; lionesses’ £5 to £25.</p> + +<p><span class="sc">Lynx.</span>—Size 45 × 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.</p> + +<p><span class="sc">Lynx Cat</span> or <span class="sc">Bay Lynx.</span>—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.</p> + +<p><span class="sc">Marmot.</span>—Size 18 × 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.</p> + +<p><span class="sc">Marten, American.</span>—See <i>Sable</i>, below.</p> + +<p><span class="sc">Marten, Baum.</span>—Size 16 × 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.</p> + +<p><span class="sc">Marten, Black.</span>—See <i>Skunk</i>, below.</p> + +<p><span class="sc">Marten, Japanese.</span>—Size 16 × 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.</p> + +<p><span class="sc">Marten, Stone.</span>—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.</p> + +<p><span class="sc">Mink.</span>—Size 16 × 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.</p> + +<p><span class="sc">Mole</span>.—Size 3½ × 2½ 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 ½d. to 2d.</p> + +<p><span class="sc">Mongolian Lambs</span>.—See <i>Lambs</i>, above.</p> + +<p><span class="sc">Monkey, Black</span>.—Size 18 × 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 (<i>Colobus satanas</i>). 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.</p> + +<p><span class="sc">Mouflon</span>.—Size 30 × 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.</p> + +<p><span class="sc">Musk-Ox</span>.—Size 6 × 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.</p> + +<p><span class="sc">Musquash</span> or <span class="sc">Musk-Rat, Brown</span> and <span class="sc">Black Russian</span>.—Size +12 × 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 5½d. to 1s. 9d.</p> + +<p>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.</p> + +<p>The Russian musquash is very small, 7 × 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.</p> + +<p><span class="sc">Nutria</span>.—Size 20 × 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.</p> + +<p><span class="sc">Ocelot</span>.—Size 36 × 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.</p> + +<p><span class="sc">Opossum, American</span>.—Size 18 × 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 2½d. to 5s. 6d.</p> + +<p><span class="sc">Opossum, Australian</span>.—Size 16 × 8 in. Is a totally different +nature of fur to the American. Although it has wool and top hair, +<span class="pagenum"><a name="page352" id="page352"></a>352</span> +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 3¾d. to 3s. 6d.</p> + +<p><span class="sc">Opossum, Ringtailed</span>.—Size 7 × 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.</p> + +<p><span class="sc">Opossum, Tasmanian</span> (grey and black).—Size 20 × 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.</p> + +<p><span class="sc">Otter, River</span>.—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 × 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.</p> + +<p><span class="sc">Otter, Sea</span>.—Size 50 × 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 £200 to £600, 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 £10 to £220. A single skin +has been known to fetch £400.</p> + +<p><span class="sc">Ounce</span>.—See <i>Leopard</i>, above.</p> + +<p><span class="sc">Persian Lambs</span>.—See <i>Lambs</i>, above.</p> + +<p><span class="sc">Platypus</span>.—Size 12 × 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.</p> + +<p><i>Pony</i> or <i>Tatar Foal</i>.—Size 36 × 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.</p> + +<p><span class="sc">Puma</span>.—Size 4½ × 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.</p> + +<p><span class="sc">Raccoon</span>.—Size 20 × 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 1½ 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.</p> + +<p><span class="sc">Sable, American</span> and <span class="sc">Canadian.</span>—Size 17 × 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.</p> + +<p><span class="sc">Sable, Chinese</span> and <span class="sc">Japanese.</span>—Size 14 × 4½ 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.</p> + +<p><span class="sc">Sable, Russian.</span>—Size 15 × 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 1½ to 2½ 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.</p> + +<p>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.</p> + +<p>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.</p> + +<p><span class="sc">Seal, Fur.</span>—Sizes range from 24 × 15 in. to 55 × 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 +<span class="pagenum"><a name="page353" id="page353"></a>353</span> +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.</p> + +<p><span class="sc">Seal, Hair.</span>—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 +¾ 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.</p> + +<p><span class="sc">Sheep.</span>—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.</p> + +<p><span class="sc">Skunk</span> or <span class="sc">Black Marten.</span>—Size 15 × 8 in. The underwool is +full and fairly close with glossy, flowing top hair about 2½ 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.</p> + +<p><span class="sc">Souslik.</span>—Size 7 in. × 2¼. 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.</p> + +<p><span class="sc">Squirrel.</span>—Size 10 × 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 2½d. to 1s. 1d.</p> + +<p><span class="sc">Tibet Lamb.</span>—Size 27 × 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.</p> + +<p><span class="sc">Tiger.</span>—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.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Value of the Indian</td> <td class="tcl">from £3 to £15.</td></tr> +<tr><td class="tcl">Value of the Chinese</td> <td class="tcl">from £10 to £65.</td></tr> +</table> + +<p><span class="sc">Vicuna</span> 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.</p> + +<p><span class="sc">Wallaby.</span>—See <i>Kangaroo</i>, above.</p> + +<p><span class="sc">Wallaroo.</span>—See <i>Kangaroo</i>, above.</p> + +<p><span class="sc">Wolf.</span>—Size 50 × 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.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Wolves</td> <td class="tcl">value 2s. 6d.</td> <td class="tcl">to 64s.</td></tr> +<tr><td class="tcl">Dog wolves</td> <td class="tcl">value 1s.</td> <td class="tcl">to 2s. 6d.</td></tr> +</table> + +<p><span class="sc">Wolverine.</span>—Size 16 × 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 2½ 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.</p> + +<p><span class="sc">Wombat</span>, <span class="sc">Koala</span> or Australian Bear.—Size 20 × 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. 8½d.</p> +</div> + +<p><i>Preparing and Dressing.</i>—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, +<span class="pagenum"><a name="page354" id="page354"></a>354</span> +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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p><i>Manufacturing Methods and Specialities.</i>—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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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 +<span class="pagenum"><a name="page355" id="page355"></a>355</span> +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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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>i.e.</i> 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.</p> + +<p><i>Hatters’ Furs and Cloths and Shawls.</i>—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.</p> + +<p><i>Frauds and Imitations.</i>—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.</p> + +<p>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.</p> + +<p>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.</p> + +<div class="condensed"> +<p>The most usual misnaming of manufactured furs is as follow:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl cl">Musquash, pulled and dyed</td> <td class="tcl cl">Sold as seal.</td></tr> +<tr><td class="tcl">Nutria, pulled and dyed</td> <td class="tcl">Sold as seal.</td></tr> +<tr><td class="tcl cl">Nutria, pulled and natural</td> <td class="tcl cl">Sold as beaver.</td></tr> +<tr><td class="tcl">Rabbit, sheared and dyed</td> <td class="tcl">Sold as seal or electric seal.</td></tr> +<tr><td class="tcl cl">Otter, pulled and dyed</td> <td class="tcl cl">Sold as seal.</td></tr> +<tr><td class="tcl">Marmot, dyed</td> <td class="tcl">Sold as mink or sable.</td></tr> +<tr><td class="tcl cl">Fitch, dyed</td> <td class="tcl cl">Sold as sable.</td></tr> +<tr><td class="tcl">Rabbit, dyed</td> <td class="tcl">Sold as sable or French sable.</td></tr> +<tr><td class="tcl cl">Hare, dyed</td> <td class="tcl cl">Sold as sable, or fox, or lynx.</td></tr> +<tr><td class="tcl">Musquash, dyed</td> <td class="tcl">Sold as mink or sable.</td></tr> +<tr><td class="tcl cl">Wallaby, dyed</td> <td class="tcl cl">Sold as skunk.</td></tr> +<tr><td class="tcl">White Rabbit</td> <td class="tcl">Sold as ermine.</td></tr> +<tr><td class="tcl cl">White Rabbit, dyed</td> <td class="tcl cl">Sold as chinchilla.</td></tr> +<tr><td class="tcl">White Hare, dyed or natural</td> <td class="tcl">Sold as fox, foxaline, and other similar names.</td></tr> +<tr><td class="tcl cl">Goat, dyed</td> <td class="tcl cl">Sold as bear, leopard, &c.</td></tr> +<tr><td class="tcl">Dyed manufactured articles of all kinds</td> <td class="tcl">Sold as “natural.”</td></tr> +<tr><td class="tcl cl">White hairs inserted in foxes and sables</td> <td class="tcl cl">Sold as real or natural furs.</td></tr> +<tr><td class="tcl">Kids</td> <td class="tcl">Sold as lamb or broadtails.</td></tr> +<tr><td class="tcl cl">American sable</td> <td class="tcl cl">Sold as real Russian sable.</td></tr> +<tr><td class="tcl">Mink</td> <td class="tcl">Sold as sable.</td></tr> +</table> +</div> + +<p><i>The Preservation of Furs.</i>—For many years raw sealskins +<span class="pagenum"><a name="page356" id="page356"></a>356</span> +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.</p> + +<p>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.</p> + +<div class="condensed"> +<p class="pt2 center"><i>Comparative Durability of Various Furs and Weight of Unlined +Skins per Square Foot.</i></p> + +<p>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:—</p> + +<p class="pt1 center"><i>The Precious Furs.</i></p> + +<table class="ws" summary="Contents"> + +<tr><td class="tccm allb"> </td> <td class="tccm allb">Points of<br />Durability.</td> <td class="tccm allb"><a name="fa2h" id="fa2h" href="#ft2h"><span class="sp">2</span></a> Weight<br />in oz. per<br />sq. ft.</td></tr> + +<tr><td class="tcl lb rb">Sable</td> <td class="tcr rb">60</td> <td class="tcc rb">2½</td></tr> +<tr><td class="tcl lb rb">Sea</td> <td class="tcr rb">75</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Fox, Silver or Black</td> <td class="tcr rb">40</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Fox, White</td> <td class="tcr rb">20</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Ermine</td> <td class="tcr rb">25</td> <td class="tcc rb">1¼</td></tr> +<tr><td class="tcl lb rb">Chinchilla</td> <td class="tcr rb">15</td> <td class="tcc rb">1½</td></tr> +<tr><td class="tcl lb rb bb">Sea-otter (for stoles or collars)</td> <td class="tcr rb bb">100</td> <td class="tcc rb bb">4¼</td></tr> +</table> + + +<p class="pt1 center"><i>The Less Valuable Furs.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb">Points of<br />Durability.</td> <td class="tccm allb">Weight<br />in oz. per<br />sq. ft.</td></tr> + +<tr><td class="tcl lb rb">Sable “topped,” <i>i.e.</i> top hairs coloured</td> <td class="tcr rb">55</td> <td class="tcc rb">2½</td></tr> +<tr><td class="tcl lb rb">Sable tinted, <i>i.e.</i> fur all coloured.</td> <td class="tcr rb">50</td> <td class="tcc rb">2½</td></tr> +<tr><td class="tcl lb rb">Baum Marten, natural</td> <td class="tcr rb">65</td> <td class="tcc rb">2¾</td></tr> +<tr><td class="tcl lb rb">Baum Marten, tinted</td> <td class="tcr rb">45</td> <td class="tcc rb">2¾</td></tr> +<tr><td class="tcl lb rb">Stone Marten</td> <td class="tcr rb">40</td> <td class="tcc rb">2¾</td></tr> +<tr><td class="tcl lb rb">Nutria</td> <td class="tcr rb">27</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Musquash, natural</td> <td class="tcr rb">37</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Musquash, water hairs removed, sheared and seal finished.</td> <td class="tcr rb">33</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Skunk</td> <td class="tcr rb">70</td> <td class="tcc rb">2¾</td></tr> +<tr><td class="tcl lb rb">Mink</td> <td class="tcr rb">70</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Lynx, natural</td> <td class="tcr rb">25</td> <td class="tcc rb">2¾</td></tr> +<tr><td class="tcl lb rb">Lynx, tinted black</td> <td class="tcr rb">20</td> <td class="tcc rb">2¾</td></tr> +<tr><td class="tcl lb rb">Marmot, tinted</td> <td class="tcr rb">10</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Fox, tinted black</td> <td class="tcr rb">25</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Fox, tinted blue</td> <td class="tcr rb">20</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Opossum</td> <td class="tcr rb">37</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Otter (with water hairs)</td> <td class="tcr rb">100</td> <td class="tcc rb">4</td></tr> +<tr><td class="tcl lb rb">Otter (water hairs removed)</td> <td class="tcr rb">95</td> <td class="tcc rb">3<span class="spp">15</span>⁄<span class="suu">16</span></td></tr> +<tr><td class="tcl lb rb">Beaver (water hairs cut level with fur)</td> <td class="tcr rb">90</td> <td class="tcc rb">4</td></tr> +<tr><td class="tcl lb rb">Beaver (water hairs removed)</td> <td class="tcr rb">85</td> <td class="tcc rb">3<span class="spp">15</span>⁄<span class="suu">16</span></td></tr> +<tr><td class="tcl lb rb">Moleskin</td> <td class="tcr rb">7</td> <td class="tcc rb">1¾</td></tr> +<tr><td class="tcl lb rb">Persian Lamb</td> <td class="tcr rb">65</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Grey Lamb</td> <td class="tcr rb">30</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Broadtail</td> <td class="tcr rb">15</td> <td class="tcc rb">2¼</td></tr> +<tr><td class="tcl lb rb">Caracul Kid</td> <td class="tcr rb">10</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Caracul Lamb</td> <td class="tcr rb">15</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Squirrel</td> <td class="tcr rb">25</td> <td class="tcc rb">1¾</td></tr> +<tr><td class="tcl lb rb">Hare</td> <td class="tcr rb">5</td> <td class="tcc rb">1¾</td></tr> +<tr><td class="tcl lb rb bb">Rabbit</td> <td class="tcr rb bb">5</td> <td class="tcc rb bb">2¼</td></tr> +</table> + +<p class="pt1 center"><i>Quantities of Fur needed, in Square Feet.</i></p> + +<p>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.</p> + +<table class="ws" summary="Contents"> +<tr><td> </td> <td class="tcc">Sq. Ft.<br />(approximate).</td></tr> + +<tr><td class="tcl cl">Straight stole ½ length (just below the waist line)</td> <td class="tcc cl">2¾</td></tr> +<tr><td class="tcl">Straight stole ¾ length (just below the knee)</td> <td class="tcc">3¾</td></tr> +<tr><td class="tcl cl">Stole, broad enough at the neck to cover the top of arm ¾ length</td> <td class="tcc cl">5</td></tr> +<tr><td class="tcl">The same, full length (to hem of skirt)</td> <td class="tcc">6</td></tr> +<tr><td class="tcl cl">Eton jacket, without collar</td> <td class="tcc cl">13</td></tr> +<tr><td class="tcl">Plain cape, 15 in. long</td> <td class="tcc">6½</td></tr> +<tr><td class="tcl cl">Deep cape, 30 in. long</td> <td class="tcc cl">15</td></tr> +<tr><td class="tcl">Full cape with broad stole front, ¾ length</td> <td class="tcc">15</td></tr> +<tr><td class="tcl cl">Inverness cape (to knee)</td> <td class="tcc cl">25</td></tr> +<tr><td class="tcl">Double-breasted, straight, semi-fitting coat, covering hips</td> <td class="tcc">16</td></tr> +<tr><td class="tcl cl">Double-breasted sacque jacket, 36 in. long, full sleeves</td> <td class="tcc cl">20</td></tr> +<tr><td class="tcl">Same, 30 in. long</td> <td class="tcc">18</td></tr> +<tr><td class="tcl cl">Same, 22 in. long</td> <td class="tcc cl">15</td></tr> +<tr><td class="tcl">Long, full, shawl cape with points at back and front, well below knee</td> <td class="tcc">15</td></tr> +<tr><td class="tcl cl">Shorter shawl cape</td> <td class="tcc cl">16</td></tr> +<tr><td class="tcl">Motoring or driving coat, ¾ length</td> <td class="tcc">22</td></tr> +<tr><td class="tcl cl">Motoring or driving coat, full length</td> <td class="tcc cl">27</td></tr> +</table> + +<p class="pt1 center"><i>Weight and Durability of Furs for Men’s Coat Linings.</i></p> + +<p>Otter with the water hairs removed, the strongest fur suited for +linings, is here taken as the standard.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb">Points of<br />Durability.</td> <td class="tccm allb">Weight<br />in oz. per<br />sq. ft.</td></tr> + +<tr><td class="tcl lb rb">Otter (the water hairs removed)</td> <td class="tcr rb">100</td> <td class="tcc rb">3<span class="spp">15</span>⁄<span class="suu">16</span></td></tr> +<tr><td class="tcl lb rb">Beaver (the water hairs removed)</td> <td class="tcr rb">90</td> <td class="tcc rb">3<span class="spp">15</span>⁄<span class="suu">16</span></td></tr> +<tr><td class="tcl lb rb">Mink</td> <td class="tcr rb">90</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Sealskin</td> <td class="tcr rb">75</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Raccoon</td> <td class="tcr rb">75</td> <td class="tcc rb">4½</td></tr> +<tr><td class="tcl lb rb">Persian lamb or astrachan</td> <td class="tcr rb">70</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Sable</td> <td class="tcr rb">65</td> <td class="tcc rb">2½</td></tr> +<tr><td class="tcl lb rb">Musquash</td> <td class="tcr rb">55</td> <td class="tcc rb">3½</td></tr> +<tr><td class="tcl lb rb">Nutria</td> <td class="tcr rb">40</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Grey Opossum</td> <td class="tcr rb">40</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Wallaby</td> <td class="tcr rb">30</td> <td class="tcc rb">3¾</td></tr> +<tr><td class="tcl lb rb">Squirrel</td> <td class="tcr rb">30</td> <td class="tcc rb">1¾</td></tr> +<tr><td class="tcl lb rb">Hamster</td> <td class="tcr rb">15</td> <td class="tcc rb">1¼</td></tr> +<tr><td class="tcl lb rb bb">Rabbit</td> <td class="tcr rb bb">10</td> <td class="tcc rb bb">2¼</td></tr> +</table> + +<p class="pt1 center"><i>Durability and Weight of Linings for Ladies’ Coats or Wraps.</i></p> + +<p>Sable gills, the strongest fur suited for ladies’ linings, is taken as +the standard.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb">Points of<br />Durability.</td> <td class="tccm allb">Weight<br />in oz. per<br />sq. ft.</td></tr> + +<tr><td class="tcl lb rb">Sable gills</td> <td class="tcr rb">100</td> <td class="tcc rb">2<span class="spp">7</span>⁄<span class="suu">8</span></td></tr> +<tr><td class="tcl lb rb">Sable</td> <td class="tcr rb">85</td> <td class="tcc rb">2½</td></tr> +<tr><td class="tcl lb rb">Sable paws</td> <td class="tcr rb">64</td> <td class="tcc rb">1<span class="spp">5</span>⁄<span class="suu">8</span></td></tr> +<tr><td class="tcl lb rb">Ermine</td> <td class="tcr rb">57</td> <td class="tcc rb">1¼</td></tr> +<tr><td class="tcl lb rb">Squirrel back</td> <td class="tcr rb">50</td> <td class="tcc rb">1¾</td></tr> +<tr><td class="tcl lb rb">Squirrel heads</td> <td class="tcr rb">36</td> <td class="tcc rb">2½</td></tr> +<tr><td class="tcl lb rb">Squirrel lock</td> <td class="tcr rb">21</td> <td class="tcc rb">1<span class="spp">3</span>⁄<span class="suu">16</span></td></tr> +<tr><td class="tcl lb rb">Hamster</td> <td class="tcr rb">10</td> <td class="tcc rb">1¼</td></tr> +<tr><td class="tcl lb rb bb">Rabbit</td> <td class="tcr rb bb">7</td> <td class="tcc rb bb">2¼</td></tr> +</table> + +<p class="pt1 center"><i>Durability and Weight of Motoring Furs made up with Fur outside.</i></p> + +<p>Otter with the water hairs, the strongest fur suited for motoring +garments, is taken as the standard.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb">Points of<br />Durability.</td> <td class="tccm allb">Weight<br />in oz. per<br />sq. ft.</td></tr> + +<tr><td class="tcl lb rb">Otter (with water hairs)</td> <td class="tcr rb">100</td> <td class="tcc rb">4</td></tr> +<tr><td class="tcl lb rb">Sealskin, marble</td> <td class="tcr rb">80</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">”Hair Sealskin” (tinted) with water hairs (a special variety of seal)</td> <td class="tcr rb">75</td> <td class="tcc rb">3¼</td></tr> +<tr><td class="tcl lb rb">Raccoon</td> <td class="tcr rb">65</td> <td class="tcc rb">4½</td></tr> +<tr><td class="tcl lb rb bb">Russian Pony</td> <td class="tcr rb bb">35</td> <td class="tcc rb bb">2<span class="spp">5</span>⁄<span class="suu">8</span></td></tr> +</table> + +<p><span class="pagenum"><a name="page357" id="page357"></a>357</span></p> + +<p class="pt1 center"><i>Durability and Weight of Furs for Rugs and Foot-sacks.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb">Points of<br />Durability.</td> <td class="tccm allb">Weight<br />in oz. per<br />sq. ft.</td></tr> + +<tr><td class="tcl lb rb">Wolverine</td> <td class="tcr rb">100</td> <td class="tcc rb">6</td></tr> +<tr><td class="tcl lb rb">Bear (black or brown natural)</td> <td class="tcr rb">94</td> <td class="tcc rb">7</td></tr> +<tr><td class="tcl lb rb">Bear (tinted black)</td> <td class="tcr rb">88</td> <td class="tcc rb">7½</td></tr> +<tr><td class="tcl lb rb">Beaver</td> <td class="tcr rb">88</td> <td class="tcc rb">4</td></tr> +<tr><td class="tcl lb rb">Raccoon</td> <td class="tcr rb">77</td> <td class="tcc rb">4½</td></tr> +<tr><td class="tcl lb rb">Opossum</td> <td class="tcr rb">61</td> <td class="tcc rb">3</td></tr> +<tr><td class="tcl lb rb">Wolf</td> <td class="tcr rb">50</td> <td class="tcc rb">6½</td></tr> +<tr><td class="tcl lb rb">Jackal</td> <td class="tcr rb">27</td> <td class="tcc rb">4½</td></tr> +<tr><td class="tcl lb rb">Australian Bear</td> <td class="tcr rb">16</td> <td class="tcc rb">6</td></tr> +<tr><td class="tcl lb rb bb">Goat</td> <td class="tcr rb bb">11</td> <td class="tcc rb bb">4<span class="spp">1</span>⁄<span class="suu">6</span></td></tr> +</table> + +<p>Wolverine, the strongest fur suited for rugs and foot-sacks, is +taken as the standard.</p> + +<p>For a rug about 20 to 25 sq. ft. of fur are needed, for a foot-sack +14½.</p> +</div> +<div class="author">(W. S. P.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1h" id="ft1h" href="#fa1h"><span class="fn">1</span></a> The measurements given are from nose to root of tail of average +large sizes after the dressing process, which has a shrinking tendency. +The depths of fur quoted are the greatest, but there are plenty of +good useful skins possessing a lesser depth.</p> + +<p><a name="ft2h" id="ft2h" href="#fa2h"><span class="fn">2</span></a> Stout, old-fashioned boxcloth is almost the only cloth that +(after a soft, heavy lining has been added to it) affords even two-thirds +as much protection against cold as does fur. It weighs +4.273 oz. per sq. ft. more than the heaviest of coat-furs, and is so +rigid as to be uncomfortable, while the subtileness of fur makes it +“kind” to the body.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FURAZANES<a name="ar78" id="ar78"></a></span> (<i>furo</i>—a.a′—<i>diazoles</i>), organic compounds obtained +by heating the glyoximes (dioximes of ortho-diketones) +with alkalis or ammonia. Dimethylfurazane is prepared by +heating dimethylglyoxime with excess of ammonia for six hours +at 165° C. (L. Wolff, <i>Ber.</i>, 1895, 28, p. 70). It is a liquid (at +ordinary temperature) which boils at 156° C. (744 mm.). +Potassium permanganate oxidizes it first to methylfurazane-carboxylic +acid and then to furazanedicarboxylic acid. Methyl-ethylfurazane +and diphenylfurazane are also known. By +warming oxyfurazane acetic acid with excess of potassium permanganate +to 100° C. oxyfurazanecarboxylic acid is obtained +(A. Hantzsch and J. Urbahn, <i>Ber.</i>, 1895, 28, p. 764). It crystallizes +in prisms, which melt at 175° C. Furazanecarboxylic +acid is prepared by the action of a large excess of potassium +permanganate on a hot solution of furazanepropionic acid. +It melts at 107º C, and dissolves in caustic soda, with a deep +yellow colour and formation of nitrosocyanacetic acid (L. Wolff +and P.F. Ganz, <i>Ber.</i>, 1891, 24, p. 1167). Furoxane is an oxide +of furazane, considered by H. Wieland to be identical with +glyoxime peroxide; Kekulés dibromnitroacetonitrile is dibromfuroxane.</p> + +<p>The formulae of the compounds above mentioned are:</p> + +<div class="center pt2"><img style="width:550px; height:74px; vertical-align: middle;" src="images/img357a.jpg" alt="" /></div> + + + +<hr class="art" /> +<p><span class="bold">FURETIÈRE, ANTOINE<a name="ar79" id="ar79"></a></span> (1619-1688), French scholar and +miscellaneous writer, was born in Paris on the 28th of December +1619. He first studied law, and practised for a time as an +advocate, but eventually took orders and after various preferments +became abbé of Chalivoy in the diocese of Bourges in +1662. In his leisure moments he devoted himself to letters, and +in virtue of his satires—<i>Nouvelle Allégorique, ou histoire des +derniers troubles arrivés au royaume d’éloquence</i> (1658); <i>Voyage de +Mercure</i> (1653)—he was admitted a member of the French +Academy in 1662. That learned body had long promised a +complete dictionary of the French tongue; and when they +heard that Furetière was on the point of issuing a work of a +similar nature, they interfered, alleging that he had purloined +from their stores, and that they possessed the exclusive privilege +of publishing such a book. After much bitter recrimination +on both sides the offender was expelled in 1685; but for this +act of injustice he took a severe revenge in his satire, <i>Couches +de l’académie</i> (Amsterdam, 1687). His <i>Dictionnaire universel</i> +was posthumously published in 1690 (Rotterdam, 2 vols.). +It was afterwards revised and improved by the Protestant +jurist, Henri Basnage de Beauval (1656-1710), who published his +edition (3 vols.) in 1701; and it was only superseded by the +compilation known as the <i>Dictionnaire de Trévoux</i> (Paris, 3 vols., +1704; 7th ed., 8 vols., 1771), which was in fact little more than a +reimpression of Basnage’s edition. Furetière is perhaps even +better known as the author of <i>Le Roman bourgeois</i> (1666). It +cast ridicule on the fashionable romances of Mlle de Scudéry +and of La Calprenède, and is of interest as descriptive of the +everyday life of his times. There is no element of burlesque, +as in Scarron’s <i>Roman comique</i>, but the author contents himself +with stringing together a number of episodes and portraits, +obviously drawn from life, without much attempt at sequence. +The book was edited in 1854 by Edward Fournier and Charles +Asselineau and by P. Jannet.</p> + +<div class="condensed"> +<p>The <i>Fureteriana</i>, which appeared in Paris eight years after +Furetière’s death, which took place on the 14th of May 1688, is a +collection of but little value.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FURFOOZ<a name="ar80" id="ar80"></a></span>, a village some 10 m. from Dinant in the Ardennes, +Belgium. Three caves containing prehistoric remains were here +excavated in 1872. Of these the <i>Trou de Frontal</i> is the most +famous. In it were found human skeletons with brachycephalic +skulls, associated with animal bones, those of the reindeer being +particularly plentiful. Among the skeletons was discovered +an oval vase of pottery. The Furfooz type of mankind is believed +to date from the close of the Quaternary age. G. de Mortillet +dates the type in the Robenhausen epoch of the Neolithic +period. His theory is that the bones are those of men of that +period buried in what had been a cave-dwelling of the Madelenian +epoch.</p> + + +<hr class="art" /> +<p><span class="bold">FURFURANE<a name="ar81" id="ar81"></a></span>, or <span class="sc">Furane</span>, C<span class="su">4</span>H<span class="su">4</span>O, a colourless liquid boiling +at 32° C., found in the distillation products of pine wood. It +was first synthetically prepared by H. Limpricht (<i>Ann.</i>, 1873, +165, p. 281) by distilling barium mucate with soda lime, pyromucic +acid C<span class="su">4</span>H<span class="su">3</span>O·CO<span class="su">2</span>H being formed, which, on further loss +of carbon dioxide, yielded furfurane. A. Henniger (<i>Ann. chim. +phys.</i>, 1886 [2], 7, p. 220), by distilling erthyrite with formic +acid, obtained a dihydrofurfurane</p> + +<p class="center">C<span class="su">4</span>H<span class="su">6</span>(OH)<span class="su">4</span> + 2H<span class="su">2</span>CO<span class="su">2</span> = C<span class="su">4</span>H<span class="su">6</span>O + CO + CO<span class="su">2</span> + 4H<span class="su">2</span>O,</p> + +<p class="noind">which, on treatment with phosphorus pentachloride, yielded +furfurane. Furfurane is insoluble in water and possesses a +characteristic smell. It does not react with sodium or with +phenylhydrazine, but yields dye-stuffs with isatin and phenanthrenequinone. +It reacts violently with hydrochloric acid, +producing a brown amorphous substance. Methyl and phenyl +derivatives have been prepared by C. Paal (<i>Ber.</i>, 1884, 17, p. +915). Paal prepared acetonyl acetophenone by condensing +sodium acetoacetate with phenacylbromide, and this substance +on dehydration yields αα′-phenylmethylfurfurane, the acetonyl +acetophenone probably reacting in the tautomeric “enolic” form,</p> + +<table class="math0" summary="math"> +<tr><td>CH<span class="su">3</span>·CO·CHNa·COOR + C<span class="su">6</span>H<span class="su">5</span>·CO·CH<span class="su">2</span>Br = + CH<span class="su">3</span>·CO·CH(CH<span class="su">2</span>COC<span class="su">6</span>H<span class="su">5</span>)·COOR.</td></tr> +</table> + +<p class="noind">This ester readily hydrolyses, and the acid formed yields acetonyl +acetophenone (by loss of carbon dioxide), which then on dehydration +yields the furfurane derivative, thus</p> + +<div class="center ptb2"><img style="width:550px; height:40px; vertical-align: middle;" src="images/img357b.jpg" alt="" /></div> + +<p class="noind">L. Knorr (<i>Ber.</i>, 1889, 22, p. 158) obtained diacetosuccinic ester +by condensing sodium acetoacetate with iodine, and by dehydrating +the ester he prepared αα′-dimethylfurfurane ββ′-dicarboxylic +acid (carbopyrotritaric acid), which on distillation +yields αα′-dimethylfurfurane as a liquid boiling at 94° C. Paal +also obtained this compound by using monochloracetone in the +place of phenacylbromide. By the distillation of mucic acid +or isosaccharic acid, furfurane-α-carboxylic acid (pyromucic +acid), C<span class="su">4</span>H<span class="su">3</span>O·CO<span class="su">2</span>H, is obtained; it crystallizes in needles or +leaflets, and melts at 134° C.</p> + +<p><i>Furfurol</i> (furol), C<span class="su">4</span>H<span class="su">3</span>O·CHO, is the aldehyde of pyromucic +acid, and is formed on distilling bran, sugar, wood and most +carbohydrates with dilute sulphuric acid, or by distilling +the pentoses with hydrochloric acid. It is a colourless liquid +which boils at 162° C., and is moderately soluble in water; +it turns brown on exposure to air and has a characteristic +aromatic smell. It shows all the usual properties of an aldehyde, +forming a bisulphite compound, an oxime and a hydrazone; +whilst it can be reduced to the corresponding furfuryl alcohol by +means of sodium amalgam, and oxidized to pyromucic acid by +means of silver oxide. It also shows all the condensation reactions +of benzaldehyde (<i>q.v.</i>); condensing with aldehydes +and ketones in the presence of caustic soda to form more +complex aldehydes and ketones with unsaturated side chains, +<span class="pagenum"><a name="page358" id="page358"></a>358</span> +such as furfuracrolein, C<span class="su">4</span>H<span class="su">3</span>O·CH:CH·CHO, and furfuracetone, +C<span class="su">4</span>H<span class="su">3</span>O·CH:CH·CO·CH<span class="su">3</span>. With alcoholic potassium cyanide +It changes to furoin, C<span class="su">4</span>H<span class="su">3</span>O·CHOH·CO·C<span class="su">4</span>H<span class="su">3</span>O, which can be +oxidized to furil, C<span class="su">4</span>H<span class="su">3</span>O·CO·CO·C<span class="su">4</span>H<span class="su">3</span>O, whilst alcoholic potash +converts it into furfuryl alcohol. With fatty acids and acid +anhydrides it gives the “Perkin” reaction (see <span class="sc"><a href="#artlinks">Cinnamic Acid</a></span>). +Furfurol is shown to have its aldehydic group in the <i>a</i> position, +by conversion into furfurpropionic acid, C<span class="su">4</span>H<span class="su">3</span>O·CH<span class="su">2</span>·CH<span class="su">2</span>·CO<span class="su">2</span>H, +which on oxidation by bromine water and subsequent reduction +of the oxidized product is converted into <i>n</i>-pimelic acid, +HO<span class="su">2</span>C(CH<span class="su">2</span>)<span class="su">5</span>CO<span class="su">2</span>H. Furfurol in minute quantities can be +detected by the red colour it forms with a solution of aniline +acetate.</p> + +<div class="condensed"> +<p>Furfurane—αα′-dicarboxylic acid or dehydromucic acid, +C<span class="su">4</span>H<span class="su">2</span>O(CO<span class="su">2</span>H)<span class="su">2</span>, is formed when mucic acid is heated with hydrochloric +acid at 100° C. On being heated, it loses carbon dioxide +and gives pyromucic acid. By digesting acetoacetic ester with +sodium succinate and acetic anhydride, methronic acid, C<span class="su">8</span>H<span class="su">8</span>O<span class="su">5</span>, +is obtained; for the constitution of this acid, see L. Knorr, <i>Ber.</i>, +1889, 22, p. 152, and R. Fittig, <i>Ann.</i>, 1889, 259, p. 166.</p> + +<p>Di- and tetrahydrofurfurane compounds are also known (see +A. Lipp, <i>Ber.</i>, 1889, 22, p. 1196; W.H. Perkin, junr. <i>Journ. Chem. +Soc.</i>, 1899, 57, p. 944; and S. Ruhemann, <i>ibid.</i>, 1896, 69, p. 1383).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FURIES<a name="ar82" id="ar82"></a></span> (Lat. <i>Furiae</i>, also called <span class="sc">Dirae</span>), in Roman mythology +an adaptation of the Greek Erinyes (<i>q.v.</i>), with whom they +are generally identical. A special aspect of them in Virgil is +that of agents employed by the higher gods to stir up mischief, +strife and hatred upon earth. Mention may here be made of +an old Italian deity Furina (or Furrina), whose worship fell +early into disuse, and who was almost forgotten in the time of +Varro. By the mythologists of Cicero’s time the name was +connected with the verb <i>furere</i> and the noun <i>furia</i>, which in the +plural (not being used in the singular in this sense) was accepted +as the equivalent of the Greek Erinyes. But it is more probably +related to <i>furvus</i>, <i>fuscus</i>, and signifies one of the spirits of darkness, +who watched over men’s lives and haunted their abodes. +This goddess had her own special priest, a grove across the Tiber +where Gaius Gracchus was slain, and a festival on the 25th of +July. Authorities differ as to the existence of more than one +goddess called Furina, and their identity with the Forinae +mentioned in two inscriptions found at Rome (<i>C.I.L.</i> vi. +422 and 10,200).</p> + + +<hr class="art" /> +<p><span class="bold">FURLONG<a name="ar83" id="ar83"></a></span> (from the O. Eng. <i>furlang</i>, <i>i.e.</i> “furrow-long”), +a measure of length, originally the length of a furrow in the +“common field” system. As the field in this system was +generally taken to be a square, 10 acres in extent, and as the +acre varied in different districts and at different times, the +“furlong” also varied. The side of a square containing 10 +statute acres is 220 yds. or 40 poles, which was the usually +accepted length of the furlong. This is also the length of <span class="spp">1</span>⁄<span class="suu">8</span>th of +the statute mile. “Furlong” was as early as the 9th century +used to translate the Latin <i>stadium</i>, <span class="spp">1</span>⁄<span class="suu">8</span>th of the Roman mile.</p> + + +<hr class="art" /> +<p><span class="bold">FURNACE<a name="ar84" id="ar84"></a></span>, a contrivance for the production and utilization +of heat by the combustion of fuel. The word is common to all +the Romance tongues, appearing in more or less modified forms +of the Latin <i>fornax</i>. But in all those languages the word has a +more extended meaning than in English, as it covers every +variety of heating apparatus; while here, in addition to furnaces +proper, we distinguish other varieties as <i>ovens</i>, <i>stoves</i> and <i>kilns</i>. +The first of these, in the form <i>Ofen</i>, is used in German as a general +term like the French <i>four</i>; but in English it has been restricted +to those apparatus in which only a moderate temperature, +usually below a red heat, is produced in a close chamber. Our +bakers’ ovens, hot-air ovens or stoves, annealing ovens for glass +or metal, &c., would all be called <i>fours</i> in French and <i>Öfen</i> in +German, in common with furnaces of all kinds. Stove, an +equivalent of oven, is from the German <i>Stube</i>, <i>i.e.</i> a heated room, +and is commonly so understood; but is also applied to open +fire-places, which appears to be somewhat of a departure from +the original signification.</p> + +<p>Furnaces are constructed according to many different patterns +with varying degrees of complexity in arrangement; but all +may be considered as combining three essential parts, namely, +the fire-place in which the fuel is consumed, the heated chamber, +laboratory, hearth or working bed, as it is variously called, +where the heat is applied to the special work for which the furnace +is designed, and the apparatus for producing rapid combustion +by the supply of air under pressure to the fire. In the simplest +cases the functions of two or more of these parts may be combined +into one, as in the smith’s forge, where the fire-place and heating +chamber are united, the iron being placed among the coals, only +the air for burning being supplied under pressure from a blowing +engine by a second special contrivance, the tuyere, tuiron, +twyer or blast-pipe; but in the more refined modern furnaces, +where great economy of fuel is an object, the different functions +are distributed over separate and distinct apparatus, the fuel +being converted into gas in one, dried in another, and heated +in a third, before arriving at the point of combustion in the +working chamber of the furnace proper.</p> + +<div class="condensed"> +<p>Furnaces may be classified according as the products of combustion +are employed (1) only for heating purposes, or (2) both for +heating and bringing about some chemical change. The furnaces +employed for steam-raising or for heating buildings are invariably +of the first type (see <span class="sc"><a href="#artlinks">Boiler</a></span> and <span class="sc"><a href="#artlinks">Heating</a></span>), while those employed +in metallurgy are generally of the second. The essential difference +in construction is that in the first class the substances heated do +not come into contact with either the fuel or the furnace gases, +whereas in the second they do. Metallurgical furnaces of the first +class are termed crucible, muffle or retort furnaces, and of the +second shaft and reverberatory furnaces. The following is a detailed +subdivision:—</p> + +<p>(1) Fuel and substance in contact.</p> + +<div class="list"> + <p>(<i>a</i>) Height of furnace greater than diameter = shaft furnaces.</p> +</div> + +<div class="list1"> + <p>(α) No blast = kilns.</p> + + <p>(β) With blast = blast furnaces.</p> +</div> + +<div class="list"> + <p>(<i>b</i>) Height not much greater than diameter = hearth furnaces.</p> +</div> + +<p>(2) Substance heated by products of combustion = reverberatory + furnaces.</p> + +<div class="list"> + <p>(<i>a</i>) Charge not melted = roasting or calcining furnaces.</p> + + <p>(<i>b</i>) Charge melted = melting furnaces.</p> +</div> + +<p>(3) Substance is not directly heated by the fuel or by the products + of combustion.</p> + +<div class="list"> + <p>(<i>a</i>) Heating chamber fixed and forming part of furnace = + muffle furnaces.</p> + + <p>(<i>b</i>) Crucible furnaces.</p> + + <p>(<i>c</i>) Retort furnaces.</p> +</div> + +<p>Another classification may be based upon the nature of the heating +agent, according as it is coal (or some similar combustible) oil, gas +or electricity. In this article the general principles of metallurgical +furnaces will be treated; the subject of gas- and oil-heated furnaces +is treated in the article Fuel, and of the electric furnace in the +article Electrometallurgy. For special furnaces reference should +be made to the articles on the industry concerned, <i>e.g.</i> <span class="sc"><a href="#artlinks">Glass</a></span>, <span class="sc"><a href="#artlinks">Gas</a></span>, +§ <span class="sc"><a href="#artlinks">Manufacture</a></span>, &c.</p> +</div> + +<p><i>Shaft, Blast and Hearth Furnaces.</i>—The blast furnace in its +simplest form is among the oldest, if not the oldest, of metallurgical +contrivances. In the old copper-smelting district of +Arabia Petraea, clay blast-pipes dating back to the earlier +dynasties of ancient Egypt have been found buried in slag heaps; +and in India the native smiths and iron-workers continue to use +furnaces of similar types. These, when reduced to their most +simple expression, are mere basin-shaped hollows in the ground, +containing ignited charcoal and the substances to be heated, +the fire being urged by a blast of air blown in through one or +more nozzles from a bellows at or near the top. They are +essentially the same as the smith’s forge. This class of furnace +is usually known as an open fire or hearth, and is represented in +a more advanced stage of development by the Catalan, German +and Walloon forges formerly used in the production of malleable +iron.</p> + +<table class="flt" style="float: right; width: 350px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:300px; height:272px" src="images/img359a.jpg" alt="" /></td></tr> +<tr><td class="caption1"> <span class="sc">Fig. 1.</span>—Elevation of Catalan +Forge.</td></tr></table> + +<div class="condensed"> +<p>Fig. 1 represents a Catalan forge. The cavity in the ground is +represented by a pit of square or rectangular section lined with +brick or stone of a kind not readily acted on by heat, about 1½ or +2 ft. deep, usually somewhat larger above than below, with a tuyere +or blast-pipe of copper penetrating one of the walls near the top, +with a considerable downward inclination, so that the air meets +the fuel some way down. In iron-smelting the ore is laid in a heap +upon the fuel (charcoal) filling up the hearth, and is gradually brought +to the metallic state by the reducing action of the carbon monoxide +formed at the tuyere. The metal sinks through the ignited fuel, +forming, in the hearth, a spongy mass or ball, which is lifted out by +the smelters at the end of each operation, and carried to the forge +hammer. The earthy matters form a fusible glass or slag melt, and +<span class="pagenum"><a name="page359" id="page359"></a>359</span> +collect at the lowest point of the hearth, whence they are removed +by opening a hole pierced through the front wall at the bottom. +The active portion of such a furnace is essentially that above the +blast-pipe, the function of the lower part being merely the collection +of the reduced metal; the fire may therefore be regarded as burning +in an unconfined space, with the waste of a large amount of its +heating power. By continuing the walls of the hearth above the +tuyere, into a shaft or stack either +of the same or some other section, +we obtain a furnace of increased +capacity, but with no greater +power of consuming fuel, in which +the material to be treated can be +heated up gradually by loading it +into the stack, alternately with +layers of fuel, the charge descending +regularly to the point of combustion, +and absorbing a proportion +of the heat of the flame +that went to waste in the open +fire. This principle is capable of +very wide extension, the blast +furnace being mainly limited in +height by the strength the column +of materials or “burden” has to +resist crushing, under the weight due to the head adopted, and the +power of the blowing engine to supply blast of sufficient density +to overcome the resistance of the closely packed materials to the +free passage of the spent gases. The consuming power of the +furnace or the rate at which it can burn the fuel supplied is measured +by the number of tuyeres and their section.</p> +</div> + +<p>The development of blast furnaces is practically the development +of iron-smelting. The profile has been very much varied +at different times. The earliest examples were square or rectangular +in horizontal section, but the general tendency of modern +practice is to substitute round sections, their construction being +facilitated by the use of specially moulded bricks which have +entirely superseded the sandstone blocks formerly used. The +vertical section, on the other hand, is subject to considerable +variation according to the work to which the furnace is applied. +Where the operation is simply one of fusion, as in the iron-founder’s +cupola, in which there is no very great change in volume +in the materials on their descent to the tuyeres, the stack is nearly +or quite straight-sided; but when, as is the case with the smelting +of iron ores with limestone flux, a large proportion of volatile +matter has to be removed in the process, a wall of varying +inclination is used, so that the body of the furnace is formed of +two dissimilar truncated cones, joined by their bases, the lower +one passing downwards into a short, nearly cylindrical, +position. For further consideration of this subject see <span class="sc"><a href="#artlinks">Iron +and Steel</a></span>.</p> + +<p><i>Hearth furnaces</i> are employed in certain metallurgical operations, +<i>e.g.</i> in the air-reduction process for smelting lead ores. +The principle is essentially that of the Catalan forge. Such +furnaces are very wasteful, and have little to recommend them +(see Schnabel, Metallurgy, 1905, vol. 1. p. 409).</p> + +<p><i>Reverberatory Furnaces.</i>—Blast furnaces are, from the intimate +contact between the burden to be smelted and the fuel, the least +wasteful of heat; but their use supposes the possibility of obtaining +fuel of good quality and free from sulphur or other substances +likely to deteriorate the metal produced. In all cases, therefore, +where it is desired to do the work out of contact with the solid +fuel, the operation of burning or heat-producing must be performed +in a special fire-place or combustion chamber, the body +of flame and heated gas being afterwards made to act upon the +surface of the material exposed in a broad thin layer in the +working bed or laboratory of the furnace by reverberation from +the low vaulted roof covering the bed. Such furnaces are known +by the general name of reverberatory or reverbatory furnaces, +also as air or wind furnaces, to distinguish them from those +worked with compressed air or blast.</p> + +<p>Originally the term cupola was used for the reverberatory +furnace, but in the course of time it has changed its meaning, +and is now given to a small blast furnace such as that used by +iron-founders—reverberatory smelting furnaces in the same +trade being called air furnaces.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:640px; height:365px" src="images/img359b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 2.</span>—Longitudinal section of Reverberatory Furnace.</td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:650px; height:384px" src="images/img359c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 3.</span>—Reverberatory Furnace (horizontal section).</td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:650px; height:715px" src="images/img359d.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 4.</span>—Reverberatory Furnace (elevation at flue end).</td></tr></table> + +<div class="condensed"> +<p>Figs. 2, 3 and 4 represent a reverberatory furnace such as is used +for the fusion of copper ores for regulus, and may be taken as generally +representing its class. The fire-place A is divided from the +working bed B by a low wall C known as the fire bridge, and at the +opposite end there is sometimes, though not invariably, a second +bridge of less height called the flue bridge D. A short diagonal flue +or up-take E conveys the current of spent flame to the chimney +F, which is of square section, diminishing by steps at two or three +different heights, and provided at the top with a covering plate or +damper G, which may be raised or lowered by a chain reaching to +the ground, and serves for regulating the speed of the exhaust gases, +and thereby the draught of air through the fire. Where several +furnaces are connected with the same chimney stack, the damper +takes the form of a sliding plate in the mouth of the connecting flue, +so that the draught in one may be modified without affecting the +others. The fire bridge is partially protected against the intense +<span class="pagenum"><a name="page360" id="page360"></a>360</span> +heat of the body of flame issuing through the fire arch by a passage +to which the air has free access. The material to be melted is +introduced into the furnace from the hoppers HH through the +charging holes in the roof. When melted the products separate on +the bed (which is made of closely packed sand or other infusible +substances), according to their density; the lighter earthy matters +forming an upper layer of slag are drawn out by the slag hole K at +the flue end into an iron wagon or bogie, while the metal subsides +to the bottom of the bed, and at the termination of the operation +is run out by the tap hole L into moulds or granulated into water. +The opposite opening M is the working door, through which the tool +for stirring the charge is introduced. It is covered by a plate +suspended to a lever, similar to that seen in the end elevation (fig. 4) +in front of the slag hole.</p> +</div> + +<p>According to the purposes to which they are applied, reverberatory +furnaces may be classed into two groups, namely, fusion +or melting furnaces, and calcining or wasting furnaces, also +called calciners. The former have a very extended application +in many branches of industry, being used by both founders and +smelters in the fusion of metals; in the concentration of poor +metallic compounds by fusion into regulus; in the reduction +of lead and tin ores; for refining copper and silver; and for +making malleable iron by the puddling processes and welding. +Calcining furnaces have a less extended application, being +chiefly employed in the conversion of metallic sulphides into +oxides by continued exposure to the action of air at a temperature +far below that of fusion, or into chlorides by roasting with common +salt. As some of these substances (for example, lead sulphide +and copper pyrites) are readily fusible when first heated, but +become more refractory as part of the sulphur is dissipated and +oxygen takes its place, it is important that the heat should be +very carefully regulated at first, otherwise the mass may become +clotted or fritted together, and the oxidizing effect of the air soon +ceases unless the fritted masses be broken small again. This is +generally done by making the bed of the furnace very long in +proportion to its breadth and to the fire-grate area, which may +be the more easily done as a not inconsiderable amount of heat +is given out during the oxidation of the ore—such increased +length being often obtained by placing two or even three working +beds one above the other, and allowing the flame to pass over them +in order from below upwards. Such calciners are used especially +in roasting zinc blende into zinc oxide, and in the conversion of +copper sulphides into chlorides in the wet extraction process. In +some processes of lead-smelting, where the minerals treated +contain sand, the long calciner is provided with a melting bottom +close to the fire-place, so that the desulphurized ore leaves the +furnace as a glassy slag or silicate, which is subsequently reduced +to the metallic state by fusion with fluxes in blast furnaces. +Reverberatory furnaces play an important part in the manufacture +of sodium carbonate; descriptions and illustrations are +given in the article <span class="sc"><a href="#artlinks">Alkali Manufacture</a></span>.</p> + +<p><i>Muffle, Crucible and Retort Furnaces.</i>—A third class of furnaces +is so arranged that the work is done by indirect heating; that +is, the material under treatment, whether subjected to calcination, +fusion or any other process, is not brought in contact either +with fuel or flame, but is raised to the proper temperature by +exposure in a chamber heated externally by the products of +combustion. These are known as muffle or chamber furnaces; +and by supposing the crucibles or retorts to represent similar +chambers of only temporary duration, the ordinary pot melting +air furnaces, and those for the reduction of zinc ores or the +manufacture of coal gas, may be included in the same category. +These are almost invariably air furnaces, though sometimes air +under pressure is used, as, for example, in the combustion of +small anthracitic coal, where a current of air from a fan-blower +is sometimes blown under the grate to promote combustion. +Types of muffle furnaces are figured in the article <span class="sc"><a href="#artlinks">Annealing, +Hardening and Tempering</a></span>.</p> + +<p><i>Furnace Materials.</i>—The materials used in the construction +of furnaces are divisible into two classes, namely, ordinary and +refractory or fire-resisting. The former are used principally as +casing, walls, pillars or other supporting parts of the structure, +and includes ordinary red or yellow bricks, clay-slate, granite +and most building stones; the latter are reserved for the parts +immediately in contact with the fuel and flame, such as the +lining of the fire-place, the arches, roof and flues, the lower part +if not the whole of the chimney lining in reverberatory furnaces, +and the whole of the internal walls of blast furnaces. Among +such substances are fireclay and firebricks, certain sandstones, +silica in the form of ganister, and Dinas stone and bricks, ferric +oxide and alumina, carbon (as coke and graphite), magnesia, +lime and chromium oxide—their relative importance being +indicated by their order, the last two or three indeed being only +of limited use.</p> + +<p>The most essential point in good fireclays, or in the bricks +or other objects made from them, is the power of resisting +fusion at the highest heat to which they may be exposed. This +supposes them to be free from metallic oxides forming easily +fusible compounds with silica, such as lime or iron, the presence +of the former even in comparatively small proportion being very +detrimental. As clays they must be sufficiently plastic to be +readily moulded, but at the same time possess sufficient stiffness +not to contract too strongly in drying, whereby the objects +produced would be liable to be warped or cracked before firing. +In most cases, however, the latter tendency is guarded against, +in making up the paste for moulding, by adding to the fresh +clay a certain proportion of burnt material of the same kind, +such as old bricks or potsherds, ground to a coarse powder. +Coke dust or graphite is used for the same purpose in crucible +making (see <span class="sc"><a href="#artlinks">Firebrick</a></span>).</p> + +<div class="condensed"> +<p>The most highly valued fireclays are derived from the Coal +Measures. Among the chief localities are the neighbourhood of +Stourbridge in Worcestershire and Stannington near Sheffield, +which supply most of the materials for crucibles used in steel and +brass melting, and the pots for glass houses; Newcastle-on-Tyne +and Glenboig near Glasgow, where heavy blast furnace and other +firebricks, gas retorts, &c., are made in large quantities. Coarse-grained +but very strong firebricks are also made of the waste of +china clay works.</p> + +<p>In Belgium the clay raised at Andenne is very largely used for +making retorts for zinc furnaces. The principal French fireclays +are derived from the Tertiary strata in the south, and more nearly +resemble porcelain clays than those of the Coal Measures. They +give wares of remarkably fine texture and surface, combined with +high refractory character.</p> + +<p>In Germany, Ips and Passau on the Danube, and Gross Almerode +in Hesse, are the best known localities producing fireclay goods, the +crucibles from the last-mentioned place, known as Hessian crucibles, +going all over the world. These, though not showing a great resistance +to extreme heat, are very slightly affected by sudden alternations +in heating, as they may be plunged cold into a strongly heated +furnace without cracking, a treatment to which French and Stourbridge +pots cannot be subjected with safety.</p> +</div> + +<p>Plumbago or graphite is largely used in the production of +crucibles, not in the pure state but in admixture with fireclay; +the proportion of the former varies with the quality from 25 to +nearly 50%. These are the most enduring of all crucibles, the +best lasting out 70 or 80 meltings in brass foundries, about 50 +with bronze, and 8 to 10 in steel-melting.</p> + +<p>Silica is used in furnace-building in the forms of sand, ganister, +a finely ground sandstone from the Coal Measures of Yorkshire, +and the analogous substance known as Dinas clay, which is +really nearly pure silica, containing at most about 2½% of bases. +Dinas clay is found at various places in the Vale of Neath in +South Wales, in the form of a loose disintegrated sandstone, +which is crushed between rollers, mixed with about 1% of lime, +and moulded into bricks that are fired in kilns at a very high +temperature. These bricks are specially used for the roof, fire +arches, and other parts subjected to intense heat in reverberatory +steel-melting furnaces, and, although infusible under +ordinary conditions, are often fairly melted by the heat without +fluxing or corrosion after a certain amount of exposure. Ganister, +a slightly plastic siliceous sand, is similarly used for the lining +of Bessemer steel converters; it is found in the neighbourhood +of Sheffield.</p> + +<p>Alumina as a refractory material is chiefly used in the form +of bauxite, but its applications are somewhat special. It has +been found to stand well for the linings of rotatory puddling +furnaces, where, under long-continued heating, it changes into +a substance as hard and infusible as natural emery. In the +<span class="pagenum"><a name="page361" id="page361"></a>361</span> +Paris Exhibition of 1878 bricks very hard and dense in character, +said to be of pure alumina, were exhibited by Muller & Co. of +Paris, as well as bricks of magnesia, the latter being specially +remarkable for their great weight. They are intended for use +at the extreme temperatures obtainable in steel furnaces, or +for the melting of platinum before the oxy-hydrogen blowpipe. +For the latter purpose, however, lime is generally used; but as +this substance has only small stability, it is usually bedded in a +casing of firebrick. Oxide of chromium and chrome iron ore +have been proposed as refractory crucible materials. The former +may be used as a bed for melting platinum in the same way as +lime or magnesia, without affecting the quality of the metal.</p> + +<p>Ferric oxide, though not strictly infusible, is largely used as a +protecting lining for furnaces in which malleable iron is made, +a portion of the ore being reduced and recovered in the process. +In an oxidizing atmosphere it is indifferent to silica, and therefore +siliceous bricks containing a considerable proportion of ferric +oxide, when used in flues of boilers, brewers’ coppers, &c. and +similar situations, are perfectly fire-resisting so long as the heated +gas contains a large proportion of unconsumed air. The red +firebricks known as Windsor bricks, which are practically +similar in composition to soft red sandstone, are of this character.</p> + +<p>The electric furnace has led to the discovery of several +important materials, which have been employed as furnace +linings. Carborundum (<i>q.v.</i>) was applied by Engels in 1899, +firebricks being washed with carborundum paste and then baked. +Siloxicon, a compound of carbon, silicon and oxygen, formed +from carbon and silica in the electric furnace, was patented by +E.G. Acheson in 1903. It is very refractory, and is applied by +mixing with water and some bond, such as sodium silicate or +gas-tar. An amorphous, soft silicon carbide, also formed in the +electric furnace, was patented by B. Talbot in 1899. For basic +linings, magnesia crystallized in the electric furnace is being +extensively used, replacing dolomite to some extent (see E. +Kilburn Scott, “Refractory Materials for Furnace Linings,” +<i>Faraday Soc.</i>, 1906, p. 289).</p> + +<div class="condensed"> +<p><i>Furnace Construction.</i>—In the construction of furnaces provision +has to be made for the unequal expansion of the different parts under +the effect of heat. This is especially necessary in the case of reverberatory +furnaces, which are essentially weak structures, and +therefore require to be bound together by complicated systems of +tie rods and uprights or buck staves. The latter are very commonly +made of old flat bottom rails, laid with the flat of the flange against +the wall. Puddling furnaces are usually entirely cased with iron +plates, and blast furnaces with hoops round each course of the stack, +or in those of thinner constructions the firebrick work is entirely +enclosed in a wrought iron casing or jacket. Such parts as may be +subjected to extreme heat and the fretting action of molten material, +as the tuyere and slag breasts of blast furnaces, and the fire bridges +and bed plates of reverberatory furnaces, are often made in cast +iron with double walls, a current of water or air being kept circulating +through the intermediate space. In this way the metal, owing to +its high conductivity and low specific heat as compared to that of +water, is kept at a temperature far below its melting point if the +water is renewed quickly enough. It is of course necessary in such +cases that the circulation shall be perfectly free, in order to prevent +the accumulation of steam under pressure in the interior of the +casting. This method has received considerable extension, notably +in furnace-smelting of iron ores containing manganese, where the +entire hearth is often completely water-cased, and in some lead +furnaces where no firebrick lining is used, the lower part of the +furnace stack being a mere double iron box cooled by water sufficiently +to keep a coating of slag adhering to the inner shell which +prevents the metal from being acted upon.</p> + +<p><i>Mechanical Furnaces.</i>—The introduction and withdrawal of the +charges in fusion furnaces is effected by gravitation, the solid masses +of raw ore, fuel and flux being thrown in at the top, and flowing +out of the furnace at the taphole or slag run at the bottom. Vertical +kilns, such as those used for burning limestone, are worked in a +similar manner—the raw stone going in at the top, and the burnt +product falling through holes in the bottom when allowed to do so. +With reverberatory calciners, however, where the work is done +upon a horizontal bed, a considerable amount of hand labour is +expended in raking out the charge when finished, and in drawing +slags from fusion furnaces; and more particularly in the puddling +process of refining iron the amount of manual exertion required is +very much greater. To diminish the item of expenditure on this +head, various kinds of mechanical furnaces have been adopted, all +of which can be classified under three heads of gravitating furnaces, +mechanical stirrers and revolving furnaces.</p> + +<p>1. In <i>gravitating furnaces</i> the bed is laid at a slope just within the +angle of repose of the charge, which is introduced at the upper end, +and is pushed down the slope by fresh material, when necessary, +in the contrary direction to the flame which enters at the lower end. +Gerstenhofer’s pyrites burner is a furnace of this class. It has a tall +vertical chamber heated from below, and traversed by numerous +narrow horizontal cross bars at different heights. The ore in fine +powder is fed in at the top, through a hopper, in a regular thin +stream, by a pair of rollers, and in falling lodges on the flats of the +bars, forming a talus upon each of the height corresponding to the +angle of rest of the material, which is, however, at short intervals +removed to lower levels by the arrival of fresh ore from above. In +this way a very large surface is exposed to the heat, and the ore, if +containing sufficient sulphur to maintain the combustion, is perfectly +burned when it arrives at the bottom; if, however, it is imperfectly +sized or damp, or if it contains much earthy matter, the result is +not very satisfactory. There are many other furnaces in which the +same principle is utilized.</p> + +<p>2. <i>Mechanical stirrers</i> constitute a second division of mechanical +furnaces, in which the labour of rabbling or stirring the charges is +performed by combinations of levers and wheel-work taking motion +from a rotating shaft, and more or less perfectly imitating the action +of hand labour. They are almost entirely confined to puddling +furnaces.</p> + +<p>3. <i>Revolving furnaces</i>, the third and most important division of +mechanical furnaces, are of two kinds. The first of these resemble +an ordinary reverberatory furnace by having a flat bed which, +however, has the form of a circular disk mounted on a central shaft, +and receives a slow movement of rotation from a water-wheel or +other motor, so that every part of the surface is brought successively +under the action of the fire, the charge being stirred and ultimately +removed by passing under a series of fixed scraper arms placed above +the surface at various points. Brunton’s calciner, used in the “burning” +of the pyritic minerals associated with tin ore, is a familiar +example of this type. The hearth may either rotate on an inclined +axis, so that the path of its surface is oblique to that of the flame, +or the working part may be a hollow cylinder, between the fireplace +and flue, with its axis horizontal or nearly so, whose inner surface +represents the working bed, mounted upon friction rollers, and +receiving motion from a special steam-engine by means of a central +belt of spur gearing. Furnaces of the second kind were first used in +alkali works for the conversion of sulphate into carbonate of sodium +in the process known as black ash fusion, but have since been applied +to other processes. As calciners they are used in tin mines and for +the chlorination of silver ores. Mechanical furnaces are figured in +the article <span class="sc"><a href="#artlinks">Alkali Manufacture</a></span>.</p> + +<p><i>Use of Heated Air.</i>—The calorific intensity of fuel is found to be +very considerably enhanced, if the combustion be effected with air +previously heated to any temperature between that of boiling water +and a dull red heat, the same effect being observed both with solid +and gaseous fuel. The latter, especially when brought to the burning +point at a high temperature, produces a heat that can be resisted +by the most refractory substances only, such as silica, alumina and +magnesia. This is attained in the regenerative furnace of Siemens, +detailed consideration of which belongs more properly to the subject +of iron.</p> + +<p><i>Economy of Waste Heat.</i>—In every system of artificial heating, the +amount of heat usefully applied is but a small proportion of that +developed by combustion. Even under the most advantageous +application, that of evaporation of water in a steam boiler where the +gases of the fire have to travel through a great length of flues bounded +by thin iron surfaces of great heat-absorbing capacity, the temperature +of the current at the chimney is generally much above that +required to maintain an active draught in the fireplace; and other +tubes containing water, often in considerable numbers, forming the +so-called fuel economizers, may often be interposed between the +boiler and the chimney with marked advantage as regards saving +of fuel. In reverberatory and air furnaces used in the different +operations of iron manufacture, where an extremely high temperature +has to be maintained in spaces of comparatively small extent, such +as the beds of puddling, welding and steel-melting furnaces, the +temperature of the exhaust gases is exceedingly high, and if allowed +to pass directly into the chimney they appear as a great body of +flame at the top. It is now general to save a portion of this heat by +passing the flame through flues of steam boilers, air-heating apparatus, +or both—so that the steam required for the necessary operations +of the forge and heated blast for the furnace itself may be obtained +without further expenditure of fuel. The most perfect method of +utilizing the waste heat hitherto applied is that of the Siemens regenerator, +in which the spent gases are made to travel through +chambers, known as regenerators or recuperators of heat, containing +a quantity of thin firebricks piled into a cellular mass so as to offer +a very large heat-absorbing surface, whereby their temperature is +very considerably reduced, and they arrive at the chimney at a heat +not exceeding 300 or 400 degrees. As soon as the bricks have become +red hot, the current is diverted to an adjacent chamber or pair of +chambers, and the acquired heat is removed by a current of cool +gas or air passing towards the furnace, where it arrives at a temperature +sufficiently high to ensure the greatest possible heating +effect in combustion.</p> + +<p><span class="pagenum"><a name="page362" id="page362"></a>362</span></p> + +<p>In iron-smelting blast furnaces the waste gases are of considerable +fuel value, and may render important services if properly applied. +Owing to the conditions of the work, which require the maintenance +of a sensibly reducing atmosphere, they contain a very notable +proportion of carbonic oxide, and are drawn off by large wrought iron +tubes near the top of the furnace and conveyed by branch pipes +to the different boilers and air-heating apparatus, which are now +entirely heated by the combustion of such gases, or mixed with air +and exploded in gas engines. Formerly they were allowed to burn +to waste at the mouth of a short chimney place above the furnace +top, forming a huge body of flame, which was one of the most +striking features of the Black Country landscape at night.</p> + +<p><i>Laboratory and Portable Furnaces.</i>—Small air-furnaces with hot +plates or sand bath flues were formerly much employed in chemical +laboratories, as well as small blast furnaces for crucibles heated with +charcoal or coke. The use of such furnaces has very considerably +diminished, owing to the general introduction of coal-gas for heating +purposes in laboratories, which has been rendered possible by the +invention of the Bunsen burner, in which the mixture of air and gas +giving the least luminous but most powerfully heating flame is +effected automatically by the effluent gas. These burners, or +modifications of them, have also been applied to muffle furnaces, +which are convenient when only a few assays have to be made—the +furnace being a mere clay shell and soon brought to a working +temperature; but the fuel is too expensive to allow of their being +used habitually or on a large scale. Petroleum, or rather the heavy +oils obtained in tar refineries, having an equal or superior heating +power to coal-gas, may also be used in laboratories for producing +high temperatures. The oil is introduced in a thin stream upon a +series of inclined and channelled bars, where it is almost immediately +volatilized and burnt by air flowing in through parallel orifices. +Furnaces of this kind may be used for melting cast iron or bronze +in small quantities, and were employed by H. Sainte Claire Deville +in experiments in the metallurgy of the platinum group of metals.</p> + +<p>Sefstrom’s blast furnace, used in Sweden for the assay of iron ores, +is a convenient form of portable furnace applied to melting in +crucibles. It consists of a sheet-iron cylinder about 8 or 9 in. in +diameter, within which is fixed one of smaller size lined with fireclay. +The space between the two cylinders serves as a heater and +distributor for the blast, which is introduced through the nozzle at +the bottom, and enters the furnace through a series of several small +tuyeres arranged round the inner lining. Charcoal is the fuel used, +and the crucibles stand upon the bottom of the clay lining. When +a large body of fuel is required, the cylinder can be lengthened by +an iron hoop which fits over the top ring. Deville’s portable blast +furnace is very similar in principle to the above, but the body of the +furnace is formed of a single cast iron cylinder lined with fireclay, +closed below by a cast iron plate perforated by a ring of small holes—a +hemispherical basin below forming the air-heating chamber.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FURNEAUX, TOBIAS<a name="ar85" id="ar85"></a></span> (1735-1781), English navigator, was +born at Swilly near Plymouth on the 21st of August 1735. He +entered the royal navy, and was employed on the French and +African coasts and in the West Indies during the latter part of the +Seven Years’ War (1760-1763). He served as second lieutenant +of the “Dolphin” under Captain Samuel Wallis on the latter’s +voyage round the globe (August 1766-May 1768); was made +a commander in November 1771; and commanded the “Adventure” +which accompanied Captain Cook (in the “Resolution”) +in Cook’s second voyage. On this expedition Furneaux +was twice separated from his leader (February 8-May 19, 1773; +October 22, 1773-July 14, 1774, the date of his return to +England). On the former occasion he explored a great part of +the south and east coasts of Tasmania, and made the earliest +British chart of the same. Most of his names here survive; +Cook, visiting this shore-line on his third voyage, confirmed +Furneaux’s account and delineation of it (with certain minor +criticisms and emendations), and named after him the islands +in Banks Straits, opening into Bass’s Straits, and the group now +known as the Low Archipelago. After the “Adventure” was +finally separated from the “Resolution” off New Zealand in +October 1773, Furneaux returned home alone, bringing with him +Omai of Ulaietea. This first South Sea Islander seen in the +British Isles returned to his home with Cook in 1776-1777. +Furneaux was made a captain in 1775, and commanded the +“Syren” in the British attack of the 28th of June 1776 upon +Charleston, South Carolina. His successful efforts to introduce +domestic animals and potatoes into the South Sea Islands are +worthy of note. He died at Swilly on the 19th of September +1781.</p> + +<div class="condensed"> +<p>See Hawkesworth’s <i>Narrative of Wallis’ Voyage</i>; Captain Cook’s +<i>Narrative of his Second Voyage</i>; also T. Furneaux’s life by Rev. +Henry Furneaux in the <i>Dictionary of National Biography</i>.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FURNES<a name="ar86" id="ar86"></a></span> (Flem. <i>Veurne</i>), an old-fashioned little town amid +the dunes near the coast in West Flanders, Belgium, about +26 m. S.W. of Bruges. Pop. (1904) 6099. It is the centre of a +considerable area extending to the French frontier, and its +market is an important one for the disposal of corn, stock, hops +and dairy produce. During the Norman raids Furnes was +destroyed, and the present town was built by Baldwin Bras de +Fer, first count of Flanders, about the year 870. At the height +of the prosperity of the Flemish communes in the 14th century +there were dependent on the barony of Furnes not fewer than +fifty-two rich villages, but these have all disappeared, partly +no doubt as the consequence of repeated French invasions down +to the end of the 18th century, but chiefly through the encroachment +of the sea followed by the accumulation of sand along the +whole of this portion of the coast. Furnes contains many +curious old houses and the church of St Walburga, which is a +fine survival of the 13th century with some older portions. The +old church and buildings, grouped round the Grand Place, which +is the scene of the weekly market, present a quaint picture +which is perhaps not to be equalled in the country. Near Furnes +on the seashore is the fashionable bathing place called La Panne.</p> + +<p>Furnes one day a year becomes a centre of attraction to all +the people of Flanders. This is the last Sunday in July, when the +fête of Calvary and the Crucifixion is celebrated. Of all popular +festivities in Belgium this is the nearest approach to the old +Passion Play. The whole story of Christ is told with great +precision by means of succeeding groups which typify the different +phases of the subject. The people of Furnes pose as Roman +soldiers or Jewish priests, as the apostles or mere spectators, +while the women put on long black veils so that they may figure +in the procession as the just women.</p> + + +<hr class="art" /> +<p><span class="bold">FURNESS, HORACE HOWARD<a name="ar87" id="ar87"></a></span> (1833-  ), American +Shakespearian scholar, was born in Philadelphia on the 2nd of +November 1833, being the son of William Henry Furness (1802-1896) +minister of the First Unitarian church in that city, a +powerful preacher and writer. He graduated at Harvard in +1854, and was admitted to the bar in 1859, but soon devoted +himself to the study of Shakespeare. He accumulated a collection +of illustrative material of great richness and extent, and brought +out in 1871 the first volume of a new Variorum edition, designed +to represent and summarize the conclusions of the best authorities +in all languages—textual, critical and annotative. The volumes +appeared as follows: <i>Romeo and Juliet</i> (1871); <i>Macbeth</i> (1873) +(revised edition, 1903); <i>Hamlet</i> (2 vols., 1877); <i>King Lear</i> +(1880); <i>Othello</i> (1886); <i>The Merchant of Venice</i> (1888); <i>As You +Like It</i> (1890); <i>The Tempest</i> (1892); <i>A Midsummer Night’s +Dream</i> (1895); <i>The Winter’s Tale</i> (1898); <i>Much Ado about +Nothing</i> (1899); <i>Twelfth Night</i> (1901); <i>Love’s Labour’s Lost</i> +(1904). The edition has been generally accepted as a thorough +and scholarly piece of work; its chief fault is that, beginning +with <i>Othello</i> (1858), the editor used the First Folio text as his +basis, while in others he makes the text of the Cambridge (Globe) +editors his foundation. His wife, Helen Kate Furness (1837-1883), +compiled <i>A Concordance to the Poems of Shakespeare</i> (1872).</p> + + +<hr class="art" /> +<p><span class="bold">FURNESS<a name="ar88" id="ar88"></a></span>, a district of Lancashire, England, separated from +the major portion of the county by Morecambe Bay. It is +bounded S.E. by this inlet of the Irish Sea, S.W. by the sea, +W. by the Duddon estuary and Cumberland, and N. and E. by +Westmorland. Its area is about 250 sq. m. It forms the greater +part of the North Lonsdale parliamentary division of Lancashire, +and contains the parliamentary borough of Barrow-in-Furness. +The surface is almost entirely hilly. The northern half is included +in the celebrated Lake District, and contains such eminences +as the Old Man of Coniston and Wetherlam. Apart from the +Duddon, which forms part of the western boundary, the principal +rivers are the Leven and Crake, flowing southward into a common +estuary in Morecambe Bay. The Leven drains Windermere +and the Crake Coniston Lake. The usage of the term “Lake +District,” however, tends to limit the name of Furness in common +thought to the district south of the Lakes, where several of the +place-names are suffixed with that of the district, as Barrow-in-Farness, +Dalton-in-Furness, Broughton-in-Furness. Between +<span class="pagenum"><a name="page363" id="page363"></a>363</span> +the Duddon and Morecambe Bay lies Walney Island, 8 m. in +length, and in the shallow strait between it and the mainland +are several smaller islands. That part of Furness which forms a +peninsula between the Leven estuary and Morecambe Bay, and +the Duddon estuary, is rich in hematite iron ore, which has been +worked from very early times. It was known and smelted by +British and Romans, and by the monks of Furness Abbey and +Conishead Priory, both in the district. It was owing to the +existence of this ore that the town of Barrow grew up in the 19th +century; at first as a port from which the ore was exported to +South Wales, while later furnaces were established on the spot, +and acquired additional importance on the introduction of the +Bessemer process, which requires a non-phosphoric ore such as +is found here. The hematite is also worked at Ulverston, Askam, +Dalton and elsewhere, but the furnaces now depend in part +upon ore imported from Spain. The supposed extension of the +ore under the sands of the Duddon estuary led to the construction +of a sea wall to facilitate the working. The district is served +by the main line of the Furness railway, from Carnforth (junction +with the London & North-Western railway), passing the pleasant +watering-place of Grange, and approximately following the +coast by Ulverston, Dalton and Barrow, with branches to Lake +Side, Windermere, and to Coniston.</p> + +<p>Apart from its industrial importance and scenic attractions, +Furness has an especial interest on account of its famous abbey. +The ruins of this, beautifully situated in a wooded +valley, are extensive, and mainly of fine transitional +<span class="sidenote">Furness Abbey.</span> +Norman and Early English date, acquiring additional +picturesqueness from the warm colour of the red sandstone +of which they are built. The abbey of Furness, otherwise +Furdenesia or the further <i>nese</i> (promontory), which was dedicated +to St Mary, was founded in 1127 by a small body of monks +belonging to the Benedictine order of Savigny. In 1124 they +had settled at Tulketh, near Preston, but migrated in 1127 to +Furness under the auspices of Stephen, count of Boulogne, +afterwards king, at that time lord of the liberty of Furness. +In 1148 the brotherhood joined the Cistercian order. Stephen +granted to the monks the lordship of Furness, and his charter +was confirmed by Henry I., Henry II. and subsequent kings. +The abbot’s power throughout the lordship was almost absolute; +he had a market and fair at Dalton, was free from service to the +county and wapentake, and held a sheriff’s tourn. By a succession +of gifts the abbey became one of the richest in England +and was the largest Cistercian foundation in the kingdom. At +the Dissolution its revenues amounted to between £750 and +£800 a year, exclusive of meadows, pastures, fisheries, mines, +mills and salt works, and the wealth of the monks enabled them +to practise a regal hospitality. The abbot was one of the twenty +Cistercian abbots summoned to the parliament of 1264, but was +not cited after 1330, as he did not hold of the king <i>in capite per +baroniam</i>. The abbey founded several offshoot houses, one of +the most important being Rushen Abbey in the Isle of Man. In +1535 the royal commissioners visited the abbey and reported +four of its inmates, including the abbot, for incontinence. In +1536 the abbot was charged with complicity in the Pilgrimage +of Grace, and on the 7th of April 1537, under compulsion, +surrendered the abbey to the king. A few monks were granted +pensions, and the abbot was endowed with the profits of the +rectory of Dalton, valued at £33, 6s. 8d. per annum. In 1540 +the estates and revenues were annexed by act of parliament to +the Duchy of Lancaster. About James I.’s reign the site and +territories were alienated to the Prestons of Preston-Patrick, +from whom they descended to the dukes of Devonshire.</p> + +<p>Conishead Priory, near Ulverston, an Augustinian foundation +of the reign of Henry II., has left no remains, but of the priory +of Cartmel (1188) the fine church is still in use. It is a cruciform +structure of transitional Norman and later dates, its central +tower having the upper storey set diagonally upon the lower. +The chancel contains some superb Jacobean carved oak screens, +with stalls of earlier date.</p> + + +<hr class="art" /> +<p><span class="bold">FURNISS, HARRY<a name="ar89" id="ar89"></a></span> (1854-  ), British caricaturist and +illustrator, was born at Wexford, Ireland, of English and Scottish +parents. He was educated in Dublin, and in his schooldays +edited a <i>Schoolboy’s Punch</i> in close imitation of the original. +He came to London when he was nineteen, and began to draw +for the illustrated papers, being for some years a regular contributor +to the <i>Illustrated London News</i>. His first drawing in <i>Punch</i> +appeared in 1880, and he joined its staff in 1884. He illustrated +Lucy’s “Diary of Toby, M.P.,” in <i>Punch</i>, where his political +caricatures became a popular feature. Among his other successes +were a series of “Puzzle Heads,” and his annual “Royal +Academy guy’d.” In <i>Royal Academy Antics</i> (1890) he published +a volume of caricatures of the work of leading artists. He +resigned from the staff of <i>Punch</i> in 1894, produced for a short +time a weekly comic paper <i>Lika Joko</i>, and in 1898 began a +humorous monthly, <i>Fair Game</i>; but these were short-lived. +Among the numerous books he illustrated were James Payn’s +<i>Talk of the Town</i>, Lewis Carroll’s <i>Sylvie and Bruno</i>, Gilbert à +Beckett’s <i>Comic Blackstone</i>, G.E. Farrow’s <i>Wallypug Book</i>, +and his own novel, <i>Poverty Bay</i> (1905). <i>Our Joe, his great Fight</i> +(1903), was a collection of original cartoons. His volume of +reminiscences, <i>Confessions of a Caricaturist</i> (1901), was followed +by <i>Harry Furniss at Home</i> (1904). In 1905 he published <i>How to +draw in Pen and Ink</i>, and produced the first number of <i>Harry +Furniss’s Christmas Annual</i>.</p> + + +<hr class="art" /> +<p><span class="bold">FURNITURE<a name="ar90" id="ar90"></a></span> (from “furnish,” Fr. <i>fournir</i>), a general term +of obscure origin, used to describe the chattels and fittings required +to adapt houses and other buildings for use. Wood, +ivory, precious stones, bronze, silver and gold have been used +from the most ancient times in the construction or for the +decoration of furniture. The kinds of objects required for +furniture have varied according to the changes of manners and +customs, as well as with reference to the materials at the command +of the workman, in different climates and countries. +Of really ancient furniture there are very few surviving examples, +partly by reason of the perishable materials of which it was usually +constructed; and partly because, however great may have been +the splendour of Egypt, however consummate the taste of Greece, +however luxurious the life of Rome, the number of household +appliances was very limited. The chair, the couch, the table, +the bed, were virtually the entire furniture of early peoples, +whatever the degree of their civilization, and so they remained +until the close of what are known in European history as the +middle ages. During the long empire-strewn centuries which +intervened between the lapse of Egypt and the obliteration of +Babylon, the extinction of Greece and the dismemberment of +Rome and the great awakening of the Renaissance, household +comfort developed but little. The Ptolemies were as well lodged +as the Plantagenets, and peoples who spent their lives in the +open air, going to bed in the early hours of darkness, and rising +as soon as it was light, needed but little household furniture.</p> + +<p>Indoor life and the growth of sedentary habits exercised a +powerful influence upon the development of furniture. From +being splendid, or at least massive, and exceedingly sparse and +costly, it gradually became light, plentiful and cheap. In the +ancient civilizations, as in the periods when our own was slowly +growing, household plenishings, save in the rudest and most +elementary forms, were the privilege of the great—no person +of mean degree could have obtained, or would have dared to +use if he could, what is now the commonest object in every +house, the chair (<i>q.v.</i>). Sparse examples of the furniture of +Egypt, Nineveh, Greece and Rome are to be found in museums; +but our chief sources of information are mural and sepulchral +paintings and sculptures. The Egyptians used wooden furniture +carved and gilded, covered with splendid textiles, and supported +upon the legs of wild animals; they employed chests and coffers +as receptacles for clothes, valuables and small objects generally. +Wild animals and beasts of the chase were carved upon the +furniture of Nineveh also; the lion, the bull and the ram were +especially characteristic. The Assyrians were magnificent in +their household appointments; their tables and couches were +inlaid with ivory and precious metals. Cedar and ebony were +much used by these great Eastern peoples, and it is probable that +they were familiar with rosewood, walnut and teak. Solomon’s +<span class="pagenum"><a name="page364" id="page364"></a>364</span> +bed was of cedar of Lebanon. Greek furniture was essentially +Oriental in form; the more sumptuous varieties were of bronze, +damascened with gold and silver. The Romans employed Greek +artists and workmen and absorbed or adapted many of their +mobiliary fashions, especially in chairs and couches. The Roman +tables were of splendid marbles or rare woods. In the later +ages of the empire, in Rome and afterwards in Constantinople, +gold and silver were plentifully used in furniture; such indeed +was the abundance of these precious metals that even cooking +utensils and common domestic vessels were made of them.</p> + +<p>The architectural features so prominent in much of the +medieval furniture begin in these Byzantine and late Roman +thrones and other seats. These features became paramount as +Pointed architecture became general in Europe, and scarcely +less so during the Renaissance. Most of the medieval furniture, +chests, seats, trays, &c., of Italian make were richly gilt and +painted. In northern Europe carved oak was more generally +used. State seats in feudal halls were benches with ends carved +in tracery, backs panelled or hung with cloths (called cloths of +estate), and canopies projecting above. Bedsteads were square +frames, the testers of panelled wood, resting on carved posts. +Chests of oak carved with panels of tracery, or of Italian cypress +(when they could be imported), were used to hold and to carry +clothes, tapestries, &c., to distant castles and manor houses; +for house furniture, owing to its scarcity and cost, had to be +moved from place to place. Copes and other ecclesiastical +vestments were kept in chests with ornamental lock plates and +iron hinges. The splendour of most feudal houses depended +on pictorial tapestries which could be packed and carried from +place to place. Wardrobes were rooms fitted for the reception +of dresses, as well as for spices and other valuable stores. Excellent +carving in relief was executed on caskets, which were of +wood or of ivory, with painting and gilding, and decorated with +delicate hinge and lock metal-work. The general subjects of +sculpture were taken from legends of the saints or from metrical +romances. Renaissance art made a great change in architecture, +and this change was exemplified in furniture. Cabinets (<i>q.v.</i>) and +panelling took the outlines of palaces and temples. In Florence, +Rome, Venice, Milan and other capitals of Italy, sumptuous +cabinets, tables, chairs, chests, &c., were made to the orders +of the native princes. Vasari (<i>Lives of Painters</i>) speaks of +scientific diagrams and mathematical problems illustrated in +costly materials, by the best artists of the day, on furniture made +for the Medici family. The great extent of the rule of Charles V. +helped to give a uniform training to artists from various countries +resorting to Italy, so that cabinets, &c., which were made in +vast numbers in Spain, Flanders and Germany, can hardly be +distinguished from those executed in Italy. Francis I. and +Henry VIII. encouraged the revived arts in their respective +dominions. <i>Pietra dura</i>, or inlay of hard pebbles, agate, lapis +lazuli, and other stones, ivory carved and inlaid, carved and gilt +wood, marquetry or veneering with thin woods, tortoise-shell, +brass, &c., were used in making sumptuous furniture during the +first period of the Renaissance. Subjects of carving or relief +were generally drawn from the theological and cardinal virtues, +from classical mythology, from the seasons, months, &c. Carved +altarpieces and woodwork in churches partook of the change in +style.</p> + +<p>The great period of furniture in almost every country was, +however, unquestionably the 18th century. That century saw +many extravagances in this, as in other forms of art, but on the +whole it saw the richest <i>floraison</i> of taste, and the widest sense +of invention. This is the more remarkable since the furniture +of the 17th century has often been criticized as heavy and coarse. +The criticism is only partly justified. Throughout the first three-quarters +of the period between the accession of James I. and +that of Queen Anne, massiveness and solidity were the distinguishing +characteristics of all work. Towards the reign of +James II., however, there came in one of the most pleasing and +elegant styles ever known in England. Nearly a generation +before then Boulle was developing in France the splendid and +palatial method of inlay which, although he did not invent it, +is inseparably associated with his name. We owe it perhaps to +the fact that France, as the neighbour of Italy, was touched +more immediately by the Renaissance than England that the +reign of heaviness came earlier to an end in that country than on +the other side of the Channel. But there is a heaviness which is +pleasing as well as one which is forbidding, and much of the +furniture made in England any time after the middle of the +17th century was highly attractive. If English furniture of +the Stuart period be not sought after to the same extent as that +of a hundred years later, it is yet highly prized and exceedingly +decorative. Angularity it often still possessed, but generally +speaking its elegance of form and richness of upholstering lent +it an attraction which not long before had been entirely lacking. +Alike in France and in England, the most attractive achievements +of the cabinetmaker belong to the 18th century—English Queen +Anne and early Georgian work is universally charming; the +regency and the reigns of Louis XV. and XVI. formed a period +of the greatest artistic splendour. The inspiration of much of +the work of the great English school was derived from France, +although the gropings after the Chinese taste and the earlier +Gothic manner were mainly indigenous. The French styles of the +century, which began with excessive flamboyance, closed before +the Revolution with a chaste perfection of detail which is perhaps +more delightful than anything that has ever been done in +furniture. In the achievements of Riesener, David Röntgen, +Gouthière, Oeben and Rousseau de la Rottière we have the high-water +mark of craftsmanship. The marquetry of the period, +although not always beautiful in itself, was executed with +extraordinary smoothness and finish; the mounts of gilded +bronze, which were the leading characteristic of most of the work +of the century, were finished with a minute delicacy of touch +which was until then unknown, and has never been rivalled since. +If the periods of Francis I. and Henry II., of Louis XIV. and +the regency produced much that was sumptuous and even elegant, +that of Louis XVI., while men’s minds were as yet undisturbed +by violent political convulsions, stands out as, on the whole, +the one consummate era in the annals of furniture. Times of +great achievement are almost invariably followed directly by +those in which no tall thistles grow and in which every little +shrub is magnified to the dimensions of a forest tree; and the +so-called “empire style” which had begun even while the last +monarch of the <i>ancien régime</i> still reigned, lacked alike the graceful +conception and the superb execution of the preceding style. +Heavy and usually uninspired, it was nurtured in tragedy and +perished amid disaster. Yet it is a profoundly interesting style, +both by reason of the classical roots from which it sprang and +the attempt, which it finally reflected, to establish new ideas in +every department of life. Founded upon the wreck of a lingering +feudalism it reached back to Rome and Greece, and even to +Egypt. If it is rarely charming, it is often impressive by its +severity. Mahogany, satinwood and other rich timbers were +characteristic of the style of the end of the 18th century; +rosewood was most commonly employed for the choicer work +of the beginning of the 19th. Bronze mounts were in high +favour, although their artistic character varied materially.</p> + +<p>Previously to the middle of the 18th century the only cabinetmaker +who gained sufficient personal distinction to have had +his name preserved was André Charles Boulle; beginning with +that period France and England produced many men whose +renown is hardly less than that of artists in other media. With +Chippendale there arose a marvellously brilliant school of English +cabinetmakers, in which the most outstanding names are those +of Sheraton, Heppelwhite, Shearer and the Adams. But if the +school was splendid it was lamentably short-lived, and the 19th +century produced no single name in the least worthy to be +placed beside these giants. Whether, in an age of machinery, +much room is left for fine individual execution may be doubted, +and the manufacture of furniture now, to a great extent, takes +place in large factories both in England and on the continent. +Owing to the necessary subdivision of labour in these +establishments, each piece of furniture passes through numerous +distinct workshops. The master and a few artificers formerly +<span class="pagenum"><a name="page365" id="page365"></a>365</span> +superintended each piece of work, which, therefore, was never +far removed from the designer’s eye. Though accomplished +artists are retained by the manufacturers of London, Paris and +other capitals, there can no longer be the same relation between +the designer and his work. Many operations in these modern +factories are carried on by machinery. This, though an economy +of labour, entails loss of artistic effect. The chisel and the knife +are no longer in such cases guided and controlled by the sensitive +touch of the human hand.</p> + +<p class="pt2 noind f90 sc">Plate I.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:183px; height:255px" src="images/img364a.jpg" alt="" /></td> +<td class="figcenter"><img style="width:179px; height:254px" src="images/img364b.jpg" alt="" /></td> +<td class="figcenter"><img style="width:179px; height:249px" src="images/img364c.jpg" alt="" /></td> +<td class="figcenter"><img style="width:180px; height:250px" src="images/img364c1.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 1.</span>—Venetian Folding Chair of +carved and gilt walnut, leather +back and seat; about 1530.</td> +<td class="caption"><span class="sc">Fig. 2.</span>—Oak Arm-chair. English, +17th century.</td> +<td class="caption"><span class="sc">Fig. 3.</span>—Arm-chair, solid seat, cane +back; about 1660.</td> +<td class="caption"><span class="sc">Fig. 4.</span>—Arm-chair, stuffed back and +seat; about 1650.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:114px; height:258px" src="images/img364d.jpg" alt="" /></td> +<td class="figcenter"><img style="width:352px; height:262px" src="images/img364e.jpg" alt="" /></td> +<td class="figcenter"><img style="width:122px; height:260px" src="images/img364f.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 5.</span>—Painted and carved High-Back +Chair; about 1660.</td> +<td class="caption"><span class="sc">Fig. 6.</span>—Carved Walnut Chairs. English, early 18th century. +The arm-chair is inlaid.</td> +<td class="caption"><span class="sc">Fig. 7.</span>—Walnut Chair; about 1710.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:166px; height:254px" src="images/img364g.jpg" alt="" /></td> +<td class="figcenter"><img style="width:194px; height:252px" src="images/img364h.jpg" alt="" /></td> +<td class="figcenter"><img style="width:161px; height:250px" src="images/img364i.jpg" alt="" /></td> +<td class="figcenter"><img style="width:160px; height:252px" src="images/img364j.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 8.</span>—Carved Mahogany Chair +in the style of Chippendale; 2nd +half of 18th century.</td> +<td class="caption"><span class="sc">Fig. 9.</span>—Carved Mahogany Arm-chair, +in the style of Chippendale, with +ribbon pattern.</td> +<td class="caption"><span class="sc">Fig. 10.</span>—Carved and Inlaid Mahogany +Chair, in the style of Hepplewhite; +late 18th century.</td> +<td class="caption"><span class="sc">Fig. 11.</span>—Mahogany Chair in the +style of Sheraton; about 1780.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:198px; height:268px" src="images/img364k.jpg" alt="" /></td> +<td class="figcenter"><img style="width:205px; height:264px" src="images/img364l.jpg" alt="" /></td> +<td class="figcenter"><img style="width:194px; height:268px" src="images/img364m.jpg" alt="" /></td> +<td class="figcenter"><img style="width:206px; height:265px" src="images/img364n.jpg" alt="" /></td></tr> +<tr><td class="caption">Fig. 12.—Painted and gilt Arm-chair +with cane seat, in the style of +Adam; about 1790.</td> +<td class="caption"><span class="sc">Fig. 13.</span>—Arm-chair of carved and gilt +wood with stuffed back, seat and +arms. French, Louis XV. style.</td> +<td class="caption"><span class="sc">Fig. 14.</span>—Mahogany Arm-chair. Empire +style, early 19th century, said to have +belonged to the Bonaparte family.</td> +<td class="caption"><span class="sc">Fig. 15.</span>—Painted and gilt Beech Chair. +English, about 1800.</td></tr></table></td></tr> +</table> + +<p class="pt2 noind f90 sc">Plate II.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:376px; height:210px" src="images/img365a.jpg" alt="" /></td> +<td class="figcenter"><img style="width:371px; height:211px" src="images/img365b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 1.</span>—Front of Oak Coffer with wrought iron bands. +French, 2nd half of 13th century.</td> +<td class="caption"><span class="sc">Fig. 2.</span>—English Oak Chest, dated 1637.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:370px; height:191px" src="images/img365c.jpg" alt="" /></td> +<td class="figcenter"><img style="width:368px; height:186px" src="images/img365d.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 3.</span>—Italian (Florentine) Coffer of Wood with gilt arabesque +stucco ornament, about 1480.</td> +<td class="caption"><span class="sc">Fig. 4.</span>—Italian “Cassone” or Marriage Coffer, 13th century. +Carved and gilt wood with painted front and ends.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:461px; height:264px" src="images/img365e.jpg" alt="" /></td> +<td class="figcenter"><img style="width:274px; height:270px" src="images/img365f.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 5.</span>—Walnut Table with expanding leaves. Swiss, 17th century.</td> +<td class="caption"><span class="sc">Fig. 6.</span>—Oak Gate-Legged Table. English, +17th century.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:369px; height:233px" src="images/img365g.jpg" alt="" /></td> +<td class="figcenter"><img style="width:366px; height:235px" src="images/img365h.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 7.</span>—Writing Table. French, end of Louis XV. period. +Riesener marquetry, ormolu mounts and Sèvres plaques.</td> +<td class="caption"><span class="sc">Fig. 8.</span>—Painted Satin-Wood Tables, in the style of Sheraton, +about 1790.</td></tr> +<tr><td class="caption f80" colspan="2">(The above are in the Victoria and Albert Museum, except Fig. 8, which were in the Bethnal Green Exhibition, 1892.)</td></tr></table></td></tr> +</table> + +<p class="pt2 noind f90 sc">Plate III.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:244px; height:316px" src="images/img366a.jpg" alt="" /></td> +<td class="figcenter"><img style="width:367px; height:316px" src="images/img366b.jpg" alt="" /></td></tr> +<tr><td class="caption">1. CARVED OAK SIDEBOARD. English, 17th century. Victoria and Albert +Museum.</td> +<td class="caption">2. CARVED OAK COURT CUPBOARD. English, early 17th +century. Victoria and Albert Museum.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:349px; height:269px" src="images/img366c.jpg" alt="" /></td> +<td class="figcenter"><img style="width:221px; height:260px" src="images/img366d.jpg" alt="" /></td></tr> +<tr><td class="caption">3. EBONY CARVED CABINET. The interior +decorated with inlaid ivory and coloured +woods; French or Dutch, middle of 17th +century. Victoria and Albert Museum.</td> +<td class="caption">4. VENEERED CHEST OF DRAWERS. About +1690. Lent to Bethnal Green Exhibition by +Sir Spencer Ponsonby-Fane, G.C.B.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:263px; height:384px" src="images/img366e.jpg" alt="" /></td> +<td class="figcenter"><img style="width:257px; height:384px" src="images/img366f.jpg" alt="" /></td></tr> +<tr><td class="caption">5. EBONY ARMOIRE. With tortoise-shell +panels inlaid with brass and other +metals, and ormolu mountings. Designed +by Bérain, and executed by André +Boulle. French, Louis XIV. period. +Victoria and Albert Museum.</td> +<td class="caption">6. GLASS-FRONTED BOOKCASE AND CABINET. Of +mahogany. In the style of Sheraton, about 1790. Lent +to the Bethnal Green Exhibition by the late Vincent J. Robinson, +C.I.E.</td></tr></table></td></tr> +</table> + +<p class="pt2 noind f90 sc">Plate IV.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:371px; height:263px" src="images/img367a.jpg" alt="" /></td> +<td class="figcenter"><img style="width:368px; height:262px" src="images/img367b.jpg" alt="" /></td></tr> +<tr><td class="caption">1. COMMODE OF PINE. With marquetry of brass, ebony, tortoise-shell, +mother-of-pearl, ivory, and green-stained bone. “Boulle” work with +designs in the style of Bérain. French, late period of Louis XIV.</td> +<td class="caption">2. COMMODE. With panels of Japanese lacquer and ormolu mountings, +in the style of Caffieri. French, Louis XV. period.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:246px; height:314px" src="images/img367c.jpg" alt="" /></td> +<td class="figcenter"><img style="width:251px; height:317px" src="images/img367d.jpg" alt="" /></td></tr> +<tr><td class="caption">3. TABLE OF KING AND TULIP WOODS. With ormolu mountings. +Louis XV. period.</td> +<td class="caption">4. ESCRITOIRE À TOILETTE. Formerly belonging to Marie Antoinette. +Of tulip and sycamore woods inlaid with other coloured woods, ormolu +mounts. Louis XV. period.</td></tr></table></td></tr> + +<tr><td> +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:296px; height:394px" src="images/img367e.jpg" alt="" /></td> +<td class="figcenter"><img style="width:290px; height:394px" src="images/img367f.jpg" alt="" /></td></tr> +<tr><td class="caption">5. FOUR-POST BEDSTEAD. Of oak inlaid +with bog-oak and holly, from the “Inlaid Room” +at Sizergh Castle, Westmorland. Latter half of +sixteenth century.</td> +<td class="caption">6. CARVED AND GILT BEDSTEAD. With +blue silk damask coverings and hangings. +French, late 18th century. Louis XVI. period.</td></tr> +<tr><td class="caption f80" colspan="2">From the Victoria and Albert Museum, S. Kensington.</td></tr></table></td></tr> +</table> + +<p class="pt2 noind f90 sc">Plate V.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:730px; height:550px" src="images/img368a.jpg" alt="" /></td></tr> +<tr><td class="figcenter"><img style="width:731px; height:544px" src="images/img368b.jpg" alt="" /></td></tr> +<tr><td class="tcl f80"><i>Photo, Mansell & Co.</i></td></tr> +<tr><td class="caption">THE “BUREAU DU ROI,” MADE FOR LOUIS XV., NOW IN THE LOUVRE. For description, see <span class="sc">Desk</span>.</td></tr></table> + +<p>A decided, if not always intelligent, effort to devise a new +style in furniture began during the last few years of the 19th +century, which gained the name of “<i>l’art nouveau</i>.” Its pioneers +professed to be free from all old traditions and to seek inspiration +from nature alone. Happily nature is less forbidding than many +of these interpretations of it, and much of the “new art” is a +remarkable exemplification of the impossibility of altogether +ignoring traditional forms. The style was not long in degenerating +into extreme extravagance. Perhaps the most striking consequence +of this effort has been, especially in England, the +revival of the use of oak. Lightly polished, or waxed, the cheap +foreign oaks often produce very agreeable results, especially +when there is applied to them a simple inlay of boxwood and +stained holly, or a modern form of pewter. The simplicity of +these English forms is in remarkable contrast to the tortured +and ungainly outlines of continental seekers after a conscious +and unpleasing “originality.”</p> + +<p>Until a very recent period the most famous collections of +historic furniture were to be found in such French museums as +the Louvre, Cluny and the Garde Meuble. Now, however, they +are rivalled, if not surpassed, by the magnificent collections of +the Victoria and Albert Museum at South Kensington, and the +Wallace collection at Hertford House, London. The latter, in +conjunction with the Jones bequest at South Kensington, forms +the finest of all gatherings of French furniture of the great +periods, notwithstanding that in the Bureau du Roi the Louvre +possesses the most magnificent individual example in existence. +In America there are a number of admirable collections representative +of the graceful and homely “colonial furniture” +made in England and the United States during the Queen Anne +and Georgian periods.</p> + +<div class="condensed"> +<p>See also the separate articles in this work on particular forms of +furniture. The literature of the subject has become very extensive, +and it is needless to multiply here the references to books. Perrot +and Chipiez, in their great <i>Histoire de l’art dans l’antiquité</i> (1882 +et seq.) deal with ancient times, and A. de Champeaux, in <i>Le Meuble</i> +(1885), with the middle ages and later period; English furniture is +admirably treated by Percy Macquoid in his <i>History of English +Furniture</i> (1905); and Lady Dilke’s <i>French Furniture in the 18th +Century</i> (1901), and Luke Vincent Lockwood’s <i>Colonial Furniture in +America</i> (1901), should also be consulted.</p> +</div> +<div class="author">(J. P. B.)</div> + + +<hr class="art" /> +<p><span class="bold">FURNIVALL, FREDERICK JAMES<a name="ar91" id="ar91"></a></span> (1825-1910), English +philologist and editor, was born at Egham, Surrey, on the 4th +of February 1825, the son of a surgeon. He was called to the bar +in 1849, but his attention was soon diverted to philological +studies and social problems. He gave Frederick Denison Maurice +valuable assistance in the Christian Socialist movement, and was +one of the founders of the Working Men’s College. For half a +century he indefatigably promoted the study of early English +literature, partly by his own work as editor, and still more +efficaciously by the agency of the numerous learned societies +of which he was both founder and director, especially the Early +English Text Society (1864), which has been of inestimable +service in promoting the study of early and middle English. +He also established and conducted the Chaucer, Ballad, New +Shakespeare and Wyclif Societies, and at a later period societies +for the special study of Browning and Shelley. He edited texts +for the Early English Text Society, for the Roxburghe Club +and the Rolls Series; but his most important labours were +devoted to Chaucer, whose study he as an editor greatly assisted +by his “Six-Text” edition of the <i>Canterbury Tales</i>, and other +publications of the Chaucer Society. He was the honorary +secretary of the Philological Society, and was one of the original +promoters of the Oxford <i>New English Dictionary</i>. He co-operated +with its first editor, Herbert Coleridge, and after his death +was for some time principal editor during the preliminary period +of the collection of material. The completion of his half-century +of labour was acknowledged in 1900 by a handsome testimonial, +including the preparation by his friends of a volume of philological +essays specially dedicated to him, <i>An English Miscellany</i> +(Oxford, 1901), and a considerable donation to the Early English +Text Society. Dr Furnivall was always an enthusiastic oarsman, +and till the end kept up his interest in rowing; with John +Beesley in 1845 he introduced the new type of narrow sculling +boat, and in 1886 started races on the Thames for sculling fours +and sculling eights. He died on the 2nd of July 1910.</p> + + +<hr class="art" /> +<p><span class="bold">FURSE, CHARLES WELLINGTON<a name="ar92" id="ar92"></a></span> (1868-1904), English +painter, born at Staines, the son of the Rev. C.W. Furse, archdeacon +of Westminster, was descended collaterally from Sir +Joshua Reynolds, and in his short span of life achieved such +rare excellence as a portrait and figure painter that he forms an +important link in the chain of British portraiture which extends +from the time when Van Dyck was called to the court of Charles I. +to our own day. His talent was precocious; at the age of seven +he gave indications of it in a number of drawings illustrating +Scott’s novels. He entered the Slade school in 1884, winning the +Slade scholarship in the following year, and completed his education +at Julian’s <i>atelier</i> in Paris. Hard worker as he was, his +activity was frequently interrupted by spells of illness, for he had +developed signs of consumption when he was still attending the +Slade school. An important canvas called “Cain” was his first +contribution (1888) to the Royal Academy, to the associateship +of which he was elected in the year of his death. For some years +before he had been a staunch supporter of the New English Art +Club, to the exhibitions of which he was a regular contributor. +He was married in October 1900 to Katherine, daughter of John +Addington Symonds. His fondness for sport and of an open-air +life found expression in his art and introduced a new, fresh and +vigorous note into portraiture. There is never a suggestion of +the studio or of the fatiguing pose in his portraits. The sitters +appear unconscious of being painted, and are generally seen in +the pursuit of their favourite outdoor sport or pastime, in the +full enjoyment of life. Such are the “Diana of the Uplands,” +the “Lord Roberts” and “The Return from the Ride” at the +Tate Gallery; the four children in the “Cubbing with the York +and Ainsty,” “The Lilac Gown,” “Mr and Mrs Oliver Fishing” +and the portrait of Lord Charles Beresford. Most of these +pictures, and indeed nearly all the work completed in the few +years of Furse’s activity, show a pronounced decorative tendency. +His sense of space, composition and decorative design can best +be judged by his admirable mural decorations for Liverpool +town hall, executed between 1899 and 1902. A memorial exhibition +of Furse’s paintings and sketches was held at the Burlington +Fine Arts Club in 1906.</p> + + +<hr class="art" /> +<p><span class="bold">FÜRST, JULIUS<a name="ar93" id="ar93"></a></span> (1805-1873), German Orientalist, was born +of Jewish parents at Zerkowo in Posen, on the 12th of May 1805. +He studied philosophy and philology at Berlin, and oriental +literature at Posen, Breslau and Halle. In 1857 he was appointed +to a lectureship at the university of Leipzig, and he was promoted +to a professorship in 1864, which he held until his death at Leipzig +on the 9th of February 1873. Among his writings may be +mentioned <i>Lehrgebäude der aramäischen Idiome</i> (Leipzig, 1835); +<i>Librorum sacrorum Veteris Testamenti concordantiae Hebraicae +atque Chaldaicae</i> (Leipzig, 1837-1840); <i>Hebräisches und chaldäisches +Wörterbuch</i> (1851, English translation by S. Davidson 1867); +<i>Kultur und Literaturgeschichte der Juden in Asien</i> (1849). Fürst +also edited a valuable <i>Bibliotheca Judaica</i> (Leipzig, 1849-1863), +and was the author of some other works of minor importance. +From 1840 to 1851 he was editor of <i>Der Orient</i>, a journal devoted +to the language, literature, history and antiquities of the Jews.</p> + + +<hr class="art" /> +<p><span class="bold">FÜRSTENBERG,<a name="ar94" id="ar94"></a></span> the name of two noble houses of Germany.</p> + +<p>1. The more important is in possession of a mediatized principality +in the district of the Black Forest and the Upper Danube, +which comprises the countship of Heiligenberg, about 7 m. to +the N. of the Lake of Constance, the landgraviates of Stühlingen +and Baar, and the lordships of Jungnau, Trochtelfingen, Hausen +<span class="pagenum"><a name="page366" id="page366"></a>366</span> +and Möskirch or Messkirch. The territory is discontinuous; +and as it lies partly in Baden, partly in Württemberg, and partly +in the Prussian province of Sigmaringen, the head of the family +is an hereditary member of the first chamber of Baden and of +the chamber of peers in Württemberg and in Prussia. The +relations of the principality with Baden are defined by the treaty +of May 1825, and its relations with Württemberg by the royal +declaration of 1839. The <i>Stammort</i> or ancestral seat of the +family is Fürstenberg in the Black Forest, about 13 m. N. of +Schaffhausen, but the principal residence of the present representatives +of the main line is at Donaueschingen.</p> + +<p>The family of Fürstenberg claims descent from a certain +Count Unruoch, a contemporary of Charlemagne, but their +authentic pedigree is only traceable to Egino II., count of +Urach, who died before 1136. In 1218 his successors inherited +the possessions of the house of Zähringen in the Baar district +of the Black Forest, where they built the town and castle of +Fürstenberg. Of the two sons of Egino V. of Urach, Conrad, +the elder, inherited the Breisgau and founded the line of the +counts of Freiburg, while the younger, Heinrich (1215-1284), +received the territories lying in the Kinzigthal and Baar, and +from 1250 onward styled himself first lord, then count, of +Fürstenberg. His territories were subsequently divided among +several branches of his descendants, though temporarily reunited +under Count Friedrich III., whose wife, Anna, heiress +of the last count of Wardenberg, brought him the countship of +Heiligenberg and lordships of Jungnau and Trochtelfingen in +1534. On Friedrich’s death (1559) his territories were divided +between his two sons, Joachim and Christof I. Of these the +former founded the line of Heiligenberg, the latter that of +Kinzigthal. The Kinzigthal branch was again subdivided in +the 17th century between the two sons of Christof II. (d. 1614), +the elder, Wratislaw II. (d. 1642), founding the line of Mösskirch, +the younger, Friedrich Rudolf (d. 1655), that of Stühlingen. +The Heiligenberg branch received an accession of dignity by the +elevation of Count Hermann Egon (d. 1674) to the rank of prince +of the Empire in 1664, but his line became extinct with the +death of his son Prince Anton Egon, favourite of King Augustus +the Strong and regent of Saxony, in 1716. The heads of both +the Mösskirch and Stühlingen lines were now raised to the +dignity of princes of the Empire (1716). The Mösskirch branch +died out with Prince Karl Friedrich (d. 1744); the territories +of the Stühlingen branch had been divided on the death of +Count Prosper Ferdinand (1662-1704) between his two sons, +Joseph Wilhelm Ernst (1699-1762) and Ludwig August Egon +(1705-1759). The first of these was created prince of the Empire +on the 10th of December 1716, and founded the princely line +of the Swabian Fürstenbergs; in 1772 he obtained from the +emperor Francis I. for all his legitimate sons and their descendants +the right to bear, instead of the style of landgrave, that of +prince, which had so far been confined to the reigning head of +the family. Ludwig, on the other hand, founded the family of +the landgraves of Fürstenberg, who, since their territories lay +in Austria and Moravia, were known as the “cadet line in +Austria.” The princely line became extinct with the death +of Karl Joachim in 1804, and the inheritance passed to the +Bohemian branch of the Austrian cadet line in the person of +Karl Egon II. (see below). Two years later the principality +was mediatized.</p> + +<p>In 1909 there were two branches of the princely house of +Fürstenberg: (1) the main branch, that of Fürstenberg-Donaueschingen, +the head of which was Prince Maximilian Egon (b. +1863), who succeeded his cousin Karl Egon III. in 1896; (2) +that of Fürstenberg-Königshof, in Bohemia, the head of which +was Prince Emil Egon (b. 1876), chamberlain and secretary of +legation to the Austro-Hungarian embassy in London (1907). +The cadet line of the landgraves of Fürstenberg is now extinct, +its last representative having been the landgrave Joseph Friedrich +Ernst of Fürstenberg-Weitra (1860-1896), son of the +landgrave Ernst (1816-1889) by a morganatic marriage. He +was not recognized as <i>ebenbürtig</i> by the family. The landgraves +of Fürstenberg were in 1909 represented only by the landgravines +Theresa (b. 1839) and Gabrielle (b. 1844), daughters of the +landgrave Johann Egon (1802-1879).</p> + +<p>From the days of Heinrich of Urach, a relative and notable +supporter of Rudolph of Habsburg, the Fürstenbergs have +played a stirring part in German history as statesmen, ecclesiastics +and notably soldiers. There was a popular saying that +“the emperor fights no great battle but a Fürstenberg falls.” +In the Heiligenberg line the following may be more particularly +noticed.</p> + +<p><span class="sc">Franz Egon</span> (1625-1682), bishop of Strassburg, was the elder +son of Egon VII., count of Fürstenberg (1588-1635), who served +with distinction as a Bavarian general in the Thirty Years’ War. +He began life as a soldier in the imperial service, but on the +elevation of his friend Maximilian Henry of Bavaria to the +electorate of Cologne in 1650, he went to his court and embraced +the ecclesiastical career. He soon gained a complete ascendancy +over the weak-minded elector, and, with his brother William +Egon (see below), was mainly instrumental in making him the +tool of the aggressive policy of Louis XIV. of France. Ecclesiastical +preferments were heaped upon him. As a child he had +been appointed to a canonry of Cologne; to these he added +others at Strassburg, Liége, Hildesheim and Spires; he became +also suffragan bishop and dean of Cologne and provost of Hildesheim, +and in 1663 bishop of Strassburg. Later he was also +prince-abbot of Lüders and Murbach and abbot of Stablo and +Malmedy. On the conclusion of a treaty between the emperor +and the elector of Cologne, on the 11th of May 1674, Franz was +deprived of all his preferments in Germany, and was compelled +to take refuge in France. He was, however, amnestied with his +brother William by a special article of the treaty of Nijmwegen +(1679), whereupon he returned to Cologne. After the French +occupation of Strassburg (1681) he took up his residence there +and died on the 1st of April 1682.</p> + +<p>His brother <span class="sc">William Egon</span> (1629-1704), bishop of Strassburg, +began his career as a soldier in the French service. He went to +the court of the elector of Cologne at the same time as Franz +Egon, whose zeal for the cause of Louis XIV. of France he shared. +In 1672 the intrigues of the two Fürstenbergs had resulted in a +treaty of offensive alliance between the French monarchy and +the electorate of Cologne, and, the brothers being regarded by +the Imperialists as the main cause of this disaster, William was +seized by imperial soldiers in the monastery of St Pantaleon at +Cologne, hurried off to Vienna and there tried for his life. He +was saved by the intervention of the papal nuncio, but was kept +in prison till the signature of the treaty of Nijmwegen (1679). +As a reward for his services Louis XIV. appointed him bishop +of Strassburg in succession to his brother in 1682, in 1686 obtained +for him from Pope Innocent XI. the cardinal’s hat, and in 1688 +succeeded in obtaining his election as coadjutor-archbishop of +Cologne and successor to the elector Maximilian Henry. At the +instance of the emperor, however, the pope interposed his veto; +the canons followed the papal lead, and, the progress of the +Allies against Louis XIV. depriving him of all prospect of +success, William Egon retired to France. Here he took up his +abode at his abbey of St Germain des Près near Paris, where he +died on the 10th of April 1704.</p> + +<p>In the Stühlingen line the most notable was <span class="sc">Karl Egon</span> +(1796-1854), prince of Fürstenberg, the son of Prince Karl +Alois of Fürstenberg, a general in the Austrian service, who was +killed at the battle of Loptingen on the 25th of March 1799. +In 1804 he inherited the Swabian principality of Fürstenberg +and all the possessions of the family except the Moravian estates. +He studied at Freiburg and Würzburg, and in 1815 accompanied +Prince Schwarzenberg to Paris as staff-officer. In 1817 he came +of age, and in the following year married the princess Amalie +of Baden. By the mediatization of his principality in 1806 the +greater part of his vast estates had fallen under the sovereignty +of the grand-duke of Baden, and Prince Fürstenberg took a +conspicuous part in the upper house of the grand-duchy. In +politics he distinguished himself by a liberalism rare in a great +German noble, carrying through by his personal influence with +his peers the abolition of tithes and feudal dues and stanchly +<span class="pagenum"><a name="page367" id="page367"></a>367</span> +advocating the freedom of the press. He was not less distinguished +by his large charities: among other foundations he +established a hospital at Donaueschingen. For the industrial +development of the country, too, he did much, and proved himself +also a notable patron of the arts. His palace of Donaueschingen, +with its collections of paintings, engravings and coins, was a +centre of culture, where poets, painters and musicians met with +princely entertainment. He died on the 14th of September +1869, and was succeeded by his son Karl Egon II. (1820-1892), +with the death of whose son, Karl Egon III., in 1896, the title +and estates passed to Prince Maximilian Egon, head of the cadet +line of Fürstenberg-Pürglitz.</p> + +<div class="condensed"> +<p>See Münch, <i>Gesch. des Hauses und des Landes Fürstenberg</i>, 4 vols. +(Aix-la-Chapelle, 1829-1847); S. Riezler, <i>Gesch. des fürstlichen +Hauses Fürstenberg bis 1507</i> (Tübingen, 1883); <i>Fürstenbergisches +Urkundenbuch</i>, edited by S. Riezler and F.L. Baumann, vols. i.-vii. +(Tübingen, 1877-1891), continued <i>s. tit. Mitteilungen aus dem +fürstlich. Fürstenbergischem Archiv</i> by Baumann and G. Tumbült, +2 vols. (ib. 1899-1902); Stokvis, <i>Manuel d’histoire</i> (Leiden, 1890-1893); +<i>Almanach de Gotha; Allgemeine deutsche Biographie</i>.</p> +</div> + +<p>2. The second Fürstenberg family has its possessions in +Westphalia and the country of the Rhine, and takes its name +from the castle of Fürstenberg on the Ruhr. The two most +remarkable men whom it has produced are Franz Friedrich +Wilhelm, freiherr von Fürstenberg, and Franz Egon, count von +Fürstenberg-Stammheim. The former (1728-1810) became +ultimately vicar-general of the prince-bishop of Münster, and +effected a great number of important reforms in the administration +of the country, besides doing much for its educational +and industrial development. The latter (1797-1859) was an +enthusiastic patron of art, who zealously advocated the completion +of the Cologne cathedral, and erected the beautiful church +of St Apollinaris near Remagen on the Rhine. He was a member +of the Prussian Upper House in 1849, collaborated in founding +the <i>Preussisches Wochenblatt</i>, and was an ardent defender of +Catholic interests. His son, Count Gisbert von Fürstenberg-Stammheim +(b. 1836), was in 1909 head of the Rhenish line of +the house of Fürstenberg.</p> + + +<hr class="art" /> +<p><span class="bold">FÜRSTENWALDE,<a name="ar95" id="ar95"></a></span> a town of Germany, in the Prussian +province of Brandenburg, on the right bank of the Spree, and +on the railway from Berlin to Frankfort-on-Oder, 28 m. E. of +the former city. Pop. (1905) 20,498. Its beautiful cathedral +church contains several old monuments. The industries are +important, including, besides brewing and malting, manufactures +of starch, vinegar, electric lamps and gas-fittings, stoves, &c., +iron-founding and wool-weaving. Fürstenwalde is one of the +oldest towns of Brandenburg. From 1385 it was the seat of +the bishop of Lebus, whose bishopric was incorporated with +the electorate of Brunswick in 1595.</p> + + +<hr class="art" /> +<p><span class="bold">FÜRTH,<a name="ar96" id="ar96"></a></span> a manufacturing town of Germany, in the kingdom +of Bavaria, at the confluence of the Pegnitz with the Regnitz, +5 m. N.W. from Nuremberg by rail, at the junction of lines to +Hof and Würzburg. Pop. (1885) 35,455; (1905) 60,638. It is +a modern town in appearance, with broad streets and palatial +business houses. Of its four Evangelical churches, the old St +Michaeliskirche is a handsome structure; but its chief edifices +are the new town hall, with a tower 175 ft. high and the +magnificent synagogue. The Jews have also a high school, +which enjoys a great reputation. There are besides a classical, +a wood-carving and an agricultural school and a library. Fürth +is the seat of several important industries; particularly, the +production of chromolithographs and picture-books, the manufacture +of mirrors and mirror-frames, bronze and gold-leaf wares, +pencils, toys, haberdashery, optical instruments, silver work, +turnery, chicory, machinery, fancy boxes and cases, and an +extensive trade is carried on in these goods as also in hops, +metals, wool, groceries and coal. A large annual fair is held +at Michaelmas and lasts for eleven days. The earliest railway +in Germany was that between Nuremberg and Fürth (opened +on the 7th of December 1835).</p> + +<p>Fürth was founded, according to tradition, by Charlemagne, +who erected a chapel there. It was for a time a <i>Vogtei</i> (advocateship) +under the burgraves of Nuremberg, but about 1314 it was +bequeathed to the see of Bamberg, and in 1806 it came into +the possession of Bavaria. In 1632 Gustavus Adolphus besieged +it in vain, and in 1634 it was pillaged and burnt by the Croats. +It owes its rise to prosperity to the tolerance it meted out to the +Jews, who found here an asylum from the oppression under +which they suffered in Nuremberg.</p> + +<div class="condensed"> +<p>See Fronmüller, <i>Chronik der Stadt Fürth</i> (1887).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FURTWÄNGLER, ADOLF<a name="ar97" id="ar97"></a></span> (1853-1907), German archaeologist, +was born at Freiburg im Breisgau, and was educated there, +at Leipzig and at Munich, where he was a pupil of H. Brunn, +whose comparative method in art-criticism he much developed. +He took part in the excavations at Olympia in 1878, became +an assistant in the Berlin Museum in 1880, and professor at +Berlin (1884) and later at Munich. His latest excavation work +was at Aegina. He was a prolific writer, with a prodigious +knowledge and memory, and a most ingenious and confident +critic; and his work not only dominated the field of archaeological +criticism but also raised its standing both at home and abroad. +Among his numerous publications the most important were a +volume on the bronzes found at Olympia, vast works on ancient +gems and Greek vases, and the invaluable <i>Masterpieces of +Greek Sculpture</i> (English translation by Eugénie Strong). He +died at Athens on the 10th of October 1907.</p> + + +<hr class="art" /> +<p><span class="bold">FURZE,<a name="ar98" id="ar98"></a></span> <span class="sc">Gorse</span> or <span class="sc">Whin</span>; botanical name <i>Ulex</i> (Ger. +<i>Stechginster</i>, Fr. <i>ajonc</i>), a genus of thorny papilionaceous +shrubs, of few species, confined to west and central Europe and +north-west Africa. Common furze, <i>U. europaeus</i>, is found on +heaths and commons in western Europe from Denmark to Italy +and Greece, and in the Canaries and Azores, and is abundant +in nearly all parts of the British Isles. It grows to a height +of 2-6 ft.; it has hairy stems, and the smaller branches end each +in a spine; the leaves, sometimes lanceolate on the lowermost +branches, are mostly represented by spines from 2 to 6 lines long, +and branching at their base; and the flowers, about three-quarters +of an inch in length, have a shaggy, yellowish-olive calyx, with two +small ovate bracts at its base, and appear in early spring and +late autumn. They are yellow and sweet-scented and visited by +bees. The pods are few-seeded; their crackling as they burst +may often be heard in hot weather. This species comprises the +varieties <i>vulgaris</i>, or <i>U. europaeus</i> proper, which has spreading +branches, and strong, many-ridged spines, and <i>strictus</i> (Irish +furze), with erect branches, and slender 4-edged spines. The +other British species of furze is <i>U. nanus</i>, dwarf furze, a native +of Belgium, Spain and the west of France; it is a procumbent +plant, less hairy than <i>U. europaeus</i>, with smaller and more +orange-coloured flowers, which spring from the primary spines, +and have a nearly smooth calyx, with minute basal bracts. +Furze, or gorse, is sometimes employed for fences.</p> + +<p>Notwithstanding its formidable spines, the young shoots +yield a palatable and nutritious winter forage for horses and +cattle. To fit it for this purpose it must be chopped and bruised +to destroy the spines. This is sometimes done in a primitive +and laborious way by laying the gorse upon a block of wood and +beating it with a mallet, flat at one end and armed with crossed +knife-edges at the other, by the alternate use of which it is +bruised and chopped. There are now a variety of machines +by which this is done rapidly and efficiently, and which are in +use where this kind of forage is used to any extent. The agricultural +value of this plant has often been over-rated by theoretical +writers. In the case of very poor, dry soils it does, however, +yield much valuable food at a season when green forage is not +otherwise to be had. It is on this account of importance to +dairymen; and to them it has this further recommendation, +that cows fed upon it give much rich milk, which is free from +any unpleasant flavour. To turn it to good account, it +must be sown in drills, kept clean by hoeing, and treated +as a regular green crop. If sown in March, on land fitly prepared +and afterwards duly cared for, it is ready for use in the +autumn of the following year. A succession of cuttings of +proper age is obtained for several years from the same field. +It is cut by a short stout scythe, and must be brought +from the field daily; for when put in a heap after being +<span class="pagenum"><a name="page368" id="page368"></a>368</span> +chopped and bruised it heats rapidly. It is given to horses and +cows in combination with chopped hay or straw. An acre will +produce about 2000 faggots of green two-year-old gorse, weighing +20 ℔ each.</p> + +<p>This plant is invaluable in mountain sheep-walks. The +rounded form of the furze bushes that are met with in such +situations shows how diligently the annual growth, as far as it +is accessible, is nibbled by the sheep. The food and shelter +afforded to them in snowstorms by clusters of such bushes is +of such importance that the wonder is our sheep farmers do not +bestow more pains to have it in adequate quantity. Young +plants of whin are so kept down by the sheep that they can +seldom attain to a profitable size unless protected by a fence +for a few years. In various parts of England it is cut for fuel. +The ashes contain a large proportion of alkali, and are a good +manure, especially for peaty land.</p> + + +<hr class="art" /> +<p><span class="bold">FUSARO, LAGO,<a name="ar99" id="ar99"></a></span> a lake of Campania, Italy, ½ m. W. of Baia, +and 1 m. S. of the acropolis of Cumae. It is the ancient <i>Acherusia +palus</i>, separated from the sea on the W. by a line of sandhills. +It may have been the harbour of Cumae in early antiquity. +In the 1st century <span class="scs">A.D.</span> an artificial outlet was dug for it at its +S. end, with a tunnel, lined with <i>opus reticulatum</i> and brick, +under the hill of Torregaveta. This hill is covered with the +remains of a large villa, which is almost certainly that of Servilius +Vatia, described by Seneca (<i>Epist.</i> 55). There are remains of +other villas on the shores of the lake. Oyster cultivation is +carried on there.</p> + +<div class="condensed"> +<p>See J. Beloch, <i>Campanien</i> (2nd ed., Breslau, 1890), 188.</p> +</div> +<div class="author">(T. As.)</div> + + +<hr class="art" /> +<p><span class="bold">FUSELI, HENRY<a name="ar100" id="ar100"></a></span> (1741-1825), English painter and writer on +art, of German-Swiss family, was born at Zürich in Switzerland +on the 7th of February 1741; he himself asserted in 1745, but +this appears to have been a mere whim. He was the second +child in a family of eighteen. His father was John Caspar +Füssli, of some note as a painter of portraits and landscapes, +and author of Lives of the <i>Helvetic Painters</i>. This parent +destined his son for the church, and with this view sent him to +the Caroline college of his native town, where he received an +excellent classical education. One of his schoolmates there +was Lavater, with whom he formed an intimate friendship.</p> + +<p>After taking orders in 1761 Fuseli was obliged to leave his +country for a while in consequence of having aided Lavater to +expose an unjust magistrate, whose family was still powerful +enough to make its vengeance felt. He first travelled through +Germany, and then, in 1765, visited England, where he supported +himself for some time by miscellaneous writing: there was a +sort of project of promoting through his means a regular literary +communication between England and Germany. He became +in course of time acquainted with Sir Joshua Reynolds, to whom +he showed his drawings. By Sir Joshua’s advice he then devoted +himself wholly to art. In 1770 he made an art-pilgrimage to +Italy, where he remained till 1778, changing his name from +Füssli to Fuseli, as more Italian-sounding. Early in 1779 he +returned to England, taking Zürich on his way. He found a +commission awaiting him from Alderman Boydell, who was then +organizing his celebrated Shakespeare gallery. Fuseli painted +a number of pieces for this patron, and about this time published +an English edition of Lavater’s work on physiognomy. He likewise +gave Cowper some valuable assistance in preparing the +translation of Homer. In 1788 Fuseli married Miss Sophia +Rawlins (who it appears was originally one of his models, and who +proved an affectionate wife), and he soon after became an +associate of the Royal Academy. Two years later he was promoted +to the grade of Academician. In 1799 he exhibited a +series of paintings from subjects furnished by the works of +Milton, with a view to forming a Milton gallery corresponding +to Boydell’s Shakespeare gallery. The number of the Milton +paintings was forty-seven, many of them very large; they were +executed at intervals within nine years. This exhibition, which +closed in 1800, proved a failure as regards profit. In 1799 also +he was appointed professor of painting to the Academy. Four +years afterwards he was chosen keeper, and resigned his professorship; +but he resumed it in 1810, and continued to hold +both offices till his death. In 1805 he brought out an edition of +Pilkington’s <i>Lives of the Painters</i>, which, however, did not add +much to his reputation. Canova, when on his visit to England, +was much taken with Fuseli’s works, and on returning to Rome +in 1817 caused him to be elected a member of the first class in +the Academy of St Luke. Fuseli, after a life of uninterrupted +good health, died at Putney Hill on the 16th of April 1825, +at the advanced age of eighty-four, and was buried in the crypt +of St Paul’s cathedral. He was comparatively rich at his death, +though his professional gains had always appeared to be meagre.</p> + +<p>As a painter, Fuseli had a daring invention, was original, +fertile in resource, and ever aspiring after the highest forms +of excellence. His mind was capable of grasping and realizing +the loftiest conceptions, which, however, he often spoiled on the +canvas by exaggerating the due proportions of the parts, and +throwing his figures into attitudes of fantastic and over-strained +contortion. He delighted to select from the region of the supernatural, +and pitched everything upon an ideal scale, believing +a certain amount of exaggeration necessary in the higher branches +of historical painting. “Damn Nature! she always puts me +out,” was his characteristic exclamation. In this theory he was +confirmed by the study of Michelangelo’s works and the marble +statues of the Monte Cavallo, which, when at Rome, he used +often to contemplate in the evening, relieved against a murky +sky or illuminated by lightning. But this idea was by him +carried out to an excess, not only in the forms, but also in the +attitudes of his figures; and the violent and intemperate action +which he often displays destroys the grand effect which many +of his pieces would otherwise produce. A striking illustration +of this occurs in his famous picture of “Hamlet breaking from +his Attendants to follow the Ghost”: Hamlet, it has been said, +looks as though he would burst his clothes with convulsive +cramps in all his muscles. This intemperance is the grand defect +of nearly all Fuseli’s compositions. On the other hand, his +paintings are never either languid or cold. His figures are full +of life and earnestness, and seem to have an object in view +which they follow with rigid intensity. Like Rubens he excelled +in the art of setting his figures in motion. Though the lofty and +terrible was his proper sphere, Fuseli had a fine perception of the +ludicrous. The grotesque humour of his fairy scenes, especially +those taken from <i>A Midsummer-Night’s Dream</i>, is in its way not +less remarkable than the poetic power of his more ambitious +works. As a colourist Fuseli has but small claims to distinction. +He scorned to set a palette as most artists do; he merely dashed +his tints recklessly over it. Not unfrequently he used his paints +in the form of a dry powder, which he rubbed up with his pencil +with oil, or turpentine, or gold size, regardless of the quantity, +and depending for accident on the general effect. This recklessness +may perhaps be explained by the fact that he did not paint +in oil till he was twenty-five years of age. Despite these drawbacks +he possessed the elements of a great painter.</p> + +<p>Fuseli painted more than 200 pictures, but he exhibited only +a minority of them. His earliest painting represented “Joseph +interpreting the Dreams of the Baker and Butler”; the first +to excite particular attention was the “Nightmare,” exhibited +in 1782. He produced only two portraits. His sketches or +designs numbered about 800; they have admirable qualities of +invention and design, and are frequently superior to his paintings.</p> + +<p>His general powers of mind were large. He was a thorough +master of French, Italian, English and German, and could write +in all these tongues with equal facility and vigour, though he +preferred German as the vehicle of his thoughts. His writings +contain passages of the best art-criticism that English literature +can show. The principal work is his series of <i>Lectures</i> in the +Royal Academy, twelve in number, commenced in 1801.</p> + +<div class="condensed"> +<p>Many interesting anecdotes of Fuseli, and his relations to contemporary +artists, are given in his <i>Life</i> by John Knowles, who also +edited his works in 3 vols. 8vo, London, 1831.</p> +</div> +<div class="author">(W. M. R.)</div> + + +<hr class="art" /> +<p><span class="bold">FUSEL OIL<a name="ar101" id="ar101"></a></span> (from the Ger. <i>Fusel</i>, bad spirits), the name applied +to the volatile oily liquids, of a nauseous fiery taste and smell, +which are obtained in the rectification of spirituous liquors made +by the fermentation of grain, potatoes, the marc of grapes, and +<span class="pagenum"><a name="page369" id="page369"></a>369</span> +other material, and which, as they are of higher boiling point +than ethyl alcohol, occur in largest quantity in the last portions +of the distillate. Besides ethyl or ordinary alcohol, and amyl +alcohol, which are present in them all, there have been found in +fusel oil several other bodies of the C<span class="su">n</span>H<span class="su">2n+1</span>·OH series, also +certain ethers, and members of the C<span class="su">n</span>H<span class="su">2n+1</span>·CO<span class="su">2</span>H series of +fatty acids. Normal propyl alcohol is contained in the fusel +oil of the marc brandy of the south of France, and isoprimary +butyl alcohol in that of beet-root molasses. The chief constituent +of the fusel oil procured in the manufacture of alcohol from +potatoes and grain, usually known as fusel oil and potato-spirit, +is isoprimary amyl alcohol, or isobutylcarbinol. Ordinary fusel +oil yields also an isomeric amyl alcohol (active amyl alcohol) +boiling at about 128°. Variable quantities of fusel oil, less or +greater according to the stage of ripening, exist in commercial +spirits (see <span class="sc"><a href="#artlinks">Spirits</a></span>).</p> + +<p>Fusel oil and its chief constituent, amyl alcohol, are direct +nerve poisons. In small doses it causes only thirst and headache, +with furred tongue and some excitement. In large doses it is +a convulsent poison. Impure beverages induce all the graver +neurotic and visceral disorders in alcoholism; and, like fusel +oil, furfurol and the essence of absinthe, are convulsent poisons. +Pure ethyl alcohol intoxication, indeed, is rarely seen, being +modified in the case of spirits by the higher alcohols contained +in fusel oil. According to Rabuteau the toxic properties of the +higher alcohols increase with their molecular weight and boiling +point. Richet considers that the fusel oil contained in spirits +constitutes the chief danger in the consumption of alcoholic +beverages. The expert can immediately detect the peculiarly +virulent characters of the mixed intoxication due to the consumption +of spirits containing a large percentage of fusel oil.</p> + + +<hr class="art" /> +<p><span class="bold">FUSIBLE METAL,<a name="ar102" id="ar102"></a></span> a term applied to certain alloys, generally +composed of bismuth, lead and tin, which possess the property of +melting at comparatively low temperatures. Newton’s fusible +metal (named after Sir Isaac Newton) contains 50 parts of +bismuth, 31.25 of lead and 18.75 of tin; that of Jean Darcet +(1725-1801), 50 parts of bismuth with 25 each of lead and tin; +and that of Valentin Rose the elder, 50 of bismuth with 28.1 of +lead and 24.1 of tin. These melt between 91° and 95° C. The +addition of cadmium gives still greater fusibility; in Wood’s +metal, for instance, which is Darcet’s metal with half the tin +replaced by cadmium, the melting point is lowered to 66°-71° C.; +while another described by Lipowitz and containing 15 parts of +bismuth, 8 of lead, 4 of tin and 3 of cadmium, softens at about +55° and is completely liquid a little above 60°. By the addition +of mercury to Darcet’s metal the melting point may be reduced +so low as 45°. These fusible metals have the peculiarity of expanding +as they cool; Rose’s metal, for instance, remains pasty +for a considerable range of temperature below its fusing point, +contracts somewhat rapidly from 80° to 55°, expands from 55° +to 35°, and contracts again from 35° to 0°. For this reason they +may be used for taking casts of anatomical specimens or making +<i>clichés</i> from wood-blocks, the expansion on cooling securing +sharp impressions. By suitable modification in the proportions +of the components, a series of alloys can be made which melt +at various temperatures above the boiling point of water; for +example, with 8 parts of bismuth, 8 of lead and 3 of tin the +melting point is 123°, and with 8 of bismuth, 30 of lead and 24 of +tin it is 172°. With tin and lead only in equal proportions it is +241°. Such alloys are used for making the fusible plugs inserted +in the furnace-crowns of steam boilers, as a safeguard in the event +of the water-level being allowed to fall too low. When this +happens the plug being no longer covered with water is heated +to such a temperature that it melts and allows the contents of +the boiler to escape into the furnace. In automatic fire-sprinklers +the orifices of the pipes are closed with fusible metal, which melts +and liberates the water when, owing to an outbreak of fire in +the room, the temperature rises above a predetermined limit.</p> + + +<hr class="art" /> +<p><span class="bold">FUSILIER,<a name="ar103" id="ar103"></a></span> originally (in French about 1670, in English about +1680) the name of a soldier armed with a light flintlock musket +called the fusil; now a regimental designation. Various forms +of flintlock small arms had been used in warfare since the middle +of the 16th century. At the time of the English civil war (1642-1652) +the term “firelock” was usually employed to distinguish +these weapons from the more common matchlock musket. The +special value of the firelock in armies of the 17th century lay +in the fact that the artillery of the time used open powder barrels +for the service of the guns, making it unsafe to allow lighted +matches in the muskets of the escort. Further, a military escort +was required, not only for the protection, but also for the +surveillance of the artillerymen of those days. Companies of +“firelocks” were therefore organized for these duties, and out of +these companies grew the “fusiliers” who were employed in +the same way in the wars of Louis XIV. In the latter part of +the Thirty Years’ War (1643) fusiliers were simply mounted +troops armed with the fusil, as carabiniers were with the carbine. +But the escort companies of artillery came to be known by the +name shortly afterwards, and the regiment of French Royal +Fusiliers, organized in 1671 by Vauban, was considered the model +for Europe. The general adoption of the flintlock musket and +the suppression of the pike in the armies of Europe put an end +to the original special duties of fusiliers, and they were subsequently +employed to a large extent in light infantry work, +perhaps on account of the greater individual aptitude for +detached duties naturally shown by soldiers who had never been +restricted to a fixed and unchangeable place in the line of battle. +The senior fusilier regiment in the British service, the (7th) +Royal Fusiliers (City of London Regiment), was formed on the +French model in 1685; the 5th foot (now Northumberland +Fusiliers), senior to the 7th in the army, was not at that time +a fusilier regiment. The distinctive head-dress of fusiliers in the +British service is a fur cap, generally resembling, but smaller +than and different in details from, that of the Foot Guards.</p> + +<p>In Germany the name “fusilier” is borne by certain infantry +regiments and by one battalion in each grenadier regiment.</p> + + +<hr class="art" /> +<p><span class="bold">FUSION,<a name="ar104" id="ar104"></a></span> the term generally applied to the melting of a solid +substance, or the change of state of aggregation from the solid +to the liquid. The term “liquefaction” is frequently employed +in the same sense, but is often restricted to the condensation +of a gas or vapour. The converse process of freezing or solidification, +the change from the liquid to the solid state, is subject to +the same laws, and must be considered together with fusion. +The solution of a solid in a foreign liquid, and the deposition or +crystallization of a solid from a solution, are so closely related +to the fusion of a pure substance, that it will also be necessary +to consider some of the analogies which they present.</p> + +<p>1. <i>General Phenomena.</i>—There are two chief varieties of the +process of fusion, namely, crystalline and amorphous, which are +in many ways distinct, although it is possible to find intermediate +cases which partake of the characteristics of both. The melting +of ice may be taken as a typical case of crystalline fusion. The +passage from rigid solid to mobile liquid occurs at a definite +surface without any intermediate stage or plastic condition. +The change takes place at a definite temperature, the fusing or +freezing point (abbreviated F.P.), and requires the addition +of a definite quantity of heat to the solid, which is called the +latent heat of fusion. There is also in general a considerable +change of volume during fusion, which amounts in the case of +ice to a contraction of 9%. Typical cases of amorphous solidification +are those of silica, glass, plastic sulphur, pitch, alcohol and +many organic liquids. In this type the liquid gradually becomes +more and more viscous as the temperature falls, and ultimately +attains the rigidity characteristic of a solid, without any definite +freezing point or latent heat. The condition of the substance +remains uniform throughout, if its temperature is uniform; +there is no separation into the two distinct phases of solid +and liquid, and there is no sudden change of volume at any +temperature.</p> + +<p>A change or transition from one crystalline form to another +may occur in the solid state with evolution or absorption of +heat at a definite temperature, and is analogous to the change +from solid to liquid, but usually takes place more slowly owing +to the small molecular mobility of the solid state. Thus +rhombic sulphur when heated passes slowly at 95.6° C. into the +<span class="pagenum"><a name="page370" id="page370"></a>370</span> +monosymmetric form which melts at 120°, but if heated rapidly +the rhombic form melts at 114.5. The two forms, rhombic and +monosymmetric, can exist in equilibrium at 95.6°, the transition +point at which they have the same vapour pressure. Similarly +a solid solution of carbon in iron, when cooled slowly, passes +at about 700° C., with considerable evolution of heat, into the +form of “pearlite,” which is soft when cold, but if rapidly chilled +the carbon remains in solution and the steel is very hard (see +also <span class="sc"><a href="#artlinks">Alloys</a></span>).</p> + +<p>In the case of crystalline fusion it is necessary to distinguish +two cases, the homogeneous and the heterogeneous. In the first +case the composition of the solid and liquid phases are the same, +and the temperature remains constant during the whole process +of fusion. In the second case the solid and liquid phases differ +in composition; that of the liquid phase changes continuously, +and the temperature does not remain constant during the fusion. +The first case comprises the fusion of pure substances, and +that of eutectics, or cryohydrates; the second is the general +case of an alloy or a solution. These have been very fully +studied and their phenomena greatly elucidated in recent +years.</p> + +<p>There is also a sub-variety of amorphous fusion, which may +be styled colloid or gelatinous, and may be illustrated by the +behaviour of solutions of water in gelatin. Many of these jellies +melt at a fairly definite temperature on heating, and coagulate or +set at a definite temperature on cooling. But in some cases the +process is not reversible, and there is generally marked hysteresis, +the temperature of setting and other phenomena depending on +the rate of cooling. This case has not yet been fully worked out; +but it appears probable that in many cases the jelly possesses +a spongy framework of solid, holding liquid in its meshes or +interstices. It might be regarded as a case of “heterogeneous” +amorphous fusion, in which the liquid separates into two phases +of different composition, one of which solidifies before the other. +The two phases cannot, as a rule, be distinguished optically, +but it is generally possible to squeeze out some of the liquid +phase when the jelly has set, which proves that the substance +is not really homogeneous. In very complicated mixtures, such +as acid lavas or slags containing a large proportion of silica, +amorphous and crystalline solidification may occur together. +In this case the crystals separate first during the process of +cooling, the mother liquor increases gradually in viscosity, and +finally sets as an amorphous ground-mass or matrix, in which +crystals of different kinds and sizes, formed at different stages +of the cooling, remain embedded. The formation of crystals +in an amorphous solid after it has set is also of frequent +occurrence. It is termed devitrification, but is a very slow +process unless the solid is in a plastic state.</p> + +<p>2. <i>Homogeneous Crystalline Fusion.</i>—The fusion of a solid of +this type is characterized most clearly by the perfect constancy +of temperature during the process. In fact, the law of constant +temperature, which is generally stated as the first of the so-called +“laws of fusion,” does not strictly apply except to this case. +The constancy of the F.P. of a pure substance is so characteristic +that change of the F.P. is often one of the most convenient tests +of the presence of foreign material. In the case of substances +like ice, which melt at a low temperature and are easily obtained +in large quantities in a state of purity, the point of fusion may +be very accurately determined by observing the temperature +of an intimate mixture of the solid and liquid while slowly +melting as it absorbs heat from surrounding bodies. But in the +majority of cases it is more convenient to observe the freezing +point as the liquid is cooled. By this method it is possible to +ensure perfect uniformity of temperature throughout the mass +by stirring the liquid continuously during the process of freezing, +whereas it is difficult to ensure uniformity of temperature in +melting a solid, however gradually the heat is supplied, unless +the solid can be mixed with the liquid. It is also possible to +observe the F.P. in other ways, as by noting the temperature +at the moment of the breaking of a wire, of the stoppage of a +stirrer, or of the maximum rate of change of volume, but these +methods are generally less certain in their indications than the +point of greatest constancy of temperature in the case of homogeneous +crystalline solids.</p> + +<div class="condensed"> +<p class="pt2 center"><i>Fusing Points of Common Metals</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Mercury</td> <td class="tcr rb">−38.8°</td> <td class="tcl">Antimony</td> <td class="tcr">630°</td></tr> +<tr><td class="tcl">Potassium</td> <td class="tcr rb">62.5°</td> <td class="tcl">Aluminium</td> <td class="tcr">655°</td></tr> +<tr><td class="tcl">Sodium</td> <td class="tcr rb">95.6°</td> <td class="tcl">Silver</td> <td class="tcr">962°</td></tr> +<tr><td class="tcl">Tin</td> <td class="tcr rb">231.9°</td> <td class="tcl">Gold</td> <td class="tcr">1064°</td></tr> +<tr><td class="tcl">Bismuth</td> <td class="tcr rb">269.2°</td> <td class="tcl">Copper</td> <td class="tcr">1082°</td></tr> +<tr><td class="tcl">Cadmium</td> <td class="tcr rb">320.7°</td> <td class="tcl">Nickel</td> <td class="tcr">1427°</td></tr> +<tr><td class="tcl">Lead</td> <td class="tcr rb">327.7°</td> <td class="tcl">Palladium</td> <td class="tcr">1535°</td></tr> +<tr><td class="tcl">Zinc</td> <td class="tcr rb">419.0°</td> <td class="tcl">Platinum</td> <td class="tcr">1710°</td></tr> +</table></div> + +<p>The above table contains some of the most recent values of +fusing points of metals determined (except the first three and +the last three) with platinum thermometers. The last three +values are those obtained by extrapolation with platinum-rhodium +and platinum-iridium couples. (See Harker, <i>Proc. +Roy. Soc.</i> A 76, p. 235, 1905.) Some doubt has recently been +raised with regard to the value for platinum, which is much +lower than that previously accepted, namely 1775°.</p> + +<p>3. <i>Superfusion, Supersaturation.</i>—It is generally possible to +cool a liquid several degrees below its normal freezing point +without a separation of crystals, especially if it is protected +from agitation, which would assist the molecules to rearrange +themselves. A liquid in this state is said to be “undercooled” +or “superfused.” The phenomenon is even more familiar in +the case of solutions (<i>e.g.</i> sodium sulphate or acetate) which may +remain in the “metastable” condition for an indefinite time +if protected from dust, &c. The introduction into the liquid +under this condition of the smallest fragment of the crystal, +with respect to which the solution is supersaturated, will produce +immediate crystallization, which will continue until the +temperature is raised to the saturation point by the liberation +of the latent heat of fusion. The constancy of temperature at +the normal freezing point is due to the equilibrium of exchange +existing between the liquid and solid. Unless both solid and +liquid are present, there is no condition of equilibrium, and the +temperature is indeterminate.</p> + +<p>It has been shown by H.A. Miers (<i>Jour. Chem. Soc.</i>, 1906, 89, +p. 413) that for a supersaturated solution in metastable equilibrium +there is an inferior limit of temperature, at which it passes +into the “labile” state, <i>i.e.</i> spontaneous crystallization occurs +throughout the mass in a fine shower. This seems to be analogous +to the fine misty condensation which occurs in a supersaturated +vapour in the absence of nuclei (see <span class="sc"><a href="#artlinks">Vaporization</a></span>) when the +supersaturation exceeds a certain limit.</p> + +<div class="condensed"> +<p>4. <i>Effect of Pressure on the F.P.</i>—The effect of pressure on the +fusing-point depends on the change of volume during fusion. Substances +which expand on <i>freezing</i>, like ice, have their freezing points +lowered by increase of pressure; substances which expand on +<i>fusing</i>, like wax, have their melting points raised by pressure. +In each case the effect of pressure is to retard increase of volume. +This effect was first predicted by James Thomson on the analogy +of the effect of pressure on the boiling point, and was numerically +verified by Lord Kelvin in the case of ice, and later by Bunsen in +the case of paraffin and spermaceti. The equation by which the +change of the F.P. is calculated may be proved by a simple application +of the Carnot cycle, exactly as in the case of vapour and liquid. +(See <span class="sc"><a href="#artlinks">Thermodynamics</a></span>.) If L be the latent heat of fusion in +mechanical units, v′ the volume of unit mass of the solid, and v″ +that of the liquid, the work done in an elementary Carnot cycle of +range dθ will be dp(v″ − v′), if dp is the increase of pressure required +to produce a change dθ in the F.P. Since the ratio of the work-difference +or cycle-area to the heat-transferred L must be equal to +dθ/θ, we have the relation</p> + +<p class="center">dθ/dp = θ (v″ − v′)/L.</p> +<div class="author">(1)</div> + +<p class="noind">The sign of dθ, the change of the F.P., is the same as that of the +change of volume (v″ − v′). Since the change of volume seldom +exceeds 0.1 c.c. per gramme, the change of the F.P. per atmosphere +is so small that it is not as a rule necessary to take account of variations +of atmospheric pressure in observing a freezing point. A +variation of 1 cm. in the height of the barometer would correspond +to a change of .0001° C. only in the F.P. of ice. This is far beyond the +limits of accuracy of most observations. Although the effect of +pressure is so small, it produces, as is well known, remarkable +results in the motion of glaciers, the moulding and regelation of +ice, and many other phenomena. It has also been employed to +explain the apparent inversion of the order of crystallization in +rocks like granite, in which the arrangement of the crystals indicates +that the quartz matrix solidified subsequently to the crystals of +<span class="pagenum"><a name="page371" id="page371"></a>371</span> +felspar, mica or hornblende embedded in it, although the quartz +has a higher melting point. It is contended that under enormous +pressure the freezing points of the more fusible constituents might +be raised above that of the quartz, if the latter is less affected by +pressure. Thus Bunsen found the F.P. of paraffin wax 1.4° C. +below that of spermaceti at atmospheric pressure. At 100 atmospheres +the two melted at the same temperature. At higher pressures +the paraffin would solidify first. The effect of pressure on the +silicates, however, is much smaller, and it is not so easy to explain +a change of several hundred degrees in the F.P. It seems more +likely in this particular case that the order of crystallization depends +on the action of superheated water or steam at high temperatures +and pressures, which is well known to exert a highly solvent and +metamorphic action on silicates.</p> + +<p>5. <i>Variation of Latent Heat.</i>—C.C. Person in 1847 endeavoured to +show by the application of the first law of thermodynamics that +the increase of the latent heat per degree should be equal to the +difference (s″ − s′) between the specific heats of the liquid and solid. +If, for instance, water at 0° C. were first frozen and then cooled to +−t° C., the heat abstracted per gramme would be (L′ + s′t) calories. +But if the water were first cooled to −t° C., and then frozen at −t°C., +by abstracting heat L″, the heat abstracted would be L″ + s″t. +Assuming that the heat abstracted should be the same in the two +cases, we evidently obtain L′ − L″ = (s″ − s′)t. This theory has been +approximately verified by Petterson, by observing the freezing of a +liquid cooled below its normal F.P. (<i>Jour. Chem. Soc.</i> 24, p. 151). +But his method does not represent the true variation of the latent +heat with temperature, since the freezing, in the case of a superfused +liquid, really takes place at the normal freezing point. A quantity +of heat s″t is abstracted in cooling to −t, (L″ − s″t) in raising to 0° +and freezing at 0°, and s’t in cooling the ice to -t. The latent heat +L″ at −t does not really enter into the experiment. In order to +make the liquid freeze at a different temperature, it is necessary to +subject it to pressure, and the effect of the pressure on the latent +heat cannot be neglected. The entropy of a liquid φ″ at its F.P. +reckoned from any convenient zero φ<span class="su">0</span> in the solid state may be +represented by the expression</p> + +<p class="center">φ″ − φ<span class="su">0</span> = ∫ s′dθ/θ + L/θ.</p> +<div class="author">(2)</div> + +<p class="noind">Since θdφ″/dθ = s″, we obtain by differentiation the relation</p> + +<p class="center">dL/dθ = s″ − s′ + L/θ,</p> +<div class="author">(3)</div> + +<p class="noind">which is exactly similar to the equation for the specific heat of a +vapour maintained in the saturated condition. If we suppose that +the specific heats s′ and s″ of the solid and liquid at equilibrium +pressure are nearly the same as those ordinarily observed at constant +pressure, the relation (3) differs from that of Person only by +the addition of the term L/θ. Since s″ is greater than s′ in all cases +hitherto investigated, and L/θ is necessarily positive, it is clear that +the latent heat of fusion must increase with rise of temperature, or +diminish with fall of temperature. It is possible to imagine the F.P. +so lowered by pressure (positive or negative) that the latent heat +should vanish, in which case we should probably obtain a continuous +passage from the liquid to the solid state similar to that which +occurs in the case of amorphous substances. According to equation +(3), the rate of change of the latent heat of water is approximately +0.80 calorie per degree at 0° C. (as compared with 0.50, Person), +if we assume s″ = 1, and s′ = 0.5. Putting (s″ − s′) = 0.5 in equation +(2), we find L = 0 at −160° C. approximately, but no stress can be +laid on this estimate, as the variation of (s″ − s′) is so uncertain.</p> +</div> + +<table class="flt" style="float: right; width: 360px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:307px; height:351px" src="images/img371a.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 1.</span>—F.P. or Solubility +Curve: simple case.</td></tr></table> + +<p>6. <i>Freezing of Solutions and Alloys.</i>—The phenomena of +freezing of heterogeneous crystalline mixtures may be illustrated +by the case of aqueous solutions and of metallic solutions or +alloys, which have been most widely studied. The usual effect +of an impurity, such as salt or sugar in solution in water, is to +lower the freezing point, so that no crystallization occurs until +the temperature has fallen below the normal F.P. of the pure +solvent, the depression of F.P. being nearly proportional to the +concentration of the solution. When freezing begins, the solvent +generally separates out from the solution in the pure state. This +separation of the solvent involves an increase in the strength +of the remaining solution, so that the temperature does not +remain constant during the freezing, but continues to fall as +more of the solvent is separated. There is a perfectly definite +relation between temperature and concentration at each stage +of the process, which may be represented in the form of a curve +as AC in fig. 1, called the freezing point curve. The equilibrium +temperature, at the surface of contact between the solid and +liquid, depends only on the composition of the liquid phase and +not at all on the quantity of solid present. The abscissa of the +F.P. curve represents the composition of that portion of the +original solution which remains liquid at any temperature. If +instead of starting with a dilute solution we start with a strong +solution represented by a point N, and cool it as shown by the +vertical line ND, a point D is generally reached at which the +solution becomes “saturated.” The dissolved substance or +“solute” then separates out as the solution is further cooled, +and the concentration diminishes with fall of temperature in +a definite relation, as indicated by the curve CB, which is called +the solubility curve. Though often called by different names, +the two curves AC and CB are +essentially of a similar nature. +To take the case of an aqueous +solution of salt as an example, +along CB the solution is saturated +with respect to salt, along +AC the solution is saturated with +respect to ice. When the point +C is reached along either curve, +the solution is saturated with +respect to both salt and ice. +The concentration cannot vary +further, and the temperature +remains constant, while the salt +and ice crystallize out together, +maintaining the exact proportions +in which they exist in the solution. The resulting solid was +termed a cryohydrate by F. Guthrie, but it is really an intimate +mixture of two kinds of crystals, and not a chemical compound +or hydrate containing the constituents in chemically equivalent +proportions. The lowest temperature attainable by means of a +freezing mixture is the temperature of the F.P. of the corresponding +cryohydrate. In a mixture of salt and ice with the least +trace of water a saturated brine is quickly formed, which dissolves +the ice and falls rapidly in temperature, owing to the absorption +of the latent heat of fusion. So long as both ice and salt are +present, if the mixture is well stirred, the solution must necessarily +become saturated with respect to both ice and salt, and this can +only occur at the cryohydric temperature, at which the two +curves of solubility intersect.</p> + +<p>The curves in fig. 1 also illustrate the simplest type of freezing +point curve in the case of alloys of two metals A and B which +do not form mixed crystals or chemical compounds. The alloy +corresponding to the cryohydrate, possessing the lowest melting +point, is called the eutectic alloy, as it is most easily cast and +worked. It generally possesses a very fine-grained structure, +and is not a chemical compound. (See <span class="sc"><a href="#artlinks">Alloys</a></span>.)</p> + +<table class="flt" style="float: right; width: 310px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:256px; height:349px" src="images/img371b.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 2.</span>—Cooling Curves +of Alloys: typical case.</td></tr></table> + +<p>To obtain a complete F.P. curve even for a binary alloy is a +laborious and complicated process, but the information contained +in such a curve is often very valuable. It is necessary to operate +with a number of different alloys of suitably chosen composition, +and to observe the freezing points of each separately. Each alloy +should also be analysed after the process if there is any risk of +its composition having been altered by oxidation or otherwise. +The freezing points are generally best +determined by observing the gradual +cooling of a considerable mass, which +is well stirred so long as it remains +liquid. The curve of cooling may most +conveniently be recorded, either photographically, +using a thermocouple and +galvanometer, as in the method of Sir +W. Roberts-Austen, or with pen and +ink, if a platinum thermometer is available, +according to the method put in +practice by C.T. Heycock and F.H. +Neville. A typical set of curves obtained +in this manner is shown in fig. 2. When +the pure metal A in cooling reaches its +F.P. the temperature suddenly becomes +stationary, and remains accurately constant for a considerable +period. Often it falls slightly below the F.P. owing to super-fusion, +but rises to the F.P. and remains constant as soon as +freezing begins. The second curve shows the cooling of A with +10% of another metal B added. The freezing begins at a lower +temperature with the separation of pure A. The temperature +<span class="pagenum"><a name="page372" id="page372"></a>372</span> +no longer remains constant during freezing, but falls more and +more rapidly as the proportion of B in the liquid increases. +When the eutectic temperature is reached there is a second +F.P. or arrest at which the whole of the remaining liquid solidifies. +With 20% of B the first F.P. is further lowered, and the temperature +falls faster. The eutectic F.P. is of longer duration, but +still at the same temperature. For an alloy of the composition +of the eutectic itself there is no arrest until the eutectic temperature +is reached, at which the whole solidifies without change of +temperature. There is a great advantage in recording these +curves automatically, as the primary arrest is often very slight, +and difficult to observe in any other way.</p> + +<div class="condensed"> +<p>7. <i>Change of Solubility with Temperature.</i>—The lowering of the +F.P. of a solution with increase of concentration, as shown by the +F.P. or solubility curves, may be explained and calculated by +equation (1) in terms of the osmotic pressure of the dissolved substance +by analogy with the effect of mechanical pressure. It is +possible in salt solutions to strain out the salt mechanically by a +suitable filter or “semi-permeable membrane,” which permits the +water to pass, but retains the salt. To separate 1 gramme of +salt requires the performance of work PV against the osmotic +pressure P, where V is the corresponding diminution in the volume +of the solution. In dilute solutions, to which alone the following +calculation can be applied, the volume V is the reciprocal of the +concentration C of the solution in grammes per unit volume, and +the osmotic pressure P is equal to that of an equal number of molecules +of gas in the same space, and may be deduced from the usual +equation of a gas,</p> + +<p class="center">P = Rθ / VM = RθC / M, </p> +<div class="author">(4)</div> + +<p class="noind">where M is the molecular weight of the salt in solution, θ the absolute +temperature, and R a constant which has the value 8.32 joules, +or nearly 2 calories, per degree C. It is necessary to consider +two cases, corresponding to the curves CB and AB in fig. 1, in +which the solution is saturated with respect to salt and water +respectively. To facilitate description we take the case of a salt +dissolved in water, but similar results apply to solutions in other +liquids and alloys of metals.</p> + +<p>(<i>a</i>) If unit mass of salt is separated in the solid state from a saturated +solution of salt (curve CB) by forcing out through a semi-permeable +membrane against the osmotic pressure P the corresponding +volume of water V in which it is dissolved, the heat evolved +is the latent heat of saturated solution of the salt Q together with +the work done PV. Writing (Q + PV) for L, and V for (v″ − v′) in +equation (1), and substituting P for p, we obtain</p> + +<p class="center">Q + PV = VθdP / dθ,</p> +<div class="author">(5)</div> + +<p class="noind">which is equivalent to equation (1), and may be established by +similar reasoning. Substituting for P and V in terms of C from +equation (4), if Q is measured in calories, R = 2, and we obtain</p> + +<p class="center">QC = 2θ²dC / dθ,</p> +<div class="author">(6)</div> + +<p class="noind">which may be integrated, assuming Q constant, with the result</p> + +<p class="center">2log<span class="su">e</span>C″ / C′ = Q / θ′ − Q / θ″,</p> +<div class="author">(7)</div> + +<p class="noind">where C′, C″ are the concentrations of the saturated solution corresponding +to the temperatures θ′ and θ″. This equation may be +employed to calculate the latent heat of solution Q from two observations +of the solubility. It follows from these equations that +Q is of the same sign as dC/dθ, that is to say, the solubility increases +with rise of temperature if heat is absorbed in the formation of the +saturated solution, which is the usual case. If, on the other hand, +heat is liberated on solution, as in the case of caustic potash or +sulphate of calcium, the solubility diminishes with rise of temperature.</p> + +<p>(<i>b</i>) In the case of a solution saturated with respect to ice (curve +AC), if one gramme of water having a volume v is separated by freezing, +we obtain a precisely similar equation to (5), but with L the latent +heat of fusion of water instead of Q, and v instead of V. If the +solution is dilute, we may neglect the external work Pv in comparison +with L, and also the heat of dilution, and may write P/t for dP/dθ, +where t is the depression of the F.P. below that of the pure solvent. +Substituting for P in terms of V from equation (4), we obtain</p> + +<p class="center">t = 2θ²v / LVM = 2θ²w / LWM,</p> +<div class="author">(8)</div> + +<p class="noind">where W is the weight of water and w that of salt in a given volume +of solution. If M grammes of salt are dissolved in 100 of water, +w = M and W = 100. The depression of the F.P. in this case is +called by van ‘t Hoff the “Molecular Depression of the F.P.” and +is given by the simple formula</p> + +<p class="center">t = .02θ² / L.</p> +<div class="author">(9)</div> + +<p class="noind">Equation (8) may be used to calculate L or M, if either is known, +from observations of t, θ and w/W. The results obtained are +sufficiently approximate to be of use in many cases in spite of the +rather liberal assumptions and approximations effected in the +course of the reasoning. In any case the equations give a simple +theoretical basis with which to compare experimental data in order +to estimate the order of error involved in the assumptions. We +may thus estimate the variation of the osmotic pressure from the +value given by the gaseous equation, as the concentration of the +solution or the molecular dissociation changes. The most uncertain +factor in the formula is the molecular weight M, since the +molecule in solution may be quite different from that denoted by +the chemical formula of the solid. In many cases the molecule of +a metal in dilute solution in another metal is either monatomic, or +forms a compound molecule with the solvent containing one atom +of the dissolved metal, in which case the molecular depression is +given by putting the atomic weight for M. In other cases, as +Cu, Hg, Zn, in solution in cadmium, the depression of the F.P. +per atom, according to Heycock and Neville, is only half as great, +which would imply a diatomic molecule. Similarly As and Au in +Cd appear to be triatomic, and Sn in Pb tetratomic. Intermediate +cases may occur in which different molecules exist together in +equilibrium in proportions which vary according to the temperature +and concentration. The most familiar case is that of an electrolyte, +in which the molecule of the dissolved substance is partly dissociated +into ions. In such cases the degree of dissociation may be estimated +by observing the depression of the F.P., but the results obtained +cannot always be reconciled with those deduced by other methods, +such as measurement of electrical conductivity, and there are many +difficulties which await satisfactory interpretation.</p> + +<p>Exactly similar relations to (8) and (9) apply to changes of boiling +point or vapour pressure produced by substances in solution (see +<span class="sc"><a href="#artlinks">Vaporization</a></span>), the laws of which are very closely connected with +the corresponding phenomena of fusion; but the consideration of +the vapour phase may generally be omitted in dealing with the fusion +of mixtures where the vapour pressure of either constituent is small.</p> +</div> + +<table class="flt" style="float: right; width: 430px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:372px; height:235px" src="images/img372.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 3.</span>—Solubility Curves of +Hydrates.</td></tr></table> + +<p>8. <i>Hydrates.</i>—The simple case of a freezing point curve, +illustrated in fig. 1, is generally modified by the occurrence +of compounds of a character analogous to hydrates of soluble +salts, in which the dissolved substance combines with one or +more molecules of the solvent. These hydrates may exist as +compound molecules in the solution, but their composition +cannot be demonstrated unless they can be separated in the solid +state. Corresponding to each crystalline hydrate there is generally +a separate branch of the solubility curve along which the +crystals of the hydrate are in equilibrium with the saturated +solution. At any given temperature the hydrate possessing the +least solubility is the most stable. If two are present in contact +with the same solution, the more soluble will dissolve, and the +less soluble will be formed at its expense until the conversion +is complete. The two hydrates cannot be in equilibrium with the +same solution except at the temperature at which their solubilities +are equal, <i>i.e.</i> at the point where the corresponding curves +of solubility intersect. This temperature is called the “Transition +Point.” In the case of ZnSO<span class="su">4</span>, as shown in fig. 3, the heptahydrate, +with seven molecules of water, is the least soluble +hydrate at ordinary temperatures, +and is generally +deposited from saturated +solutions. Above 39° C., +however, the hexahydrate, +with six molecules, is less +soluble, and a rapid conversion +of the hepta- into the +hexahydrate occurs if the +former is heated above the +transition point. The solubility +of the hexahydrate is +greater than that of the heptahydrate below 39°, but increases +more slowly with rise of temperature. At about 80° C. +the hexahydrate gives place to the monohydrate, which +dissolves in water with evolution of heat, and diminishes in +solubility with rise of temperature. Intermediate hydrates +exist, but they are more soluble, and cannot be readily isolated. +Both the mono- and hexahydrates are capable of existing in +equilibrium with saturated solutions at temperatures far below +their transition points, provided that the less soluble hydrate +is not present in the crystalline form. The solubility curves can +therefore be traced, as in fig. 3, over an extended range of temperature. +The equilibrium of each hydrate with the solvent, +considered separately, would present a diagram of two branches +similar to fig. 1, but as a rule only a small portion of each curve +can be realized, and the complete solubility curve, as experimentally +determined, is composed of a number of separate +pieces corresponding to the ranges of minimum solubility of +different hydrates. Failure to recognize this, coupled with the +<span class="pagenum"><a name="page373" id="page373"></a>373</span> +fact that in strong and viscous solutions the state of equilibrium +is but slowly attained, is the probable explanation of the remarkable +discrepancies existing in many recorded data of solubility.</p> + +<div class="condensed"> +<p class="pt2 center"><i>Transition Points of Hydrates.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Na<span class="su">2</span>CrO<span class="su">4</span>·10H<span class="su">2</span>O</td> <td class="tcc rb">19.9°</td> + <td class="tcl">NaBr·2H<span class="su">2</span>0</td> <td class="tcc">50.7°</td></tr> + +<tr><td class="tcl">Na<span class="su">2</span>SO<span class="su">4</span>·10H<span class="su">2</span>O</td> <td class="tcc rb">32.4°</td> + <td class="tcl">MnCl<span class="su">2</span>·4H<span class="su">2</span>O</td> <td class="tcc">57.8°</td></tr> + +<tr><td class="tcl">Na<span class="su">2</span>CO<span class="su">3</span>·10H<span class="su">2</span>O</td> <td class="tcc rb">35.1°</td> + <td class="tcl">Na<span class="su">3</span>PO<span class="su">4</span>·12H<span class="su">2</span>O</td> <td class="tcc">73.4°</td></tr> + +<tr><td class="tcl">Na<span class="su">2</span>S<span class="su">2</span>O<span class="su">3</span>·5H<span class="su">2</span>O</td> <td class="tcc rb">48.0°</td> + <td class="tcl">Ba(OH)<span class="su">2</span>·8H<span class="su">2</span>O</td> <td class="tcc">77.9°</td></tr> +</table></div> + +<p>The transition points of the hydrates given in the above list +(Richards, <i>Proc. Amer. Acad.</i>, 1899, 34, p. 277) afford well-marked +constant temperatures which can be utilized as fixed +points for experimental purposes.</p> + +<p>9. <i>Formation of Mixed Crystals.</i>—An important exception +to the general type already described, in which the addition of a +dissolved substance lowers the F.P. of the solvent, is presented +by the formation of mixed crystals, or “solid solutions,” in +which the solvent and solute occur mixed in varying proportions. +This isomorphous replacement of one substance by another, in +the same crystal with little or no change of form, has long been +known and studied in the case of minerals and salts, but the +relations between composition and melting-point have seldom +been investigated, and much still remains obscure. In this case +the process of freezing does not necessitate the performance of +work of separation of the constituents of the solution, the F.P. +is not necessarily depressed, and the effect cannot be calculated +by the usual formula for dilute solutions. One of the simplest +types of F.P. curve which may result from the occurrence of +mixed crystals is illustrated by the case of alloys of gold and +silver, or gold and platinum, in which the F.P. curve is nearly +a straight line joining the freezing-points of the constituents. +The equilibrium between the solid and liquid, in both of which +the two metals are capable of mixing in all proportions, bears in +this case an obvious and close analogy to the equilibrium between +a mixed liquid (<i>e.g.</i> alcohol and water) and its vapour. In the +latter case, as is well known, the vapour will contain a larger +proportion of the more volatile constituent. Similarly in the case +of the formation of mixed crystals, the liquid should contain +a larger proportion of the more fusible constituent than the solid +with which it is in equilibrium. The composition of the crystals +which are being deposited at any moment will, therefore, +necessarily change as solidification proceeds, following the +change in the composition of the liquid, and the temperature +will fall until the last portions of the liquid to solidify will consist +chiefly of the more fusible constituent, at the F.P. of which the +solidification will be complete. If, however, as seems to be +frequently the case, the composition of the solid and liquid phases +do not greatly differ from each other, the greater part of the +solidification will occur within a comparatively small range of +temperature, and the initial F.P. of the alloy will be well marked. +It is possible in this case to draw a second curve representing +the composition of the <i>solid</i> phase which is in equilibrium with +the liquid at any temperature. This curve will not represent the +average composition of the crystals, but that of the outer coating +only which is in equilibrium with the liquid at the moment. +H.W.B. Roozeboom (<i>Zeit. Phys. Chem.</i> xxx. p. 385) has +attempted to classify some of the possible cases which may +occur in the formation of mixed crystals on the basis of J.W. +Gibbs’s thermodynamic potential, the general properties of which +may be qualitatively deduced from a consideration of observed +phenomena. But although this method may enable us to classify +different types, and even to predict results in a qualitative +manner, it does not admit of numerical calculation similar to +equation (8), as the Gibbs’s function itself is of a purely abstract +nature and its form is unknown. There is no doubt that the +formation of mixed crystals may explain many apparent +anomalies in the study of F.P. curves. The whole subject has +been most fruitful of results in recent years, and appears full of +promise for the future.</p> + +<div class="condensed"> +<p>For further details in this particular branch the reader may consult +a report by Neville (<i>Brit. Assoc. Rep.</i>, 1900), which contains numerous +references to original papers by Roberts-Austen, Le Chatelier, +Roozeboom and others. For the properties of solutions see <span class="sc"><a href="#artlinks">Solution</a></span>.</p> +</div> +<div class="author">(H. L. C.)</div> + + +<hr class="art" /> +<p><span class="bold">FÜSSEN,<a name="ar105" id="ar105"></a></span> a town of Germany, in the kingdom of Bavaria, at +the foot of the Alps (Tirol), on the Lech, 2500 ft. above the sea, +with a branch line to Oberdorf on the railway to Augsburg. Pop. +4000. It has six Roman Catholic churches, a Franciscan monastery +and a castle. Rope-making is an important industry. +The castle, lying on a rocky eminence, is remarkable for the +peace signed here on the 22nd of April 1745 between the elector +Maximilian III., Joseph of Bavaria and Maria Theresa. Two +miles to the S.E., immediately on the Austrian frontier, romantically +situated on a rock overlooking the Schwanensee, is the +magnificent castle of Hohenschwangau, and a little to the north, +on the site of an old castle, that of Neuschwanstein, built by +Louis II. of Bavaria.</p> + +<div class="condensed"> +<p>See H. Feistle, <i>Füssen und Umgebung</i> (1898).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUST, JOHANN<a name="ar106" id="ar106"></a></span> (  ?-1466), early German printer, belonged +to a rich and respectable burgher family of Mainz, which is known +to have flourished from 1423, and to have held many civil and +religious offices. The name was always written Fust, but in +1506 Johann Schöffer, in dedicating the German translation of +Livy to the emperor Maximilian, called his grandfather Faust, +and thenceforward the family assumed this name, and the Fausts +of Aschaffenburg, an old and quite distinct family, placed +Johann Fust in their pedigree. Johann’s brother Jacob, a +goldsmith, was one of the burgomasters in 1462, when Mainz +was stormed and sacked by the troops of Count Adolf of Nassau, +on which occasion he seems to have perished (see a document, +dated May 8, 1463, published by Wyss in <i>Quartalbl. des hist. +Vereins für Hessen</i>, 1879, p. 24). There is no evidence that, as +is commonly asserted, Johann Fust was a goldsmith, but he +appears to have been a money-lender or banker. On account of +his connexion with Gutenberg (<i>q.v.</i>), he has been represented +by some as the inventor of printing, and the instructor as well as +the partner of Gutenberg, by others as his patron and benefactor, +who saw the value of his discovery and supplied him with means +to carry it out, whereas others paint him as a greedy and +crafty speculator, who took advantage of Gutenberg’s necessity +and robbed him of the fruits of his invention. However this may +be, the Helmasperger document of November 6, 1455, shows +that Fust advanced money to Gutenberg (apparently 800 +guilders in 1450, and another 800 in 1452) for carrying on his +work, and that Fust, in 1455, brought a suit against Gutenberg +to recover the money he had lent, claiming 2020 (more correctly +2026) guilders for principal and interest. It appears that he had +not paid in the 300 guilders a year which he had undertaken to +furnish for expenses, wages, &c., and, according to Gutenberg, +had said that he had no intention of claiming interest. The suit +was apparently decided in Fust’s favour, November 6, 1455, +in the refectory of the Barefooted Friars of Mainz, when Fust +made oath that he himself had borrowed 1550 guilders and +given them to Gutenberg. There is no evidence that Fust, as +is usually supposed, removed the portion of the printing materials +covered by his mortgage to his own house, and carried on printing +there with the aid of Peter Schöffer, of Gernsheim (who is known +to have been a scriptor at Paris in 1449), to whom, probably +about 1455,<a name="fa1i" id="fa1i" href="#ft1i"><span class="sp">1</span></a> he gave his only daughter Dyna or Christina in +marriage. Their first publication was the Psalter, August 14, +1457, a folio of 350 pages, the first printed book with a complete +date, and remarkable for the beauty of the large initials printed +each in two colours, red and blue, from types made in two +pieces.<a name="fa2i" id="fa2i" href="#ft2i"><span class="sp">2</span></a> The Psalter was reprinted with the same types, 1459 +(August 29), 1490, 1502 (Schöffer’s last publication) and 1516. +Fust and Schöffer’s other works are given below.<a name="fa3i" id="fa3i" href="#ft3i"><span class="sp">3</span></a> In 1464 Adolf +<span class="pagenum"><a name="page374" id="page374"></a>374</span> +of Nassau appointed for the parish of St Quintin three <i>Baumeisters</i> +(master-builders) who were to choose twelve chief parishioners +as assistants for life. One of the first of these “Vervaren,” +who were named on May-day 1464, was Johannes Fust, and in +1467 Adam von Hochheim was chosen instead of “the late” +(<i>selig</i>) Johannes Fust. Fust is said to have gone to Paris in 1466 +and to have died of the plague, which raged there in August and +September. He certainly was in Paris on the 4th of July, when +he gave Louis de Lavernade of the province of Forez, then +chancellor of the duke of Bourbon and first president of the +parliament of Toulouse, a copy of his second edition of Cicero, +as appears from a note in Lavernade’s own hand at the end of +the book, which is now in the library of Geneva. But nothing +further is known than that on the 30th of October, probably +in 1471, an annual mass was instituted for him by Peter Schöffer, +Conrad Henlif (for Henekes, or Henckis, Schöffer’s partner? +who married Fust’s widow about 1468<a name="fa4i" id="fa4i" href="#ft4i"><span class="sp">4</span></a>) and Johann Fust (the +son), in the abbey-church of St Victor of Paris, where he was +buried; and that Peter Schöffer founded a similar memorial +service for Fust in 1473 in the church of the Dominicans at +Mainz (Bockenheimer, <i>Gesch. der Stadt Mainz</i>, iv. 15).</p> + +<p>Fust was formerly often confused with the famous magician +Dr Johann Faust, who, though an historical figure, had nothing +to do with him (see <span class="sc"><a href="#artlinks">Faust</a></span>).</p> + +<div class="condensed"> +<p>See further the articles <span class="sc"><a href="#artlinks">Gutenberg</a></span> and <span class="sc"><a href="#artlinks">Typography</a></span>.</p> +</div> +<div class="author">(J. H. H.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1i" id="ft1i" href="#fa1i"><span class="fn">1</span></a> This date is uncertain; some place the marriage in 1453 or soon +after, others about 1464. It is probable that Fust alluded to this +relationship when he spoke of Schöffer as <i>pueri mei</i> in the colophons +of Cicero’s <i>De officiis</i> of 1465 and 1466.</p> + +<p><a name="ft2i" id="ft2i" href="#fa2i"><span class="fn">2</span></a> This method was patented in England by Solomon Henry in +1780, and by Sir William Congreve in 1819.</p> + +<p><a name="ft3i" id="ft3i" href="#fa3i"><span class="fn">3</span></a> (3) Durandus, <i>Rationale divinorum officiorum</i> (1459), folio, 160 +leaves; (4) the <i>Clementine Constitutions</i>, with the gloss of Johannes +Andreae (1460), 51 leaves; (5) <i>Biblia Sacra Latina</i> (1462), folio, +2 vols., 242 and 239 leaves, 48 lines to a full page; (6) the Sixth +Book of Decretals, with Andreae’s gloss, 17th December 1465, folio, +141 leaves; (7) Cicero, <i>De officiis</i> (1465). 4to, 88 leaves, the first +edition of a Latin classic and the first book containing Greek characters, +while in the colophon Fust for the first time calls Schöffer +“puerum suum”; (8) the same, 4th February 1466; (9) <i>Grammatica +rhytmica</i> (1466), folio, 11 leaves. They also printed in 1461-1462 +several papal bulls, proclamations of Adolf of Nassau, &c. Nothing +is known to have appeared for three years after the storming and +capture of Mainz in 1462.</p> + +<p><a name="ft4i" id="ft4i" href="#fa4i"><span class="fn">4</span></a> Some confusion in the history of the Fust family has arisen +since the publication of Bernard’s <i>Orig. de l’imprimerie</i> (1853). +On p. 262, vol. i. he gave an extract from the correspondence between +Oberlin and Bodmann (now preserved in the Paris Nat. Library), +from which it would appear that Peter Schöffer was the son-in-law, +not of Johann Fust, but of a brother of his, Conrad Fust. Of the +latter, however, no other trace has been found, and he is no doubt +a fiction of F.J. Bodmann, who, partly basing himself on the +“Conrad” (Henlif, or Henckis) mentioned above, added the rest +to gratify Oberlin (see Wyss in <i>Quartalblätter des hist. Vereins für +Hessen</i>, 1879, p. 17).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUSTEL DE COULANGES, NUMA DENIS<a name="ar107" id="ar107"></a></span> (1830-1889), French +historian, was born in Paris on the 18th of March 1830, of Breton +descent. After studying at the École Normale Supérieure he +was sent to the French school at Athens in 1853, directed some +excavations in Chios, and wrote an historical account of the +island. After his return he filled various educational offices, +and took his doctor’s degree with two theses, <i>Quid Vestae cultus +in institutis veterum privatis publicisque valuerit</i> and <i>Polybe, +ou la Grèce conquise par les Romains</i> (1858). In these works +his distinctive qualities were already revealed. His minute +knowledge of the language of the Greek and Roman institutions, +coupled with his low estimate of the conclusions of contemporary +scholars, led him to go direct to the original texts, which he read +without political or religious bias. When, however, he had +succeeded in extracting from the sources a general idea that +seemed to him clear and simple, he attached himself to it as if to +the truth itself, employing dialectic of the most penetrating, +subtle and even paradoxical character in his deduction of the +logical consequences. From 1860 to 1870 he was professor of +history at the faculty of letters at Strassburg, where he had a +brilliant career as a teacher, but never yielded to the influence +exercised by the German universities in the field of classical and +Germanic antiquities.</p> + +<p>It was at Strassburg that he published his remarkable volume +<i>La Cité antique</i> (1864), in which he showed forcibly the part +played by religion in the political and social evolution of Greece +and Rome. Although his making religion the sole factor of this +evolution was a perversion of the historical facts, the book was +so consistent throughout, so full of ingenious ideas, and written +in so striking a style, that it ranks as one of the masterpieces of +the French language in the 19th century. By this literary +merit Fustel set little store, but he clung tenaciously to his +theories. When he revised the book in 1875, his modifications +were very slight, and it is conceivable that, had he recast it, +as he often expressed the desire to do in the last years of his life, +he would not have abandoned any part of his fundamental +thesis. The work is now largely superseded.</p> + +<p>Fustel de Coulanges was the most conscientious of men, the +most systematic and uncompromising of historians. Appointed +to a lectureship at the École Normale Supérieure in February +1870, to a professorship at the Paris faculty of letters in 1875, +and to the chair of medieval history created for him at the +Sorbonne in 1878, he applied himself to the study of the political +institutions of ancient France. The invasion of France by +the German armies during the war of 1870-71 attracted his +attention to the Germanic invasions under the Roman Empire. +Pursuing the theory of J.B. Dubos, but singularly transforming +it, he maintained that those invasions were not marked by the +violent and destructive character usually attributed to them; +that the penetration of the German barbarians into Gaul was a +slow process; that the Germans submitted to the imperial +administration; that the political institutions of the Merovingians +had their origins in the Roman laws at least as much as, if not +more than, in German usages; and, consequently, that there was +no conquest of Gaul by the Germans. This thesis he sustained +brilliantly in his <i>Histoire des institutions politiques de l’ancienne +France</i>, the first volume of which appeared in 1874. It was the +author’s original intention to complete this work in four volumes, +but as the first volume was keenly attacked in Germany as well +as in France, Fustel was forced in self-defence to recast the book +entirely. With admirable conscientiousness he re-examined +all the texts and wrote a number of dissertations, of which, +though several (<i>e.g.</i> those on the Germanic mark and on the +<i>allodium</i> and <i>beneficium</i>) were models of learning and sagacity, +all were dominated by his general idea and characterized by a +total disregard for the results of such historical disciplines as +diplomatic. From this crucible issued an entirely new work, +less well arranged than the original, but richer in facts and +critical comments. The first volume was expanded into three +volumes, <i>La Gaule romaine</i> (1891), <i>L’Invasion germanique et +la fin de l’empire</i> (1891) and <i>La Monarchie franque</i> (1888), followed +by three other volumes, <i>L’Alleu et le domaine rural pendant +l’époque mérovingienne</i> (1889), <i>Les Origines du système féodal: +le bénéfice et le patronat ...</i> (1890) and <i>Les Transformations de +la royauté pendant l’époque carolingienne</i> (1892). Thus, in six +volumes, he had carried the work no farther than the Carolingian +period. The result of this enormous labour, albeit worthy of a +great historian, clearly showed that the author lacked all sense +of historical proportion. He was a diligent seeker after the truth, +and was perfectly sincere when he informed a critic of the exact +number of “truths” he had discovered, and when he remarked +to one of his pupils a few days before his death, “Rest assured +that what I have written in my book is the truth.” Such superb +self-confidence can accomplish much, and it undoubtedly helped +to form Fustel’s talent and to give to his style that admirable +concision which subjugates even when it fails to convince; +but a student instinctively distrusts an historian who settles the +most controverted problems with such impassioned assurance. +The dissertations not embodied in his great work were collected +by himself and (after his death) by his pupil, Camille Jullian, +and published as volumes of miscellanies: <i>Recherches sur +quelques problèmes d’histoire</i> (1885), dealing with the Roman +colonate, the land system in Normandy, the Germanic mark, and +the judiciary organization in the kingdom of the Franks; +<i>Nouvelles recherches sur quelques problèmes d’histoire</i> (1891); +and <i>Questions historiques</i> (1893), which contains his paper on +Chios and his thesis on Polybius.</p> + +<p>His life was devoted almost entirely to his teaching and his +books. In 1875 he was elected member of the Académie des +Sciences Morales, and in 1880 reluctantly accepted the post +of director of the École Normale. Without intervening personally +in French politics, he took a keen interest in the questions of +administration and social reorganization arising from the fall +of the imperialist régime and the disasters of the war. He wished +<span class="pagenum"><a name="page375" id="page375"></a>375</span> +the institutions of the present to approximate more closely to +those of the past, and devised for the new French constitution a +body of reforms which reflected the opinions he had formed +upon the democracy at Rome and in ancient France. But these +were dreams which did not hold him long, and he would have +been scandalized had he known that his name was subsequently +used as the emblem of a political and religious party. He died +at Massy (Seine-et-Oise) on the 12th of September 1889. Throughout +his historical career—at the École Normale and the Sorbonne +and in his lectures delivered to the empress Eugénie—his sole +aim was to ascertain the truth, and in the defence of truth his +polemics against what he imagined to be the blindness and +insincerity of his critics sometimes assumed a character of harshness +and injustice. But, in France at least, these critics were +the first to render justice to his learning, his talents and his +disinterestedness.</p> + +<div class="condensed"> +<p>See Paul Guiraud, <i>Fustel de Coulanges</i> (1896); H. d’Arbois de +Jubainville, <i>Deux Manières d’écrire l’histoire: critique de Bossuet, +d’Augustin Thierry et de Fustel de Coulanges</i> (1896); and Gabriel +Monod, <i>Portraits et souvenirs</i> (1897).</p> +</div> +<div class="author">(C. B.*)</div> + + +<hr class="art" /> +<p><span class="bold">FUSTIAN,<a name="ar108" id="ar108"></a></span> a term which includes a variety of heavy woven +cotton fabrics, chiefly prepared for men’s wear. It embraces +plain twilled cloth called jean, and cut fabrics similar to velvet, +known as velveteen, moleskin, corduroy, &c. The term was +once applied to a coarse cloth made of cotton and flax; now, +fustians are usually of cotton and dyed various colours. In the +reign of Edward III. the name was given to a woollen fabric. +The name is said to be derived from El-Fustat, a suburb of Cairo, +where it was first made; and certainly a kind of cloth has long +been known under that name. In a petition to parliament, +<i>temp.</i> Philip and Mary, “fustian of Naples” is mentioned. In +the 13th and 14th centuries priests’ robes and women’s dresses +were made of fustian, but though dresses are still made from +some kinds the chief use is for labourers’ clothes.</p> + + +<hr class="art" /> +<p><span class="bold">FUSTIC<a name="ar109" id="ar109"></a></span> (Fr. <i>fustoc</i>, from Arab. <i>fustuq</i>, Gr. <span class="grk" title="pistakê">πιστάκη</span>, pistachio) +<span class="sc">Yellow Wood</span> or <span class="sc">Old Fustic</span>, a dye-stuff consisting of the +wood of <i>Chlorophora tinctoria</i>, a large tree of the natural order +Moraceae, growing in the West Indies and tropical America. +Fustic occurs in commerce in blocks, which are brown without, +and of a brownish-yellow within. It is sometimes employed for +inlaid work. The dye-stuff termed young fustic or Zante fustic, +and also Venetian sumach, is the wood of <i>Rhus cotinus</i> (fustet, +or smoke tree), a southern European and Asiatic shrub of the +natural order Anacardiaceae, called by Gerarde “red sumach,” +and apparently the “coccygia” and “cotinus” of Pliny (<i>Nat. +Hist.</i> xiii. 41, xvi. 30). Its colouring matter is fisetin, C<span class="su">15</span>H<span class="su">10</span>O<span class="su">6</span>, +which was synthesized by S. von Kostanecki (<i>Ber.</i>, 1904, 37, +p. 384). (See <span class="sc"><a href="#artlinks">Dyeing</a></span>.)</p> + + +<hr class="art" /> +<p><span class="bold">FUTURES,<a name="ar110" id="ar110"></a></span> a term used in the produce markets for purchases +or sales of commodities to be completed at a future date, as +opposed to cash or “spot” transactions, which are settled +immediately. See <span class="sc"><a href="#artlinks">Market</a></span>, and (for a detailed discussion of +the question as affecting cotton) <span class="sc"><a href="#artlinks">Cotton</a></span>: <i>Marketing and Supply</i>.</p> + + +<hr class="art" /> +<p><span class="bold">FUX, JOHANN JOSEPH<a name="ar111" id="ar111"></a></span> (1660-1741), Austrian musician, +was born at Hirtenfeld (Styria) in 1660. Of his youth and +early training nothing is known. In 1696 he was organist at one +of the principal churches of Vienna, and in 1698 was appointed +by the emperor Leopold I. as his “imperial court-composer,” +with a salary of about £6 a month. At the court of Leopold and +of his successors Joseph I. and Charles VI., Fux remained for +the rest of his life. To his various court dignities that of organist +at St Stephen’s cathedral was added in 1704. He married the +daughter of the government secretary Schnitzbaum. As a +proof of the high favour in which he was held by the art-loving +Charles VI., it is told that at the coronation of that emperor +as king of Bohemia in 1723 an opera, <i>La Constanza e la Fortezza</i>, +especially composed by Fux for the occasion, was given at +Prague in an open-air theatre. Fux at the time was suffering +from gout, but the emperor had him carried in a litter all the +way from Vienna, and gave him a seat in the imperial box. +Fux died at Vienna on the 13th of February 1741. His life, +although passed in the great world, was eventless, and his only +troubles arose from the intrigues of his Italian rivals at court. +Of the numerous operas which Fux wrote it is unnecessary to +speak. They do not essentially differ from the style of the +Italian <i>opera seria</i> of the time. Of greater importance are his +sacred compositions, psalms, motets, oratorios and masses, +the celebrated <i>Missa Canonica</i> amongst the latter. It is an all +but unparalleled <i>tour de force</i> of learned musicianship, being +written entirely in that most difficult of contrapuntal devices—the +canon. As a contrapuntist and musical scholar generally, +Fux was unsurpassed by any of his contemporaries, and his +great theoretical work, the <i>Gradus ad Parnassum</i>, long +remained by far the most thorough treatment of counterpoint +and its various developments. The title of the original +Latin edition is <i>Gradus ad Parnassum sive manuductio ad +compositionem musicae regularem, methoda nova ac certa nondum +ante tam exacta ordine in lucem edita, elaborata a Joanne Josepho +Fux</i> (Vienna, 1715). It was translated into most European +languages during the 18th century, and is still studied by +musicians interested in the history of their art. The expenses +of the publication were defrayed by the emperor Charles VI.</p> + +<div class="condensed"> +<p>Fux’s biography was published by Ludwig von Köchel (Vienna, +1871). It is based on minute original research and contains, amongst +other valuable materials, a complete catalogue of the composer’s +numerous works.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">FUZE<a name="ar112" id="ar112"></a></span> or <span class="sc">Fuse</span>, an appliance for firing explosives in blasting +operations, military shells, &c. (see <span class="sc"><a href="#artlinks">Blasting</a></span> and <span class="sc"><a href="#artlinks">Ammunition</a></span>, +§ <i>Shell</i>). The spelling is not governed by authority, but modern +convenience has dictated the adoption of the “z” by military +engineers as a general rule, in order to distinguish this sense +from that of melting by heat (see below). The word, according +to the <i>New English Dictionary</i>, is one of the forms in which the +Lat. <i>fusus</i>, spindle, has been adapted through Romanic into +English, the ordinary fuze taking the shape of a spindle-like +tube. Similarly the term “fusee” (Fr. <i>fusée</i>, spindle full of tow, +Late Lat. <i>fusata</i>) is applied to a coned spindle sometimes used in +the wheel train of watches and spring clocks to equalize the action +of the mainspring (see <span class="sc"><a href="#artlinks">Watch</a></span>); and the application of the same +term to a special kind of match may also be due to its resemblance +to a spindle. Again, in heraldry, another form, “fusil,” derived +through the French from a Late Lat. diminutive (<i>fusillus</i> or +<i>fusellus</i>) of this same <i>fusus</i>, is used of a bearing, an elongated +lozenge. According to other etymological authorities, however +(see Skeat, <i>Etym. Dict.</i>, 1898), “fuze” or “fuse,” and “fusee” +in the sense of match, are all forms derived through the Fr. fusil, +from Late Lat. <i>focile</i>, steel for striking fire from a flint, from Lat. +<i>focus</i>, hearth. The Fr. <i>fusil</i> and English “fusil” were thus +transferred to the “firelock,” <i>i.e.</i> the light musket of the 17th +century (see <span class="sc"><a href="#artlinks">Fusilier</a></span>).</p> + +<p>In electrical engineering a “fuse” (always so spelled) is a +safety device, commonly consisting of a strip or wire of easily +fusible metal, which melts and thus interrupts the circuit of +which it forms part, whenever that circuit, through some accident +or derangement, is caused to carry a current larger than that +for which it is intended. In this sense the word must be connected +with <i>fusus</i>, the past participle of Lat. <i>fundere</i>, to pour, +whence comes the verb “fuse,” to melt by heat, often used +figuratively in the sense of blend, mix.</p> + + +<hr class="art" /> +<p><span class="bold">FYNE, LOCH,<a name="ar113" id="ar113"></a></span> an inlet of the sea, Argyllshire, Scotland. +From the head, 6 m. above Inveraray, to the mouth on the Sound +of Bute, it has a south-westerly and then southerly trend and +is 44 m. long, its width varying from ¼ m. to 6 m. It receives the +Fyne, Shira, Aray and many other streams, and, on the western +side, gives off Lochs Shira, Gair, Gilp (with Ardrishaig, the +Crinan Canal and Lochgilphead) and East Tarbert (with Tarbert +village). The glens debouching on the lake are Fyne, Shira, +Aray, Kinglas and Hell’s Glen. The coast generally is picturesque +and in many parts well wooded. All vessels using the Crinan +Canal navigate the loch to and from Ardrishaig, and there are +daily excursions during the season, as far up as Inveraray. +There are ferries at St Catherine’s and Otter, and piers at Tarbert, +Ardrishaig, Kilmory, Crarae, Furnace, Inveraray, Strachur and +elsewhere. The industries comprise granite quarrying at Furnace +<span class="pagenum"><a name="page376" id="page376"></a>376</span> +and Crarae, distilling at Ardrishaig, gunpowder-making at +Furnace and Kilfinan, and, above all, fishing. Haddock, whiting +and codling are taken, and the famous “Loch Fyne herrings” +command the highest price in the market.</p> + + +<hr class="art" /> +<p><span class="bold">FYRD,<a name="ar114" id="ar114"></a></span> the name given to the English army, or militia, during +the Anglo-Saxon period (see <span class="sc"><a href="#artlinks">Army</a></span>, 60). It is first mentioned +in the <i>Anglo-Saxon Chronicle</i> under the date 605. The ealdorman, +or sheriff, of the shire was probably charged with the duty of +calling out and leading the fyrd, which appears always to have +retained a local character, as during the time of the Danish +invasions we read of the fyrd of Kent, of Somerset and of +Devon. As attendance at the fyrd was included in the <i>trinoda +necessitas</i> it was compulsory on all holders of land; but that +it was not confined to them is shown by the following extract +from the laws of Ine, king of the West Saxons, dated about +690, which prescribes the penalty for the serious offence of +neglecting the fyrd: “If a <i>gesithcund</i> man owning land neglect +the fyrd, let him pay 120 shillings, and forfeit his land; one not +owning land 60 shillings; a ceorlish man 30 shillings as <i>fyrdwite</i>.” +The fyrd was gradually superseded by the gathering of the +thegns and their retainers, but it was occasionally called out for +defensive purposes even after the Norman Conquest.</p> + + +<hr class="art" /> +<p><span class="bold">FYT, JOHANNES<a name="ar115" id="ar115"></a></span> (1609-1661), Belgian animal painter, was +born at Antwerp and christened on the 19th of August 1609. +He was registered apprentice to Hans van den Berghe in 1621. +Professionally van den Berghe was a restorer of old pictures +rather than a painter of new ones. At twenty Johannes Fyt +entered the gild of St Luke as a master, and from that time +till his death in 1661 he produced a vast number of pictures +in which the bold facility of Snyders is united to the powerful +effects of Rembrandt, and harmonies of gorgeous tone are not +less conspicuous than freedom of touch and a true semblance +of nature. There never was such a master of technical processes +as Fyt in the rendering of animal life in its most varied forms. +He may have been less correct in outline, less bold in action +than Snyders, but he was much more skilful and more true in +the reproduction of the coat of deer, dogs, greyhounds, hares +and monkeys, whilst in realizing the plumage of peacocks, +woodcocks, ducks, hawks, and cocks and hens, he had not his +equal, nor was any artist even of the Dutch school more effective +in relieving his compositions with accessories of tinted cloth, +porcelain ware, vases and fruit. He was not clever at figures, +and he sometimes trusted for these to the co-operation of Cornelius +Schut or Willeborts, whilst his architectural backgrounds +were sometimes executed by Quellyn. “Silenus amongst +Fruit and Flowers,” in the Harrach collection at Vienna, “Diana +and her Nymphs with the Produce of the Chase,” in the Belvedere +at Vienna, and “Dead Game and Fruit in front of a Triumphal +Arch,” belonging to Baron von Rothschild at Vienna, are +specimens of the co-operation respectively of Schut, Willeborts +and Quellyn. They are also Fyt’s masterpieces. The earliest +dated work of the master is a cat grabbing at a piece of dead +poultry near a hare and birds, belonging to Baron Cetto at +Munich, and executed in 1644. The latest is a “Dead Snipe +with Ducks,” of 1660, sold with the Jäger collection at Cologne +in 1871. Great power is shown in the bear and boar hunts at +Munich and Ravensworth castle. A “Hunted Roedeer with +Dogs in the Water,” in the Berlin Museum, has some of the life +and more of the roughness of Snyders, but lacks variety of tint +and finish. A splendid specimen is the Page and Parrot near a +table covered with game, guarded by a dog staring at a monkey, +in the Wallace collection. With the needle and the brush +Fyt was equally clever. He etched 16 plates, and those representing +dogs are of their kind unique.</p> + + +<hr class="art" /> +<p><span class="bold">FYZABAD,<a name="ar116" id="ar116"></a></span> or <span class="sc">Faizabad</span>, a city, district and division of +British India in the United Provinces. The city stands on the +left bank of the river Gogra, 78 m. by rail E. of Lucknow. Pop. +(1901) 75,085. To the E. of Fyzabad, and now forming a +suburb, is the ancient site of Ajodhya(<i>q.v.</i>). Fyzabad was +founded about 1730 by Sa’adat Ali Khan, the first nawab +wazir of Oudh, who built a hunting-lodge here. It received its +present name in the reign of his successor; and Shuja-ud-daula, +the third nawab, laid out a large town and fortified it, and here +he was buried. It was afterwards the residence of the Begums +of Oudh, famous in connexion with the impeachment of Warren +Hastings. When the court of Oudh was removed to Lucknow +in 1775 all the leading merchants and bankers abandoned the +place. At the census of 1869 Fyzabad contained only 37,804 +inhabitants; but it is now again advancing in prosperity and +population. On the outbreak of the Mutiny in 1857, the cantonment +contained two regiments of infantry, a squadron of cavalry, +and a light field battery of artillery—all natives. Owing to +their threatening demeanour after the Meerut massacre, many +of the European women and children were sheltered by one of +the great landholders of Oudh, and others were sent to less +disturbed parts of the country. The troops rose, as was anticipated, +and although they at first permitted their officers to take +boats and proceed towards Dinapur, a message was afterwards +sent to a rebel force lower down the river to intercept the fugitives. +Of four boats, one, having passed the rebels unnoticed, succeeded +in reaching Dinapur safely. Of those in the other three boats, +one alone escaped. Fyzabad is now a station for European +as well as for native troops. It is the headquarters of a brigade +in the 8th division of the northern army. There is a government +college. Sugar-refining and trade in agricultural produce are +important.</p> + +<p><span class="sc">The District of Fyzabad</span>, lying between the two great rivers +Gogra and Gumti, has an area of 1740 sq. m. It is entirely +alluvial and well wooded, and has a good climate. Pop. (1901) +1,225,374, an increase of .7% in the decade. The district is +traversed throughout its length by the Oudh and Rohilkhand +railway from Lucknow to Benares, with a branch to Allahabad. +Tanda, with a population in 1901 of 19,853, has the largest +production of cotton goods in Oudh.</p> + +<p>The <span class="sc">Division of Fyzabad</span> has an area of 12,113 sq. m., and +comprises the six districts of Fyzabad, Gonda, Bahraich, +Sultanpur, Partabgarh and Bara Banki. Pop. (1901) 6,855,991, +an increase of 2% in the decade.</p> + +<hr class="art" /> + + + + + + + + +<pre> + + + + + +End of the Project Gutenberg EBook of Encyclopaedia Britannica, 11th +Edition, Volume 11, Slice 3, by Various + +*** END OF THIS PROJECT GUTENBERG EBOOK ENCYC. 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