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Lamb's time, mainly through the appreciative criticisms of S. T. Coleridge, Robert Southey and others, Fuller's works have received much attention.

There is an elaborate account of the life and writings of Fuller by William Oldys in the Biographia Britannica, vol. iii. (1750), based on Fuller's own works and the anonymous Life of Dr Thomas Fuller (1661; reprinted in a volume of selections by A. L. J. Gosset, 1893). The completest account of him is The Life of Thomas Fuller, with Notices of his Books, his Kinsmen and his Friends (1874), by J. E. Bailey, who gives a detailed bibliography (pp. 713-762) of his works. The Worthies of England was reprinted by John Nichols (1811) and by P. A. Nuttall (1840). His Collected Sermons were edited by J. E. Bailey and W. E. A. Axon in 1891. Fuller's quaint wit lends itself to selection, and there are several modern volumes of extracts from his works.

FULLER, WILLIAM (1670-c. 1717), English impostor, was born at Milton in Kent on the 20th of September 1670. His

paternity is doubtful, but he was related to the family of Herbert. After 1688 he served James II.'s queen, Mary of Modena, and the Jacobites, seeking at the same time to gain favour with William III.; and after associating with Titus Oates, being imprisoned for debt and pretending to reveal Jacobite plots, the House of Commons in 1692 declared he was an " imposter, cheat and false accuser." Having stood in the pillory he was again imprisoned until 1695, when he was released; and at this time he took the opportunity to revive the old and familiar story that Mary of Modena was not the mother of the prince of Wales. In 1701 he published his autobiographical Life of William Fuller and some Original Letters of the late King James. Unable to prove the assertions made in his writings he was put in the pillory, whipped and fined. He died, probably in prison, about 1717. Fuller's other writings are Mr William Fuller's trip to Bridewell, with a full account of his barbarous usage in the pillory; The sincere and hearty confession of Mr William Fuller (1704); and An humble appeal to the impartial judgment of all parties in Great Britain (1716).

He must be distinguished from WILLIAM FULLER (1608-1675), dean of St Patrick's (1660), bishop of Limerick (1663), and bishop of Lincoln (1667), the friend of Samuel Pepys; and also from William Fuller (c. 1580-1659), dean of Ely and later dean of Durham.

FULLER'S EARTH (Ger. Walkererde, Fr. terre à foulon, argile smectique)-s --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

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(Analysis by P. G. Sanford, Geol. Mag., 1889, 6, pp. 456, 526.) Of other published analyses, not a few show a lower silica content

(44%, 50%), along with a higher proportion of alumina (11%, 23%).

Fuller's earth may occur on any geological horizon; at Nutfield in Surrey, England, it is in the Cretaceous formations; at Midford near Bath it is of Jurassic age; at Bala, North Wales, it occurs in Ordovician strata; in Saxony it appears to be the decomposition product of a diabasic rock. In America it is found in California in rocks ranging from Cretaceous to Pleistocene age; in S. Dakota, Custer county and elsewhere a yellow, gritty earth of Jurassic age is worked; in Florida and Georgia occurs a brittle, whitish earth of Oligocene age. Other deposits are worked in Arkansas, Texas, Colorado, Massachusetts and South Carolina. Fuller's earth is either mined or dug in the open according to local circumstances. It is then dried in the sun or by artificial heat and transported in small lumps in sacks. In other cases it is ground to a fine powder after being dried; or it is first roughly ground and made into a slurry with water, which is allowed to carry off the finer from the coarser particles and deposit them in a creamy state in suitable tanks. After consolidation this fine material is dried artificially on drying floors, broken into lumps, and packed for transport. The use of fuller's earth for cleansing wool and cloth has greatly decreased, but the demand for the material is as great or greater than it ever was. It is now used very largely in the filtration of mineral oils, and also for decolourizing certain vegetable oils. It is employed in the formation of certain soaps and cleansing preparations.

The term "Fuller's Earth" has a special significance in geology, for it was applied by W. Smith in 1799 to certain clays in the neighbourhood of Bath, and the use of the expression is still retained by English geologists, either in this form or in the generalized "Fullonian." The Fullonian lies at the base of the Great Oolite or Bathonian series, but its palaeontological characters place it between that series and the underlying Inferior Oolite. The zonal fossils are Perisphinctes arbustigerus and Macrocephalus subcontractus with Ostrea acuminata, Rhynchonella concinna and Goniomya angulifera. The formation is in part the equivalent of the "Vesulien" of J. Marcou (Vesoul in Haute-Saône). In Dorsetshire and Somersetsha, where it is best developed, it is represented by an Upper Tuber'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 castern Gloucestershire and northern Oxfordshire the Fuller's Earth passes downwards without break into the Inferior Oolite; northward it dies out about Chipping Norton in Oxfordshire and passes laterally into the Stonesfield Slates series; in the midland counties it may perhaps be represented by the "Upper Estuarine Series." In parts of Dorsetshire the clays have been used for brickmaking and the limestone (rock) for local buildings.

See H. B. Woodward, "Jurassic Rocks of Great Britain," vol. iv. (1894), Mem. Geol. Survey (London). [J. A. H.J

FULLERTON, LADY GEORGIANA CHARLOTTE (1812-1885), English novelist and philanthropist, youngest daughter of the 1st Earl Granville, was born at Tixall Hall in Staffordshire on the 23rd of September 1812. In 1833 she married Alexander George Fullerton, then an Irish officer in the guards. After living in Paris for some eight years she and her husband accompanied Lord Granville to Cannes and thence to Rome. In 1843

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her husband entered the Roman Catholic church, and in the following year Lady Georgiana Fullerton published her first novel, Ellen Middleton, which attracted W. E. Gladstone's attention in the English Review. In 1846 she entered the Roman Catholic church. The death of her only son in 1854 plunged her in grief, and she continued to wear mourning until the end of her life. In 1856 she became one of the third order of St Francis, and thenceforward devoted herself to charitable work. In conjunction with Miss Taylor she founded the religious community known as "The Poor Servants of the Mother of God Incarnate," and she also took an active part in bringing to England the sisters of St Vincent of Paul. Her philanthropic work is described in Mrs Augustus Craven's work Lady Georgiana Fullerton, sa vie et ses œuvres (Paris, 1888), which was translated into English by Henry James Coleridge. She died at Bournemouth on the 19th | of January 1885. Among her other novels were Grantley Manor (1847), Lady Bird (1852), and Too Strange not to be True (1864). FULMAR, from the Gaelic Fulmaire, the Fulmarus glacialis of modern ornithologists, one of the largest of the petrels (Procellariidae) of the northern hemisphere, being about the size of the common gull (Larus canus) and not unlike it in general coloration, except that its primaries are grey instead of black. This bird, which ranges over the North Atlantic, is seldom seen on the European side below lat. 53° 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, F. glupischa. It has been commonly believed to have two breeding-places in the British Islands, namely, St Kilda and South Barra; but, according to Robert Gray (Birds of the West of Scotland, p. 499), it has abandoned the latter since 1844, though still breeding in Skye. Northward it established itself about 1838 on Myggenaes Holm, one of the Faeroes, while it has several stations off the coast of Iceland and Spitsbergen, as well as at Bear Island. Its range towards the pole seems to be only bounded by open water, and it is the constant attendant upon all who are employed in the whale and scal 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" (corrupted from Mallemuck), and is extremely well known to them, its flight, as it skims over the waves, first with a few beats of the wings and then gliding for a long way, being very peculiar. It only visits the land to deposit its single white egg, which is laid on a rocky ledge, where a shallow nest is made in the turf and lined with a little dried grass. Many of its breeding-places are a most valuable property to those who live near them and take the eggs and young, which, from the nature of the locality, are only to be had at a hazardous risk of life. In St Kilda a large number of the young are killed in one week of August, the only time when, by the custom of the community, they are allowed to be taken. These, after the oil is extracted from them, serve the islanders with food for the winter. The oil has been chemically analysed and found to be a fish-oil, and to possess nearly all the qualities of that obtained from the liver of the cod, with a lighter specific gravity. It, however, has an extremely strong scent, which is said by those who have visited St Kilda to pervade every thing and person on the island, and is certainly retained by an egg or skin of the bird for many years. Whenever a live example is scized in the hand it ejects a considerable quantity of this oil from its mouth.

FULMINIC ACID, HCNO or H2C2N2O2, an organic acid isomeric with cyanic and cyanuric acids; its salts, termed fulminales, 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 A name misapplied in the southern hemisphere to Diomedea melanophrys, one of the albatrosses.

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silver" obtained by C. L. Berthollet in 1788 by acting with ammonia on precipitated silver oxide. The next salt to be obtained was the mercuric salt, which was prepared in 1799 by Edward Charles Howard, who substituted mercury for silver in Brugnatelli's process. A similar method is that of J. von Liebig (1823), who heated a mixture of alcohol, nitric acid and mercuric nitrate; the salt is largely manufactured by processes closely resembling the last. A laboratory method is to mix solutions of sodium nitromethane, CH2: NO(ONa), and mercuric chloride, a yellow basic salt being formed at the same time. Mercuric fulminate is less explosive than the silver salt, and forms white needles (with 4H-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).

The

The constitution of fulminic acid has been investigated by many experimenters, but apparently without definitive results. 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), since it gave mixed and acid salts. Kekulé, in 1858, concluded that it was_nitroacetonitrile, NO, CH, CN, a view opposed by Steiner (1883), E. Divers and M. Kawakita (1884), R. Scholl (1890), and by J. U. Nef (1894), who proposed the formulae: C: NOH N: CH CH:NO C: N.OH. N: COH, CH: N-O, Divers, Scholl, Nef.

C: NOH, Steiner,

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 of falminic acid. The existence of double salts, and the observations formula is correct. of L. Wöhler and K. Theodorovits (Ber., 1905, 38, p. 345), that only compounds containing two carbon atoms yielded fulminates, points to (HCNO),; on the other hand, Wöhler (loc. cit. p. 1351) found that cryoscopic and electric conductivity measurements showed sodium fulminate to be NaCNO. Nef based his formula, which involves bivalent carbon, on many reactions; in particular, that silver fulminate with hydrochloric acid gave salts of formylchloridoxime, which with water gave hydroxylamine and formic acid, thus

There is some doubt as to the molecular formula

C:NO OAg→HCNOAg→HC< NOHH.CO2H+H2N OH,

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OH

and also on the production from sodium nitromethane and mercuric chloride, thus CH2: NO-Ohg→H2O+C: NOhg (hg=4Hg). H. Wieland and F. C. Palazzo (1907) support this formula, finding that methyl nitrolic acid, NO, CH: N-OII, yielded under certain conditions fulminic acid, and vice versa (Palazzo, 1907). M.Z. Jowitschitsch (Ann., 1906, 347, p. 233) inclines to Scholl's formula; he found that the synthetic silver salt of glyoxime peroxide resembled silver fulminate in yielding hydroxylamine with hydrochloric acid, but differed in being less explosive, and in being soluble in nitric acid. H. Wieland and his collaborators regard "glyoxime peroxide as an oxide of furazane (q.v.), and have shown that a close relationship exists between the nitrile oxides, furoxane, and fulminic acid (see Ann. Rep., London Chem. Soc., 1909, p. 84). Fulminuric acid, (CNO), obtained by Liebig by boiling mercuric fulminate with water, was synthesized in 1905 by C. Ulpiani and L. Bernardini (Gazetta, iii. 35, p. 7), who regard it as NO2 CH(CN) CO NH2. It deflagrates at 145°, and forms a characteristic cuprammonium salt. The early history of mercuric fulminate and a critical account of its application as a detonator is given in The Rise and Progress of the British Explosives Industry International Congress of Applied Chemistry, 1909). The manufacture and modern aspects are treated in Oscar Guttmann, The Manufacture of Explosives, and Manu facture of Explosives, Twenty Years' Progress (1909).

settler came in 1793, and the first survey (which included only a fort within the present limits of Fulton. The first civilian a part of the subsequent village) was made in 1815. Fulton combined with the village of Oswego Falls (pop. in 1900, 2925) was incorporated as a village in 1835, and in April 1902 was and was chartered as a city.

which in Chinese mythology are believed to keep watch and ward FUM, or FUNJ HWANG, one of the four symbolical creatures 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, and perches only on the woo-tung tree. The appearance of Fum the face of a swallow, the beak of a cock, is about six cubits high, heralds an age of universal virtue. Its figure is that which is embroidered on the dresses of some mandarins.

FULTON, ROBERT (1765-1815), American engineer, was born | about 700) under De Villiers. Soon after this, Bradstreet built in 1765 in Little Britain (now Fulton, Lancaster county), Pa. His parents were Irish, and so poor that they could afford him only a very scanty education. At an early age he was bound apprentice to a jeweller in Philadelphia, but subsequently adopted portrait and landscape painting as his profession. In his twenty-second year, with the object of studying with his countryman, Benjamin West, he went to England, and there became acquainted with the duke of Bridgewater, Earl Stanhope and James Watt. Partly by their influence he was led to devote his attention to engineering, especially in connexion with canal construction; he obtained an English patent in 1794 for superseding canal locks by inclined planes, and in 1796 he published a Treatise on the Improvement of Canal Navigation. He then took up his residence in Paris, where he projected the first panorama ever exhibited in that city, and constructed a submarine boat, the "Nautilus," which was tried in Brest harbour in 1801 before a commission appointed by Napoleon I., and by the aid of which he was enabled to blow up a small vessel with a torpedo. It was at Paris also in 1803 that he first succeeded in propelling a boat by steam-power, thus realizing a design which he had conceived ten years previously. Returning to America he continued his experiments with submarine explosives, but failed to convince either the English, French or United States governments of the adequacy of his methods. With steam navigation | he had more success. In association with Robert R. Livingston (q.v.), who in 1798 had been granted the exclusive right to navigate the waters of New York state with steam-vessels, he❘ constructed the 66 'Clermont," which, engined by Boulton & Watt of Birmingham, began to ply on the Hudson between New York and Albany in 1807. The privilege obtained by Livingston in 1798 was granted jointly to Fulton and Livingston in 1803, and by an act passed in 1808 the monopoly was secured to them and their associates for a period depending on the number of steamers constructed, but limited to a maximum of thirty years. In 1814-1815, on behalf of the United States government, he constructed the "Fulton," a vessel of 38 tons with central paddle-wheels, which was the first steam warship. He died at New York on the 24th of February 1815. Among Fulton's inventions were machines for spinning flax, for making ropes, and for sawing and polishing marble.

See C. D. Colden, Life of Robert Fulton (New York, 1817); Robert H. Thurston, History of the Growth of the Steam-Engine (New York, 1878); George H. Preble, Chronological History of Steam Navigation (Philadelphia, 1883); and Mrs A. Ĉ. Sutcliffe, Robert Fulton and the Clermont (New York, 1909).

FULTON, a city and the county-seat of Callaway county, Missouri, U.S.A., 25 m. N.E. of Jefferson City. Pop. (1890) 4314; (1900) 4883 (1167 negroes); (1910) 5228. It is served by the Chicago & Alton railway. The city has an important stock market and manufactures fire-brick and pottery. At Fulton are the Westminster College (Presbyterian, founded in 1853), the Synodical College for Young Women (Pres., founded in 1871), the William Woods College for Girls (Christian Church, 1890), and the Missouri school for the deaf (1851). Here, too, is a state hospital for the insane (1847), the first institution of the kind in Missouri. The place was laid out as a town in 1825 and named Volney, but in honour of Robert Fulton the present name was adopted a little later. Fulton was incorporated in 1859.

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FULTON, a city of Oswego county, New York, U.S.A., on the right bank of the Oswego river, about 10 m. S. by E. of Oswego. Pop. (1900) 5281; (1905, state census) 8847; (1910) 10,480. Fulton is served by the Delaware, Lackawanna & Western, the New York Central & Hudson River, and the New York, Ontario & Western railways, by electric railway to Oswego and Syracuse and by the Oswego Canal. The city has a Carnegie library. Ample water-power is furnished by the Oswego river, which here flows in a series of rapids, and the manufactures are many in kind. On the 3rd of July 1756, on an island (afterward called Battle Island) 4 m. N. of the present city of Fulton, a British force of about 300 under Captain John Bradstreet (1711-1774). defeated an attacking force of French and Indians (numbering

FUMARIC AND MALEIC ACIDS, two isomeric unsaturated (Fumaria officinalis), in various fungi (Agaricus piperatus, &c.), acids of composition C4H4O4. Fumaric acid is found in fumitory and in Iceland moss. to 150° C., or by heating it with hydrochloric acid (V. Dessaignes, Jahresb., 1856, p. 463) or with a large quantity of hydrobromic It is obtained by heating malic acid alone acids (A. Kekulé, Ann., 1864, 130, p. 21). It may also be obtained by boiling monobromsuccinic acid with water; by the action of dichloracetic acid and water on silver malonate (T. Komnenos, Ann., 1883, 218, p. 169); by the cyanide synthesis from acetylene di-iodide; and by heating maleic acid to 210° C. (Z. Skraup, Monats. f. Chemie, 1891, 12, p. 112). It crystallizes in small prisms or needles, and is practically insoluble in cold water. It sublimes to some extent at about 200° C., being partially conpractically quantitative if dehydrating agents be used. Reducing verted into maleic anhydride and water, the reaction becoming 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, Ber., 1881, 14, p. 713). By long-continued heating with caustic soda at 100° C. it is converted into inactive malic acid.

by heating fumaric acid with acetyl chloride to 100° C.; or by the hydrolysis of trichlorphenomalic acid (B-trichloracetoMaleic acid is obtained by distilling malic or fumaric acids; acrylic acid) (A. Kekulé, Ann., 1884, 223, p. 185]. It crystallizes at 130° C., and boil at 160° C., decomposing into water and in monoclinic prisms, which are easily soluble in water, melt maleic anhydride. When heated with concentrated hydrobromic or hydriodic acids, it is converted into fumaric acid. It yields anhydride is obtained by distilling fumaric acid with phosphorus an anilide; oxidation converts it into mesotartaric acid. Maleic pentoxide. It forms triclinic crystals which melt at 60° C. and boil at 196° C.

their silver salts, and the maleic ester is readily transformed into the Both acids are readily esterified by the action of alkyl halides on fumaric ester by warming with iodine, the same result being obtained hydrochloric acid. Both acids yield acetylene by the electrolysis of aqueous solutions of their alkali salts, and on reduction both by esterification of maleic acid in alcoholic solution by means of yield succinic acid, whilst by the addition of hydrobromic acid they identical, and the isomerism has consequently to be explained on both yield monobromsuccinic acid (R. Fittig, Ann., 1877, 188, p. 98). other grounds. This was accomplished by W. Wislicenus [“ Über From these results it follows that the two acids are structurally die räumliche Anordnung der Atome," &c., Trans. of the Saxon Acad. of Sciences (Math. Phys. Section), 1887, p. 14] by an extension of the van't Hoff hypothesis (sce STEREO-ISOMERISM). The formulae These account for maleic acid readily yielding an anhydride, whereas HC.COH fumaric acid does not, and for the behaviour of the acids towards HC.CO,H HC COH HO2C.C.H Fumaric acid. bromine, fumaric acid yielding ordinary dibromsuccinic acid, and maleic acid the isomeric isodibromsuccinic acid.

of the acids are written thus:

Maleic acid.

named indirectly from the Lat. fumariolum, a smoke-hole. FUMAROLE, a vent from which volcanic vapours issue,

The vapours from fumaroles were studied first by R. W. Bunsen, | sugar-cane plantations occupy the surrounding heights. Three on his visit to Iceland, and afterwards by H. Sainte-Claire Deville mountain streams traverse the city through deep channels, and other chemists and geologists in France, who examined the which in summer are dry, owing to the diversion of the water vapours from Santorin, Etna, &c. The hottest vapours issue for irrigation. A small fort, on an isolated rock off shore, from dry fumaroles, at temperatures of at least 500° C., and guards the entrance to the bay, and a larger and more powerfully consist chiefly of anhydrous chlorides, notably sodium chloride. armed fort crowns an eminence inland. The chief buildings The acid fumaroles yield vapours of lower temperature (300° to include the cathedral, Anglican and Presbyterian churches, 400°) containing much water vapour, with hydrogen chloride hospitals, opera-house, museum and casino. There are small and sulphur dioxide. The alkaline fumaroles are still cooler, public gardens and a meteorological observatory. In the steep though above 100°, and evolve ammonium chloride with other and narrow streets, which are lighted by electricity, wheeled vapours. Cold fumaroles, below 100°, discharge principally traffic is impossible; sledges drawn by oxen, and other primitive aqueous vapour, with carbon dioxide, and perhaps hydrogen conveyances are used instead (see MADEIRA). In winter the fine sulphide. The fumaroles of Mont Pelé in Martinique during the climate and scenery attract numerous invalids and other visitors, eruption of 1902 were examined by A. Lacroix, and the vapours for whose accommodation there are good hotels; many foreigners analysed by H. Moissan, who found that they consisted chiefly engaged in the coal and wine trades also reside here permanently. of water vapour, with hydrogen chloride, sulphur, carbon dioxide, The majority of these belong to the British community, which carbon monoxide, methane, hydrogen, nitrogen, oxygen and was first established here in the 18th century. Funchal is the argon. These vapours issued at a temperature of about 400°. headquarters of Madeiran industry and commerce (see Madeira). Armand Gautier has pointed out that these gases are practically It has no docks and no facilities for landing passengers or goods; of the same composition as those which he obtained on heating vessels are obliged to anchor in the roadstead, which, however, granite and certain other rocks. (See VOLCANO). 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).

FUMIGATION (from Lat. fumigarė, 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.

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FUMITORY, in botany, the popular name for the British species of Fumaria, a genus of small, branched, often climbing annual herbs with much-divided leaves and racemes of small flowers. The flowers are tubular with a spurred base, and in the British species are pink to purplish in colour. They are weeds of cultivation growing in fields and waste places. F. capreolata climbs by means of twisting petioles. In past times fumitory was in esteem for its reputed cholagogue and other medicinal properties; and in England, boiled in water, milk or whey, it was used as a cosmetic. The root of the allied species (Corydalis cava or tuberosa) is known as radix aristolochia, and has been used medicinally for various cutaneous and other disorders, in doses of 10 to 30 grains. Some eleven alkaloids have been isolated from it. The herbage of Fumaria officinalis and F. racemosa is used in China under the name of Tsze-hwa-ti-ling as an application for glandular swellings, carbuncles and abscesses, and was formerly valued in jaundice, and in cases of accidental swallowing of the beard of grain (see F. Porter Smith, Contrib. towards the Mat. Medica... of China, p. 99, 1871). The name fumitory, Latin fumus terrae, has been supposed to be derived from the fact that its juice irritates the eyes like smoke (see Fuchs, De historia stirpium, p. 338, 1542); but The Grete Herball, cap. clxix., 1529, fol., following the De simplici medicina of Platearius, fo. xciii. (see in Nicolai Praepositi dispensatorium ad aromatarios, 1536), says: "It is called Fumus terre fume or smoke of the erthe bycause it is engendred of a cours fumosyte rysynge frome the erthe in grete quantyte lyke smoke: this grosse or cours fumosyte of the erthe wyndeth and wryeth out: and by workynge of the ayre and sonne it turneth into this herbe.

FUNCHAL, the capital of the Portuguese archipelago of the Madeiras; on the south coast of Madeira, in 32° 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 (quintas), with terraced gardens, vineyards and

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FUNCTION,' in mathematics,, a variable number the value of which depends upon the values of one or more other variable numbers. The theory of functions is conveniently divided into (I.) Functions of Real Variables, wherein real, and only real, numbers are involved, and (II.) Functions of Complex Variables, wherein complex or imaginary numbers are involved.

I. FUNCTIONS OF REAL VARIABLES

1. Historical.-The word function, defined in the above sense, was introduced by Leibnitz in a short note of date 1694 concerning the construction of what we now call an envelope (Leibnizens mathematische Schriften, edited by C. I. Gerhardt, Bd. v. p. 306), and was there used to denote a variable length In 1698 related in a defined way to a variable point of a curve. James Bernoulli used the word in a special sense in connexion with some isoperimetric problems (Joh. Bernoulli, Opera, t. i. p. 255). He said that when it is a question of selecting from an infinite set of like curves that one which best fulfils some function, then of two curves whose intersection determines the thing sought one is always the "line of the function" (Linea functionis). In 1718 John Bernoulli (Opera, t. ii. p. 241) defined a "function of a variable magnitude as a quantity made up in any way of this variable magnitude and constants; and in 1730 (Opera, t. iii. p. 174) he noted a distinction between "algebraic" and "tran scendental" functions. By the latter he meant integrals of algebraic functions. The notation f(x) for a function of a variable was introduced by Leonhard Euler in 1734 (Comm. Acad. Petropol. t. vii. p. 186), in connexion with the theorem of the interchange of the order of differentiations. The notion of functionality or functional relation of two magnitudes was thus of geometrical origin; but a function soon came to be regarded as an analytical expression, not necessarily an algebraic expression, containing the variable or variables. Thus we may have rational integral algebraic functions such as ax2 + bx + c, or rational algebraic functions which are not integral, such as a1x*+a2x"¬1+..... tan,

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or irrational algebraic functions, such as √x, or, more generally the algebraic functions that are determined implicitly by an algebraic equation, as, for instance,

fn(x,y) +fn-1(x,y) +...+fo=0

1 The word "function" (from Lat. fungi, to perform) has many uses, with the fundamental sense of an activity special or proper to an office, business or profession, or to an organ of an animal or plant, the definite work for which the organ is an apparatus. From the use of the word, as in the Italian funzione, for a ceremony of the Roman Church, "function" is often employed for a public ceremony of any kind, and loosely of a social entertainment or gathering.

where f. (x,y),.. ... mean homogeneous expressions in x and y | the case the number is said to be "continuously variable." having constant coefficients, and having the degrees indicated by the suffixes, and fo is a constant. Or again we may have trigonometrical functions, such as sin x and tan x, or inverse trigonometrical functions, such as sinx, or exponential functions, such as e and a2, 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.

2. Graphic Representation.—In the case of a function of one variable x, any value of x and the corresponding value y of the function can be the co-ordinates of a point in a plane. To any value of x there corresponds a point N on the axis of x, in accordance with the rule that x is the abscissa of N. The corresponding value of y determines a point P in accordance with the rule that x is the abscissa and y the ordinate of P. The ordinate y gives the value of the function which corresponds to that value of the variable x which is specified by N; and it may be described as" the value of the function at N." Since there is a one-to-one correspondence of the points N and the numbers x, we may also describe the ordinate as "the value of the function at x." In simple cases the aggregate of the points P which are determined by any particular function (of one variable) is a curve, called the "graph of the function" (see § 14). In like manner a function

of two variables defines a surface.

3. The Variable.-Graphic methods of representation, such as those just described, enabled mathematicians to deal with irrational values of functions and variables at the time when there was no theory of irrational numbers other than Euclid's theory of incommensurables. In that theory an irrational number was the ratio of two incommensurable geometric magnitudes. In the modern theory of number irrational numbers are defined in a purely arithmetical manner, independent of the measurement of any quantities or magnitudes, whether geometric or of any other kind. The definition is effected by means of the system of ordinal numbers (see NUMBER). When this formal system is established, the theory of measurement may be founded upon it; and, in particular, the co-ordinates of a point are defined as numbers (not lengths), which are assigned in accordance with a rule. This rule involves the measurement of lengths. The theory of functions can be developed without any reference to graphs, or co-ordinates or lengths. The process by which analysis has been freed from any consideration of measurable quantities has been called the "arithmetization of analysis." In the theory so developed, the variable upon which a function depends is always to be regarded as a number, and the corresponding value of the function is also a number. Any reference to points or coordinates is to be regarded as a picturesque mode of expression, pointing to a possible application of the theory to geometry. The development of "arithmetized analysis" in the 19th century is associated with the name of Karl Weierstrass.

All possible values of a variable are numbers. In what follows we shall confine our attention to the case where the numbers are real. When complex numbers are introduced, instead of real ones, the theory of functions receives a wide extension, which is accompanied by appropriate limitations (see below, II. Functions of Complex Variables). The set of all

real numbers forms a continuum. In fact the notion of a onedimensional continuum first becomes precise in virtue of the establishment of the system of real numbers.

4. Domain of a Variable.-Theory of Aggregales.-The notion of a "variable" is that of a number to which we may assign at pleasure any one of the values that belong to some chosen set, or aggregate, of numbers; and this set, or aggregate, is called the "domain of the variable." This domain may be an "interval," that is to say it may consist of two terminal numbers, all the numbers between them and no others. When this is

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When the domain consists of all real numbers, the variable is said to be "unrestricted." A domain which consists of all the real numbers which exceed some fixed number may be described as an "interval unlimited towards the right "; similarly we may have an interval" unlimited towards the left." In more complicated cases we must have some rule or process for assigning the aggregate of numbers which constitute the domain of gates, and the theorems relating to them, form a branch of analysis a variable. The methods of definition of particular types of aggrecalled the "theory of aggregates" (Mengenlehre, Théorie des ensembles, Theory of sets of points). The notion of an "aggregate in general underlies the system of ordinal numbers. An aggregate is said to be "infinite when it is possible to effect a one-to-one correspondence of all its elements to some of its elements. For example, we may make all the integers correspond to the even integers, by making I correspond to 2, 2 to 4, and generally n to 2n. The aggregate of positive integers is an infinite aggregate. The aggregates of all other examples of infinite aggregates. An aggregate whose elements rational numbers and of all real numbers and of points on a line are 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 (or point) a for a positive number h" is the aggregate of all numbers aggregate is always infinite. The neighbourhood of a number (or points) x for which the absolute value of x-a denoted by x-al, does not exceed h.

5. General Notion of Functionality.-A function of one variable was for a long time commonly regarded as the ordinate of a curve; and the two notions (1) that which is determined by a curve supposed drawn, and (2) that which is determined by an analytical expression supposed written down, were not for a long time clearly distinguished. It was for this reason that Fourier's discovery that a single analytical expression is capable of representing (in different parts of an interval) what would in his time have been called different functions so profoundly struck mathematicians (§ 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.

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For the specification of any particular function two things are requisite: (1) a statement of the values of the variable, or of the aggregate of points, to which values of the function are to be made to correspond, i.e. of the "domain of the argument "; (2) a.rule any point in this domain. We may refer to the second of these two for assigning the value or values of the function that correspond to 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 valuc," or at all points of a definite aggregate included in the domain of the again more generally, "Give the function the value of the expression argument." The former of these is the rule of those among the earlier analysts who regarded an analytical expression and a function confusion and with the advantage of brevity, the analytical expres as the same thing, and their usage may be retained without causing sion serving to specify the domain of the argument as well as the rule of calculation, e.g. we may speak of "the function 1/x." This function is defined by the analytical expression 1/x at all points except the point x=o. But in complicated cases separate statements of the domain of the argument and the rule of calculation cannot be dispensed with. In general, when the rule of calculation is determined as above by an analytical expression at any aggregate of points, the function is said to be " represented "by the expression at those points.

When the rule of calculation assigns a single definite value for a function at each point in the domain of the argument the function is "uniform" or "one-valued." In what follows it is to be understood that all the functions considered are one-valued, and the values

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