VOLUME XI SLICE III Frost to Fyzabad (continued 5)

About 1640 he had married Eleanor, daughter of Hugh Grove of Chisenbury, Wiltshire. She died in 1641. Their eldest child, John, baptized at Broadwindsor by his father, 6th June 1641, was afterwards rector of Sidney Sussex College, edited the Worthies of England, 1662, and became rector of Great Wakering, Essex, where he died in 1687.

At Broadwindsor, early in the year 1641, Thomas Fuller, his curate Henry Sanders, the church wardens, and others, nine persons altogether, certified that their parish, represented by 242 grown-up male persons, had taken the Protestation ordered by the speaker of the Long Parliament. Fuller was not formally dispossessed of his living and prebend on the triumph of the Presbyterian party, but he relinquished both preferments about this time. For a short time he preached with success at the Inns of Court, and thence removed, at the invitation of the master of the Savoy (Dr Balcanqual) and the brotherhood of that foundation, to be lecturer at their chapel of St Mary Savoy. Some of the best discourses of the witty preacher were delivered at the Savoy to audiences which extended into the chapel-yard. In one he set forth with searching and truthful minuteness the hindrances to peace, and urged the signing of petitions to the king at Oxford, and to the parliament, to continue their care in advancing an accommodation. In his Appeal of Injured Innocence Fuller says that he was once deputed to carry a petition to the king at Oxford. This has been identified with a petition entrusted to Sir Edward Wardour, clerk of the pells, Dr Dukeson, “Dr Fuller,” and four or five others from the city of Westminster and the parishes contiguous to the Savoy. A pass was granted by the House of Lords, on the 2nd of January 1643, for an equipage of two coaches, four or six horses and eight or ten attendants. On the arrival of the deputation at Uxbridge, on the 4th of January, officers of the Parliamentary army stopped the coaches and searched the gentlemen; and they found upon the latter “two scandalous books arraigning the proceedings of the House,” and letters with ciphers to Lord Viscount Falkland and the Lord Spencer. Ultimately a joint order of both Houses remanded the party; and Fuller and his friends suffered a brief imprisonment. The Westminster Petition, notwithstanding, reached the king’s hands; and it was published with the royal reply (see J.E. Bailey, Life of Thomas Fuller, pp. 245 et seq.). When it was expected, three months later, that a favourable result would attend the negotiations at Oxford, Fuller preached a sermon at Westminster Abbey, on the 27th of March 1643, on the anniversary of Charles I.’s accession, on the text, “Yea, let him take all, so my Lord the King return in peace.” On Wednesday, the 26th of July, he preached on church reformation, satirizing the religious reformers, and maintaining that only the Supreme Power could initiate reforms.

He was now obliged to leave London, and in August 1643 he joined the king at Oxford. He lived in a hired chamber at Lincoln College for 17 weeks. Thence he put forth a witty and effective reply to John Saltmarsh, who had attacked his views on ecclesiastical reform. Fuller subsequently published by royal request a sermon preached on the 10th of May 1644, at St Mary’s, Oxford, before the king and Prince Charles, called Jacob’s Vow.

The spirit of Fuller’s preaching, always characterized by calmness and moderation, gave offence to the high royalists, who charged him with lukewarmness in their cause. To silence unjust censures he became chaplain to the regiment of Sir Ralph Hopton. For the first five years of the war, as he said, when excusing the non-appearance of his Church History, “I had little list or leisure to write, fearing to be made a history, and shifting daily for my safety. All that time I could not live to study, who did only study to live.” After the defeat of Hopton at Cheriton Down, Fuller retreated to Basing House. He took an active part in its defence, and his life with the troops caused him to be afterwards regarded as one of “the great cavalier parsons.” In his marches with his regiment round about Oxford and in the west, he devoted much time to the collection of details, from churches, old buildings, and the conversation of ancient gossips, for his Church-History and Worthies of England. He compiled in 1645 a small volume of prayers and meditations,—the Good Thoughts in Bad Times,—which, set up and printed in the besieged city of Exeter, whither he had retired, was called by himself “the first fruits of Exeter press.” It was inscribed to Lady Dalkeith, governess to the infant princess, Henrietta Anne (b. 1644), to whose household he was attached as chaplain. The corporation gave him the Bodleian lectureship on the 21st of March 1645/6, and he held it until the 17th of June following, soon after the surrender of the city to the parliament. The Fear of losing the Old Light (1646) was his farewell discourse to his Exeter friends. Under the Articles of Surrender Fuller made his composition with the government at London, his “delinquency” being that he had been present in the king’s garrisons. In Andronicus, or the Unfortunate Politician (1646), partly authentic and partly fictitious, he satirized the leaders of the Revolution; and for the comfort of sufferers by the war he issued (1647) a second devotional manual, entitled Good Thoughts in Worse Times, abounding in fervent aspirations, and drawing moral lessons in beautiful language out of the events of his life or the circumstances of the time. In grief over his losses, which included his library and manuscripts (his “upper and nether millstone”), and over the calamities of the country, he wrote his work on the Cause and Cure of a Wounded Conscience (1647). It was prepared at Boughton House in his native county, where he and his son were entertained by Edward Lord Montagu, who had been one of his contemporaries at the university and had taken the side of the parliament.

For the next few years of his life Fuller was mainly dependent upon his dealings with booksellers, of whom he asserted that none had ever lost by him. He made considerable progress in an English translation from the MS. of the Annales of his friend Archbishop Ussher. Amongst his benefactors it is curious to find Sir John Danvers of Chelsea, the regicide. Fuller in 1647 began to preach at St Clement’s, Eastcheap, and elsewhere in the capacity of lecturer. While at St Clement’s he was suspended; but speedily recovering his freedom, he preached wherever he was invited. At Chelsea, where also he occasionally officiated, he covertly preached a sermon on the death of Charles I., but he did not break with his Roundhead patrons. James Hay, 2nd earl of Carlisle, made him his chaplain, and presented him in 1648 or 1649 to the curacy of Waltham Abbey. His possession of the living was in jeopardy on the appointment of Cromwell’s “Tryers”; but he evaded their inquisitorial questions by his ready wit. He was not disturbed at Waltham in 1655, when the Protector’s edict prohibited the adherents of the late king from preaching. Lionel, 3rd earl of Middlesex, who lived at Copt Hall, near Waltham, gave him what remained of the books of the lord treasurer his father; and through the good offices of the marchioness of Hertford, part of his own pillaged library was restored to him. Fuller was thus able to prosecute his literary labours, producing successively his descriptive geography of the Holy Land, called A Pisgah-Sight of Palestine (1650), and his Church-History of Britain (1655), from the birth of Jesus Christ until the year 1648. With the Church-History was printed The History of the University of Cambridge since the Conquest and The History of Waltham Abbey. These works were furthered in no slight degree by his connexion with Sion College, London, where he had a chamber, as well for the convenience of the press as of his city lectureships. The Church-History was angrily attacked by Dr P. Heylyn, who, in the spirit of High-Churchmanship, wished, as he said, to vindicate the truth, the church and the injured clergy. About 1652 Fuller married his second wife, Mary Roper, youngest sister of Thomas, Viscount Baltinglass, by whom he had several children. At the Oxford Act of 1657, Robert South, who was Terrae filius, lampooned Fuller, whom he described in this Oratio as living in London, ever scribbling and each year bringing forth new folia like a tree. At length, continues South, the Church-History came forth with its 166 dedications to wealthy and noble friends; and with this huge volume under one arm, and his wife (said to be little of stature) on the other, he ran up and down the streets of London, seeking at the houses of his patrons invitations to dinner, to be repaid by his dull jests at table.

His last and best patron was George Berkeley, 1st Earl Berkeley (1628-1698), of Cranford House, Middlesex, whose chaplain he was, and who gave him Cranford rectory (1658). To this nobleman Fuller’s reply to Heylyn’s Examen Historicum, called The Appeal of Injured Innocence (1659), was inscribed. At the end of the Appeal is an epistle “to my loving friend Dr Peter Heylyn,” conceived in the admirable Christian spirit which characterized all Fuller’s dealings with controversialists. “Why should Peter,” he asked, “fall out with Thomas, both being disciples to the same Lord and Master? I assure you, sir, whatever you conceive to the contrary, I am cordial to the cause of the English Church, and my hoary hairs will go down to the grave in sorrow for her sufferings.”

In An Alarum to the Counties of England and Wales (1660) Fuller argued for a free and full parliament—free from force, as he expressed it, as well as from abjurations or previous engagements. Mixt Contemplations in Better Times (1660), dedicated to Lady Monk, tendered advice in the spirit of its motto, “Let your moderation be known to all men: the Lord is at hand.” There is good reason to suppose that Fuller was at the Hague immediately before the Restoration, in the retinue of Lord Berkeley, one of the commissioners of the House of Lords, whose last service to his friend was to interest himself in obtaining him a bishopric. A Panegyrick to His Majesty on his Happy Return was the last of Fuller’s verse-efforts. On the 2nd of August, by royal letters, he was admitted D.D. at Cambridge. He resumed his lectures at the Savoy, where Samuel Pepys heard him preach; but he preferred his conversation or his books to his sermons. Fuller’s last promotion was that of chaplain in extraordinary to Charles II. In the summer of 1661 he visited the west in connexion with the business of his prebend, which had been restored to him. On Sunday, the 12th of August, while preaching at the Savoy, he was seized with typhus fever, and died at his new lodgings in Covent Garden on the 16th of August. He was buried in Cranford church, where a mural tablet was afterwards set up on the north side of the chancel, with an epitaph which contains a conceit worthy of his own pen, to the effect that while he was endeavouring (viz. in The Worthies) to give immortality to others, he himself attained it.

Fuller’s wit and vivacious good-humour made him a favourite with men of both sides, and his sense of humour kept him from extremes. Probably Heylyn and South had some excuse for their attitude towards his very moderate politics. “By his particular temper and management,” said Echard (Hist. of England, iii. 71), “he weathered the late great storm with more success than many other great men.” He was known as “a perfect walking library.” The strength of his memory was proverbial, and some amusing anecdotes are connected with it.

His writings were the product of a highly original mind. He had a fertile imagination and a happy faculty of illustration. Antithetic and axiomatic sentences abound in his pages, embodying literally the wisdom of the many in the wit of one. He was “quaint,” and something more. “Wit,” said Coleridge, in a well-known eulogy, “was the stuff and substance of Fuller’s intellect. It was the element, the earthen base, the material which he worked in; and this very circumstance has defrauded him of his due praise for the practical wisdom of the thoughts, for the beauty and variety of the truths, into which he shaped the stuff. Fuller was incomparably the most sensible, the least prejudiced, great man of an age that boasted a galaxy of great men” (Literary Remains, vol. ii. (1836), pp. 389-390). This opinion was formed after the perusal of the Church-History. That work and The History of the Worthies of England are unquestionably Fuller’s greatest efforts. They embody the collections of an entire life; and since his day they have been the delight of many readers. The Holy State has taken rank amongst the best books of “characters.” Charles Lamb made some selections from Fuller, and had a profound admiration for the “golden works” of the “dear, fine, silly old angel.” Since Lamb’s time, mainly through the appreciative criticisms of S.T. Coleridge, Robert Southey and others, Fuller’s works have received much attention.

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

FULLER, WILLIAM (1670-c. 1717), English impostor, was born at Milton in Kent on the 20th of September 1670. His paternity is doubtful, but he was related to the family of Herbert. After 1688 he served James II.’s queen, Mary of Modena, and the Jacobites, seeking at the same time to gain favour with William III.; and after associating with Titus Oates, being imprisoned for debt and pretending to reveal Jacobite plots, the House of Commons in 1692 declared he was an “imposter, cheat and false accuser.” Having stood in the pillory he was again imprisoned until 1695, when he was released; and at this time he took the opportunity to revive the old and familiar story that Mary of Modena was not the mother of the prince of Wales. In 1701 he published his autobiographical Life of William Fuller and some Original Letters of the late King James. Unable to prove the assertions made in his writings he was put in the pillory, whipped and fined. He died, probably in prison, about 1717. Fuller’s other writings are Mr William Fuller’s trip to Bridewell, with a full account of his barbarous usage in the pillory; The sincere and hearty confession of Mr William Fuller (1704); and An humble appeal to the impartial judgment of all parties in Great Britain (1716).

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

FULLER’S EARTH (Ger. Walkererde, Fr. terre à foulon, argile smectique)—so named from its use by fullers as an absorbent of the grease and oil of cloth,—a clay-like substance, which from its variability is somewhat difficult to define. In colour it is most often greenish, olive-green or greenish-grey; on weathering it changes to a brown tint or it may bleach. As a rule it falls to pieces when placed in water and is not markedly plastic; when dry it adheres strongly to the tongue; since, however, these properties are possessed by many clays that do not exhibit detergent qualities, the only test of value lies in the capacity to absorb grease or clarify oil. Fuller’s earth has a specific gravity of 1.7-2.4, and a shining streak; it is usually unctuous to the touch. Microscopically, it consists of minute irregular-shaped particles of a mineral that appears to be the result of a chloritic or talcose alteration of a felspar. The small size of most of the grains, less than .07 mm., makes their determination almost impossible. Chemical analysis shows that the peculiar properties of this earth are due to its physical rather than its chemical nature.

The following analyses of the weathered and unweathered condition of the earth from Nutfield, Surrey, represent the composition of one of the best known varieties:—

Blue Earth (dried at 100° C.).

Insoluble residue 69.96 Insoluble residue—

Fe2O3 2.48 SiO2 62.81

Al2O3 3.46 Al2O3 3.46

CaO 5.87 Fe2O3 1.30

MgO 1.41 CaO 1.53

P2O5 0.27 MgO 0.86

SO3 0.05 ———

NaCl 0.05 69.96

K2O 0.74 ———

H2O (combined) 15.57

———

99.86

———

Yellow Earth (dried at 100° C.).

Insoluble residue 76.13 Insoluble residue—

Fe2O3 2.41 SiO2 59.37

Al2O3 1.77 Al2O3 10.05

CaO 4.31 Fe2O3 3.86

MgO 1.05 CaO 1.86

P2O5 0.14 MgO 1.04

SO3 0.07 ———

NaCl 0.14 76.18

K2O 0.84 ———

H2O (combined) 13.19

———

100.05

———

(Analysis by P.G. Sanford, Geol. Mag., 1889, 6, pp. 456, 526.)

Of other published analyses, not a few show a lower silica content (44%, 50%), along with a higher proportion of alumina (11%, 23%).

Fuller’s earth may occur on any geological horizon; at Nutfield in Surrey, England, it is in the Cretaceous formations; at Midford near Bath it is of Jurassic age; at Bala, North Wales, it occurs in Ordovician strata; in Saxony it appears to be the decomposition product of a diabasic rock. In America it is found in California in rocks ranging from Cretaceous to Pleistocene age; in S. Dakota, Custer county and elsewhere a yellow, gritty earth of Jurassic age is worked; in Florida and Georgia occurs a brittle, whitish earth of Oligocene age. Other deposits are worked in Arkansas, Texas, Colorado, Massachusetts and South Carolina.

Fuller’s earth is either mined or dug in the open according to local circumstances. It is then dried in the sun or by artificial heat and transported in small lumps in sacks. In other cases it is ground to a fine powder after being dried; or it is first roughly ground and made into a slurry with water, which is allowed to carry off the finer from the coarser particles and deposit them in a creamy state in suitable tanks. After consolidation this fine material is dried artificially on drying floors, broken into lumps, and packed for transport. The use of fuller’s earth for cleansing wool and cloth has greatly decreased, but the demand for the material is as great or greater than it ever was. It is now used very largely in the filtration of mineral oils, and also for decolourizing certain vegetable oils. It is employed in the formation of certain soaps and cleansing preparations.

The term “Fuller’s Earth” has a special significance in geology, for it was applied by W. Smith in 1799 to certain clays in the neighbourhood of Bath, and the use of the expression is still retained by English geologists, either in this form or in the generalized “Fullonian.” The Fullonian lies at the base of the Great Oolite or Bathonian series, but its palaeontological characters place it between that series and the underlying Inferior Oolite. The zonal fossils are Perisphinctes arbustigerus and Macrocephalus subcontractus with Ostrea acuminata, Rhynchonella concinna and Goniomya angulifera. The formation is in part the equivalent of the “Vesulien” of J. Marcou (Vesoul in Haute-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.

See H.B. Woodward, “Jurassic Rocks of Great Britain,” vol. iv. (1894), Mem. Geol. Survey (London).

[J. A. H.]

FULLERTON, LADY GEORGIANA CHARLOTTE (1812-1885), English novelist and philanthropist, youngest daughter of the 1st Earl Granville, was born at Tixall Hall in Staffordshire on the 23rd of September 1812. In 1833 she married Alexander George Fullerton, then an Irish officer in the guards. After living in Paris for some eight years she and her husband accompanied Lord Granville to Cannes and thence to Rome. In 1843 her husband entered the Roman Catholic church, and in the following year Lady Georgiana Fullerton published her first novel, Ellen Middleton, which attracted W.E. Gladstone’s attention in the English Review. In 1846 she entered the Roman Catholic church. The death of her only son in 1854 plunged her in grief, and she continued to wear mourning until the end of her life. In 1856 she became one of the third order of St Francis, and thenceforward devoted herself to charitable work. In conjunction with Miss Taylor she founded the religious community known as “The Poor Servants of the Mother of God Incarnate,” and she also took an active part in bringing to England the sisters of St Vincent of Paul. Her philanthropic work is described in Mrs Augustus Craven’s work Lady Georgiana Fullerton, sa vie et ses 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 seal fisheries, showing the greatest boldness in approaching boats and ships, and feeding on the offal obtained from them. By British seamen it is commonly called the “molly mawk”1 (corrupted from Mallemuck), and is extremely well known to them, its flight, as it skims over the waves, first with a few beats of the wings and then gliding for a long way, being very peculiar. It only visits the land to deposit its single white egg, which is laid on a rocky ledge, where a shallow nest is made in the turf and lined with a little dried grass. Many of its breeding-places are a most valuable property to those who live near them and take the eggs and young, which, from the nature of the locality, are only to be had at a hazardous risk of life. In St Kilda a large number of the young are killed in one week of August, the only time when, by the custom of the community, they are allowed to be taken. These, after the oil is extracted from them, serve the islanders with food for the winter. The oil has been chemically analysed and found to be a fish-oil, and to possess nearly all the qualities of that obtained from the liver of the cod, with a lighter specific gravity. It, however, has an extremely strong scent, which is said by those who have visited St Kilda to pervade every thing and person on the island, and is certainly retained by an egg or skin of the bird for many years. Whenever a live example is seized in the hand it ejects a considerable quantity of this oil from its mouth.

1 A name misapplied in the southern hemisphere to Diomedea melanophrys, one of the albatrosses.

FULMINIC ACID, HCNO or H2C2N2O2, an organic acid isomeric with cyanic and cyanuric acids; its salts, termed fulminates, are very explosive and are much employed as detonators. The free acid, which is obtained by treating the salts with acids, is an oily liquid smelling like prussic acid; it is very explosive, and the vapour is poisonous to about the same degree as that of prussic acid. The first fulminate prepared was the “fulminating silver” of L.G. Brugnatelli, who found in 1798 that if silver be dissolved in nitric acid and the solution added to spirits of wine, a white, highly explosive powder was obtained. This substance is to be distinguished from the black “fulminating silver” obtained by C.L. Berthollet in 1788 by acting with ammonia on precipitated silver oxide. The next salt to be obtained was the mercuric salt, which was prepared in 1799 by Edward Charles Howard, who substituted mercury for silver in Brugnatelli’s process. A similar method is that of J. von Liebig (1823), who heated a mixture of alcohol, nitric acid and mercuric nitrate; the salt is largely manufactured by processes closely resembling the last. A laboratory method is to mix solutions of sodium nitromethane, CH2 : NO(ONa), and mercuric chloride, a yellow basic salt being formed at the same time. Mercuric fulminate is less explosive than the silver salt, and forms white needles (with ½H2O) 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 constitution of fulminic acid has been investigated by many experimenters, but apparently without definitive results. The researches of Liebig (1823), Liebig and Gay-Lussac (1824), and of Liebig again in 1838 showed the acid to be isomeric with cyanic acid, and probably (HCNO)2, since it gave mixed and acid salts. Kekulé, in 1858, concluded that it was nitroacetonitrile, NO2·CH2·CN, a view opposed by Steiner (1883), E. Divers and M. Kawakita (1884), R. Scholl (1890), and by J.U. Nef (1894), who proposed the formulae:

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 (Ber., 1905, 38, p. 345), that only compounds containing two carbon atoms yielded fulminates, points to (HCNO)2; 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

and also on the production from sodium nitromethane and mercuric chloride, thus CH2 : NO·Ohg → H2O + C : NOhg(hg = ½Hg). H. Wieland and F.C. Palazzo (1907) support this formula, finding that methyl nitrolic acid, NO2·CH : N·OH, yielded under certain conditions fulminic acid, and vice versa (Palazzo, 1907). M.Z. Jowitschitsch (Ann., 1906, 347, p. 233) inclines to Scholl’s formula; he found that the synthetic silver salt of glyoxime peroxide resembled silver fulminate in yielding hydroxylamine with hydrochloric acid, but differed in being less explosive, and in being soluble in nitric acid. H. Wieland and his collaborators regard “glyoxime peroxide” as an oxide of furazane (q.v.), and have shown that a close relationship exists between the nitrile oxides, furoxane, and fulminic acid (see Ann. Rep., London Chem. Soc., 1909, p. 84). Fulminuric acid, (HCNO)3, obtained by Liebig by boiling mercuric fulminate with water, was synthesized in 1905 by C. Ulpiani and L. Bernardini (Gazetta, iii. 35, p. 7), who regard it as NO2·CH(CN)·CO·NH2. It deflagrates at 145°, and forms a characteristic cuprammonium salt.

The early history of mercuric fulminate and a critical account of its application as a detonator is given in The Rise and Progress of the British Explosives Industry (International Congress of Applied Chemistry, 1909). The manufacture and modern aspects are treated in Oscar Guttmann, The Manufacture of Explosives, and Manufacture of Explosives, Twenty Years’ Progress (1909).

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

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

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

FULTON, a city of Oswego county, New York, U.S.A., on the right bank of the Oswego river, about 10 m. S. by E. of Oswego. Pop. (1900) 5281; (1905, state census) 8847; (1910) 10,480. Fulton is served by the Delaware, Lackawanna & Western, the New York Central & Hudson River, and the New York, Ontario & Western railways, by electric railway to Oswego and Syracuse and by the Oswego Canal. The city has a Carnegie library. Ample water-power is furnished by the Oswego river, which here flows in a series of rapids, and the manufactures are many in kind. On the 3rd of July 1756, on an island (afterward called Battle Island) 4 m. N. of the present city of Fulton, a British force of about 300 under Captain John Bradstreet (1711-1774) defeated an attacking force of French and Indians (numbering about 700) under De Villiers. Soon after this, Bradstreet built a fort within the present limits of Fulton. The first civilian settler came in 1793, and the first survey (which included only a part of the subsequent village) was made in 1815. Fulton was incorporated as a village in 1835, and in April 1902 was combined with the village of Oswego Falls (pop. in 1900, 2925) and was chartered as a city.

FUM, or Funj Hwang, one of the four symbolical creatures which in Chinese mythology are believed to keep watch and ward over the Celestial Empire. It was begotten by fire, was born in the Hill of the Sun’s Halo, and its body bears inscribed on it the five cardinal virtues. It has the breast of a goose, the hindquarters of a stag, a snake’s neck, a fish’s tail, a fowl’s forehead, a duck’s down, the marks of a dragon, the back of a tortoise, the face of a swallow, the beak of a cock, is about six cubits high, and perches only on the woo-tung tree. The appearance of Fum heralds an age of universal virtue. Its figure is that which is embroidered on the dresses of some mandarins.

FUMARIC AND MALEIC ACIDS, two isomeric unsaturated acids of composition C4H4O4. Fumaric acid is found in fumitory (Fumaria officinalis), in various fungi (Agaricus piperatus, &c.), and in Iceland moss. It is obtained by heating malic acid alone to 150° C., or by heating it with hydrochloric acid (V. Dessaignes, Jahresb., 1856, p. 463) or with a large quantity of hydrobromic 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 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, Ber., 1881, 14, p. 713). By long-continued heating with caustic soda at 100° C. it is converted into inactive malic acid.

Maleic acid is obtained by distilling malic or fumaric acids; by heating fumaric acid with acetyl chloride to 100° C; or by the hydrolysis of trichlorphenomalic acid (β-trichloraceto-acrylic acid) [A. Kekulé, Ann., 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.

Both acids are readily esterified by the action of alkyl halides on their silver salts, and the maleic ester is readily transformed into the fumaric ester by warming with iodine, the same result being obtained by esterification of maleic acid in alcoholic solution by means of hydrochloric acid. Both acids yield acetylene by the electrolysis of aqueous solutions of their alkali salts, and on reduction both yield succinic acid, whilst by the addition of hydrobromic acid they both yield monobromsuccinic acid (R. Fittig, Ann., 1877, 188, p. 98). From these results it follows that the two acids are structurally identical, and the isomerism has consequently to be explained on other grounds. This was accomplished by W. Wislicenus [”Über 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 (see Stereo-Isomerism). The formulae of the acids are written thus:

These account for maleic acid readily yielding an anhydride, whereas fumaric acid does not, and for the behaviour of the acids towards bromine, fumaric acid yielding ordinary dibromsuccinic acid, and maleic acid the isomeric isodibromsuccinic acid.

FUMAROLE, a vent from which volcanic vapours issue, named indirectly from the Lat. fumariolum, a smoke-hole. The vapours from fumaroles were studied first by R.W. Bunsen, on his visit to Iceland, and afterwards by H. Sainte-Claire Deville and other chemists and geologists in France, who examined the vapours from Santorin, Etna, &c. The hottest vapours issue from dry fumaroles, at temperatures of at least 500° 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 Volcano).

FUMIGATION (from Lat. fumigare, to smoke), the process of producing smoke or fumes, as by burning sulphur, frankincense, tobacco, &c., whether as a ceremony of incantation, or for perfuming a room, or for purposes of disinfection or destruction of vermin. In medicine the term has been used of the exposure of the body, or a portion of it, to fumes such as those of nitre, sal-ammoniac, mercury, &c.; fumigation, by the injection of tobacco smoke into the great bowel, was a recognized procedure in the 18th century for the resuscitation of the apparently drowned. “Fumigated” or “fumed” oak is oak which has been darkened by exposure to ammonia vapour.

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

FUNCHAL, the capital of the Portuguese archipelago of the Madeiras; on the south coast of Madeira, in 32° 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 sugar-cane plantations occupy the surrounding heights. Three mountain streams traverse the city through deep channels, which in summer are dry, owing to the diversion of the water for irrigation. A small fort, on an isolated rock off shore, guards the entrance to the bay, and a larger and more powerfully armed fort crowns an eminence inland. The chief buildings include the cathedral, Anglican and Presbyterian churches, hospitals, opera-house, museum and casino. There are small public gardens and a meteorological observatory. In the steep and narrow streets, which are lighted by electricity, wheeled traffic is impossible; sledges drawn by oxen, and other primitive conveyances are used instead (see Madeira). In winter the fine climate and scenery attract numerous invalids and other visitors, for whose accommodation there are good hotels; many foreigners engaged in the coal and wine trades also reside here permanently. The majority of these belong to the British community, which was first established here in the 18th century. Funchal is the headquarters of Madeiran industry and commerce (see Madeira). It has no docks and no facilities for landing passengers or goods; vessels are obliged to anchor in the roadstead, which, however, is sheltered from every wind except the south. Funchal is connected by cable with Carcavellos (for Lisbon), Porthcurnow (for Falmouth, England) and St Vincent in the Cape Verde Islands (for Pernambuco, Brazil).

FUNCTION,1 in mathematics, a variable number the value of which depends upon the values of one or more other variable numbers. The theory of functions is conveniently divided into (I.) Functions of Real Variables, wherein real, and only real, numbers are involved, and (II.) Functions of Complex Variables, wherein complex or imaginary numbers are involved.

I. Functions of Real Variables

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

a1xn + a2xn−1 + ... + an ,

b1xm + b2xm−1 + ... + bm

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

ƒn(x, y) + ƒn−1(x, y) + ... + ƒ0 = 0

where ƒn(x, y), ... mean homogeneous expressions in x and y having constant coefficients, and having the degrees indicated by the suffixes, and ƒ0 is a constant. Or again we may have trigonometrical functions, such as sin x and tan x, or inverse trigonometrical functions, such as sin−1x, or exponential functions, such as ex and ax, 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 co-ordinates is to be regarded as a picturesque mode of expression, pointing to a possible application of the theory to geometry. The development of “arithmetized analysis” in the 19th century is associated with the name of Karl Weierstrass.

All possible values of a variable are numbers. In what follows we shall confine our attention to the case where the numbers are real. When complex numbers are introduced, instead of real ones, the theory of functions receives a wide extension, which is accompanied by appropriate limitations (see below, II. Functions of Complex Variables). The set of all real numbers forms a continuum. In fact the notion of a one-dimensional continuum first becomes precise in virtue of the establishment of the system of real numbers.

4. Domain of a Variable.—Theory of Aggregates.—The notion of a “variable” is that of a number to which we may assign at pleasure any one of the values that belong to some chosen set, or aggregate, of numbers; and this set, or aggregate, is called the “domain of the variable.” This domain may be an “interval,” that is to say it may consist of two terminal numbers, all the numbers between them and no others. When this is the case the number is said to be “continuously variable.” When the domain consists of all real numbers, the variable is said to be “unrestricted.” A domain which consists of all the real numbers which exceed some fixed number may be described as an “interval unlimited towards the right”; similarly we may have an interval “unlimited towards the left.”

In more complicated cases we must have some rule or process for assigning the aggregate of numbers which constitute the domain of a variable. The methods of definition of particular types of aggregates, and the theorems relating to them, form a branch of analysis called the “theory of aggregates” (Mengenlehre, 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 1 correspond to 2, 2 to 4, and generally n to 2n. The aggregate of positive integers is an infinite aggregate. The aggregates of all rational numbers and of all real numbers and of points on a line are other examples of infinite aggregates. An aggregate whose elements are real numbers is said to “extend to infinite values” if, after any number N, however great, is specified, it is possible to find in the aggregate numbers which exceed N in absolute value. Such an aggregate is always infinite. The “neighbourhood of a number (or point) a for a positive number h” is the aggregate of all numbers (or points) x for which the absolute value of x − a denoted by |x − a|, does not exceed h.

5. General Notion of Functionality.—A function of one variable was for a long time commonly regarded as the ordinate of a curve; and the two notions (1) that which is determined by a curve supposed drawn, and (2) that which is determined by an analytical expression supposed written down, were not for a long time clearly distinguished. It was for this reason that Fourier’s discovery that a single analytical expression is capable of representing (in different parts of an interval) what would in his time have been called different functions so profoundly struck mathematicians (§ 23). The analysts who, in the middle of the 19th century, occupied themselves with the theory of the convergence of Fourier’s series were led to impose a restriction on the character of a function in order that it should admit of such representation, and thus the door was opened for the introduction of the general notion of functional dependence. This notion may be expressed as follows: We have a variable number, y, and another variable number, x, a domain of the variable x, and a rule for assigning one or more definite values to y when x is any point in the domain; then y is said to be a “function” of the variable x, and x is called the “argument” of the function. According to this notion a function is, as it were, an indefinitely extended table, like a table of logarithms; to each point in the domain of the argument there correspond values for the function, but it remains arbitrary what values the function is to have at any such point.

For the specification of any particular function two things are requisite: (1) a statement of the values of the variable, or of the aggregate of points, to which values of the function are to be made to correspond, i.e. of the “domain of the argument”; (2) a rule for assigning the value or values of the function that correspond to any point in this domain. We may refer to the second of these two essentials as “the rule of calculation.” The relation of functions to analytical expressions may then be stated in the form that the rule of calculation is: “Give the function the value of the expression at any point at which the expression has a determinate value,” or again more generally, “Give the function the value of the expression at all points of a definite aggregate included in the domain of the argument.” The former of these is the rule of those among the earlier analysts who regarded an analytical expression and a function as the same thing, and their usage may be retained without causing confusion and with the advantage of brevity, the analytical expression serving to specify the domain of the argument as well as the rule of calculation, e.g. we may speak of “the function 1/x.” This function is defined by the analytical expression 1/x at all points except the point x = 0. But in complicated cases separate statements of the domain of the argument and the rule of calculation cannot be dispensed with. In general, when the rule of calculation is determined as above by an analytical expression at any aggregate of points, the function is said to be “represented” by the expression at those points.

When the rule of calculation assigns a single definite value for a function at each point in the domain of the argument the function is “uniform” or “one-valued.” In what follows it is to be understood that all the functions considered are one-valued, and the values assigned by the rule of calculation real. In the most important cases the domain of the argument of a function of one variable is an interval, with the possible exception of isolated points.

6. Limits.—Let 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.

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 Ltx=aƒ(x), or by limx=aƒ(x).

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,” u1, u2, ... un, ..., and this sequence can have a limit at n = ∞.

The principle common to the above definitions and theorems is called, after P. du Bois Reymond, “the general principle of convergence to a limit.”

It must be understood that the phrase “x = ∞” does not mean that x takes some particular value which is infinite. There is no such value. The phrase always refers to a limiting process in which, as the process is carried out, the variable number x increases without limit: it may, as in the above example of a sequence, increase by taking successively the values of all the integral numbers; in other cases it may increase by taking the values that belong to any domain which “extends to infinite values.”

A very important type of limits is furnished by infinite series. When a sequence of numbers u1, u2, ... un, ... is given, we may form a new sequence s1, s2, ... sn, ... from it by the rules s1 = u1, s2 = u1 + u2, ... sn = u1 + u2 + ... + un or by the equivalent rules s1 = u, sn − sn−1 = un(n = 2, 3, ...). If the new sequence has a limit at n = ∞, this limit is called the “sum of the infinite series” u1 + u2 + ..., and the series is said to be “convergent” (see Series).

A function which has not a limit at a point a may be such that, if a certain aggregate of points is chosen out of the domain of the argument, and the points x in the neighbourhood of a are restricted to belong to this aggregate, then the function has a limit at a. For example, sin(1/x) has limit zero at 0 if x is restricted to the aggregate 1/π, 1/2π, ... 1/nπ, ... or to the aggregate 1/2π, 2/5π, ... n/(n2 + 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).

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 Infinitesimal Calculus) are examples.

7. Superior and Inferior Limits; Infinities.—The value of a function at every point in the domain of its argument is finite, since, by definition, the value can be assigned, but this does not necessarily imply that there is a number N which exceeds all the values (or is less than all the values). It may happen that, however great a number N we take, there are among the values of the function numbers which exceed N (or are less than −N).

If a number can be found which is greater than every value of the function, then either (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.”

All that has been said here may be adapted to the description of greatest values, superior limits, &c., of a function in a restricted domain contained in the domain of the argument. In particular, the domain of the argument may contain an interval; and therein the function may have a superior limit, or an inferior limit, which is attained. Such a limit is a maximum value or a minimum value of the function.

Again, if, after any number N, however great, has been specified, it is possible to find points of the domain of the argument at which the value of the function exceeds N, the values of the function are said to have an “infinite superior limit,” and then there must be a point a which has the property that there are points of the domain, in the neighbourhood of a for any h, at which the value of the function exceeds N. If the point a is in the domain of the argument the function is said to “tend to become infinite” at a; it has of course a finite value at a. If the point a is not in the domain of the argument the function is said to “become infinite” at a; it has of course no value at a. In like manner we may have a (negatively) infinite inferior limit. Again, after any number N, however great, has been specified and a number h found, so that all the values of the function, at points in the neighbourhood of a for h, exceed N in absolute value, all these values may have the same sign; the function is then said to become, or to tend to become, “determinately (positively or negatively) infinite”; otherwise it is said to become or to tend to become, “indeterminately infinite.”

All the infinities that occur in the theory of functions are of the nature of variable finite numbers, with the single exception of the infinity of an infinite aggregate. The latter is described as an “actual infinity,” the former as “improper infinities.” There is no “actual infinitely small” corresponding to the actual infinity. The only “infinitely small” is zero. All “infinite values” are of the nature of superior and inferior limits which are not attained.

8. Increasing and Decreasing Functions.—A function 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 mutatis mutandis 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.

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 diminishes when 1/e > x > 0 and x diminishes towards zero, and it never becomes negative. It therefore has a limit on the right at x = 0. This limit is zero. The function represented by x sin (1/x) does not continually diminish towards zero as x diminishes towards zero, but is sometimes greater than zero and sometimes less than zero in any neighbourhood of x = 0, however small. Nevertheless, the function has the limit zero at x = 0.

9. Continuity of Functions.—A function 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.

A function is said to be “continuous throughout an interval” when it is continuous at every point of the interval. This implies continuity to the right at the smaller end-value and continuity to the left at the greater end-value. When these conditions at the ends are not satisfied the function is said to be continuous “within” the interval. By a “continuous function” of one variable we always mean a function which is continuous throughout an interval.

The principal properties of a continuous function are:

1. The function is practically constant throughout sufficiently small intervals. This means that, after any point a of the interval has been chosen, and any positive number 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.

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.

3. The function has a greatest value and a least value in the interval, and these are superior and inferior limits which are attained.

4. There is at least one point of the interval at which the function takes any value between its greatest and least values in the interval.

5. If the interval is unlimited towards the right (or towards the left), the function has a limit at ∞ (or at −∞).

10. Discontinuity of Functions.—The discontinuities of a function of one variable, defined in an interval with the possible exception of isolated points, may be classified as follows:

(1) The function may become infinite, or tend to become infinite, at a point.

(2) The function may be undefined at a point.

(3) The function may have a limit on the left and a limit on the right at the same point; these may be different from each other, and at least one of them must be different from the value of the function at the point.

(4) The function may have no limit at a point, or no limit on the left, or no limit on the right, at a point.

In case a function 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 ƒ(a + 0) and ƒ(a + 0), and they are called the “limits of indefiniteness” on the right. Similar limits on the left are denoted by ƒ(a − 0) and ƒ(a − 0). 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, e.g. the superior limit of the inferior limits in diminishing neighbourhoods on the right of a may be negatively infinite; this happens if, after any number N, however great, has been specified, it is possible to find a positive number h, so that all the values of the function in the interval between a and a + h (a excluded) are less than −N; in such a case 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.

11. Oscillation of Functions.—The difference between the greatest and least of the numbers ƒ(a), ƒ(a + 0), ƒ(a + 0), ƒ(a − 0), ƒ(a − 0), 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.

For the further classification of functions we divide the domain of the argument into partial intervals by means of points between the end-points. Suppose that the domain is the interval between a and b. Let intermediate points x1, x2 ... xn−1, be taken so that b > xn−1 > xn−2 ... > x1 > a. We may devise a rule by which, as n increases indefinitely, all the differences b − xn−1, xn−1 − xn−2, ... x1 − a tend to zero as a limit. The interval is then said to be divided into “indefinitely small partial intervals.”

A function defined in an interval with the possible exception of isolated points may be such that the interval can be divided into a set of finite partial intervals within each of which the function is monotonous (§ 8). When this is the case the sum of the oscillations of the function in those partial intervals is finite, provided the function does not tend to become infinite. Further, in such a case the sum of the oscillations will remain below a fixed number for any mode of dividing the interval into indefinitely small partial intervals. A class of functions may be defined by the condition that the sum of the oscillations has this property, and such functions are said to have “restricted oscillation.” Sometimes the phrase “limited fluctuation” is used. It can be proved that any function with restricted oscillation is capable of being expressed as the sum of two monotonous functions, of which one never increases and the other never diminishes throughout the interval. Such a function has a limit on the right and a limit on the left at every point of the interval. This class of functions includes all those which have a finite number of maxima and minima in a finite-interval, and some which have an infinite number. It is to be noted that the class does not include all continuous functions.

12. Differentiable Function.—The idea of the differentiation of a continuous function is that of a process for measuring the rate of growth; the increment of the function is compared with the increment of the variable. If ƒ(x) is defined in an interval containing the point a, and a − k and a + k are points of the interval, the expression

ƒ(a + h) − ƒ(a)

h

(1)

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 φ(+0), φ(+0), φ(−0), φ(−0) exist, and two or more of them may be equal. When the first two are equal either of them is the “progressive differential coefficient” of 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

ƒ′ + (a), ƒ′ + (a), ƒ′ − (a), ƒ′ − (a).

A function which has a finite differential coefficient at all points of an interval is continuous throughout the interval, but if the differential coefficient becomes infinite at a point of the interval the function may or may not be continuous throughout the interval; on the other hand a function may be continuous without being differentiable. This result, comparable in importance, from the point of view of the general theory of functions, with the discovery of Fourier’s theorem, is due to G.F.B. Riemann; but the failure of an attempt made by 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 Infinitesimal Calculus.)

13. Analytic Function.—If ƒ(x) and its first n differential coefficients, denoted byƒ′(x), ƒ″(x), ... ƒ(n) (x), are continuous in the interval between a and a + h, then

ƒ(a + h) = ƒ(a) + hƒ′(a) + h² ƒ″(a) + ... + hn−1 ƒ(n−1)(a) + Rn,

2! (n − 1)!

where Rn may have various forms, some of which are given in the article Infinitesimal Calculus. This result is known as “Taylor’s theorem.”

When Taylor's theorem leads to a representation of the function by means of an infinite series, the function is said to be “analytic” (cf. § 21).

14. Ordinary Function.—The idea of a curve representing a continuous function in an interval is that of a line which has the following properties: (1) the co-ordinates of a point of the curve are a value x of the argument and the corresponding value y of the function; (2) at every point the curve has a definite tangent; (3) the interval can be divided into a finite number of partial intervals within each of which the function is monotonous; (4) the property of monotony within partial intervals is retained after interchange of the axes of co-ordinates x and y. According to condition (2) y is a continuous and differentiable function of x, but this condition does not include conditions (3) and (4): there are continuous partially monotonous functions which are not differentiable, there are continuous differentiable functions which are not monotonous in any interval however small; and there are continuous, differentiable and monotonous functions which do not satisfy condition (4) (cf. § 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.

15. Integrable Function.—The idea of integration is twofold. We may seek the function which has a given function as its differential coefficient, or we may generalize the question of finding the area of a curve. The first inquiry leads directly to the indefinite integral, the second directly to the definite integral. Following the second method we define “the definite integral of the function ƒ(x) through the interval between a and b” to be the limit of the sum

Σn1 ƒ(x′r) (xr − xr−1)

when the interval is divided into ultimately indefinitely small partial intervals by points x1, x2, ... xn−1. Here x′r denotes any point in the rth partial interval, x0 is put for a, and xn for b. It can be shown that the limit in question is finite and independent of the mode of division into partial intervals, and of the choice of the points such as x′r, provided (1) the function is defined for all points of the interval, and does not tend to become infinite at any of them; (2) for any one mode of division of the interval into ultimately indefinitely small partial intervals, the sum of the products of the oscillation of the function in each partial interval and the difference of the end-values of that partial interval has limit zero when n is increased indefinitely. When these conditions are satisfied the function is said to be “integrable” in the interval. The numbers a and b which limit the interval are usually called the “lower and upper limits.” We shall call them the “nearer and further end-values.” The above definition of integration was introduced by Riemann in his memoir on trigonometric series (1854). A still more general definition has been given by Lebesgue. As the more general definition cannot be made intelligible without the introduction of some rather recondite notions belonging to the theory of aggregates, we shall, in what follows, adhere to Riemann’s definition.

We have the following theorems:—

1. Any continuous function is integrable.

2. Any function with restricted oscillation is integrable.

3. A discontinuous function is integrable if it does not tend to become infinite, and if the points at which the oscillation of the function exceeds a given number σ, however small, can be enclosed in partial intervals the sum of whose breadths can be diminished indefinitely.

These partial intervals must be a set chosen out of some complete set obtained by the process used in the definition of integration.

4. The sum or product of two integrable functions is integrable.

As regards integrable functions we have the following theorems:

1. If S and I are the superior and inferior limits (or greatest and least values) of ƒ(x) in the interval between a and b, ∫ ba ƒ(x)dx is intermediate between S(b − a) and I(b − a).

2. The integral is a continuous function of each of the end-values.

3. If the further end-value b is variable, and if ∫ xa ƒ(x)dx = F(x), then if ƒ(x) is continuous at b, F(x) is differentiable at b, and F′(b) = ƒ(b).

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.

5. In case ƒ′(x) is continuous throughout the interval between a and b,

∫ ba ƒ′(x)dx = ƒ(b) − ƒ(a).

6. In case ƒ(x) is discontinuous at one or more points of the interval between a and b, in which it is integrable,

∫ xa ƒ(x)dx

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.

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

F(x) = ∫ xa F′(x)dx + const.,

where a is a fixed point, and x a variable point, of the interval. Similarly, if any one of the four derivates of a function is integrable in an interval, all are integrable, and the integral of either differs from the original function by a constant only.

The theorems (4), (6), (7) show that there is some discrepancy between the indefinite integral considered as the function which has a given function as its differential coefficient, and as a definite integral with a variable end-value.

We have also two theorems concerning the integral of the product of two integrable functions 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

M ∫ ba φ(x)dx = ∫ ba ƒ(x)φ(x)dx

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

∫ ba ƒ(x) φ(x)dx = ƒ(a) ∫ ξa φ(x)dx + ƒ(b) ∫ bξ φ(x)dx.

(See Fourier’s Series.)

16. Improper Definite Integrals.—We may extend the idea of integration to cases of functions which are not defined at some point, or which tend to become infinite in the neighbourhood of some point, and to cases where the domain of the argument extends to infinite values. If c is a point in the interval between a and b at which ƒ(x) is not defined, we impose a restriction on the points x′r of the definition: none of them is to be the point c. This comes to the same thing as defining ∫ ba ƒ(x)dx to be

Lt ε=0∫ c−εa ƒ(x)dx + Lt ε′=0∫ bc+ε′ ƒ(x)dx,

(1)

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.

Again, when the interval of integration is unlimited to the right, or extends to positively infinite values, we have as a definition

∫ ∞a ƒ(x)dx = Lt h=∞∫ ha ƒ(x)dx,

provided this limit exists. Similar definitions apply to

∫ −∞a ƒ(x)dx, and to ∫ ∞−∞ ƒ(x)dx.

All such definite integrals as the above are said to be “improper.” For example, ∫ ∞0 sin x / x dx is improper in two ways. It means

Lt h=∞ Lt ε=0 ∫ hε sinx/x dx,

in which the positive number ε is first diminished indefinitely, and the positive number h is afterwards increased indefinitely.

The “theorems of the mean” (§ 15) require modification when the integrals are improper (see Fourier’s Series).

When the improper definite integral of a function which becomes, or tends to become, infinite, exists, the integral is said to be “convergent.” If 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 ∫ ba ƒ(x)dx, which has no value, is called a “divergent integral, “and it may happen that there is a definite value for

Lt { ∫ c−εa ƒ(x) dx + ∫ bc+ε′ ƒ(x) dx }

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.e. by taking the omitted interval so that the infinity is at its middle point. A divergent integral which has one or more principal values is sometimes described as “semi-convergent.”

17. Domain of a Set of Variables.—The numerical continuum of n dimensions (Cn) is the aggregate that is arrived at by attributing simultaneous values to each of n variables x1, x2, ... xn, these values being any real numbers. The elements of such an aggregate are called “points,” and the numbers x1, x2 ... xn the “co-ordinates” of a point. Denoting in general the points (x1, x2, ... xn) and (x′1, x′2 ... x′n) by x and x′, the sum of the differences |x1 − x′1| + |x2 − x′2| + ... + |xn − x′n| may be denoted by |x − x′| and called the “difference of the two points.” We can in various ways choose out of the continuum an aggregate of points, which may be an infinite aggregate, and any such aggregate can be the “domain” of a “variable point.” The domain is said to “extend to an infinite distance” if, after any number N, however great, has been specified, it is possible to find in the domain points of which one or more co-ordinates exceed N in absolute value. The “neighbourhood” of a point a for a (positive) number h is the aggregate constituted of all the points x, which are such that the “difference” denoted by |x − a| < h. If an infinite aggregate of points does not extend to an infinite distance, there must be at least one point a, which has the property that the points of the aggregate which are in the neighbourhood of a for any number h, however small, themselves constitute an infinite aggregate, and then the point a is called a “limiting point” of the aggregate; it may or may not be a point of the aggregate. An aggregate of points is “perfect” when all its points are limiting points of it, and all its limiting points are points of it; it is “connected” when, after taking any two points a, b of it, and choosing any positive number ε, however small, a number m and points x′, x″, ... x(m) of the aggregate can be found so that all the differences denoted by |x′ − a|, |x″ − x′|, ... |b − x(m)| are less than ε. A perfect connected aggregate is a continuum. This is G. Cantor’s definition.

The definition of a continuum in Cn 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” Hn of Cn. Such a part would correspond to an interval in C1, or to an area bounded by a simple closed contour in C2; and, besides being perfect and connected, it would have the following properties: (1) There are points of Cn, which are not points of Hn; these form a complementary aggregate H′n. (2) There are points “within” Hn; this means that for any such point there is a neighbourhood consisting exclusively of points of Hn. (3) The points of Hn which do not lie “within” Hn are limiting points of H′n; they are not points of H′n, but the neighbourhood of any such point for any number h, however small, contains points within Hn and points of H′n: the aggregate of these points is called the “boundary” of Hn. (4) When any two points a, b within Hn are taken, it is possible to find a number ε and a corresponding number m, and to choose points x′, x″, ... x(m), so that the neighbourhood of a for ε contains x′, and consists exclusively of points within Hn, and similarly for x′ and x″, x″ and x″′, ... x(m) and b. Condition (3) would exclude such an aggregate as that of the points within and upon two circles external to each other and a line joining a point on one to a point on the other, and condition (4) would exclude such an aggregate as that of the points within and upon two circles which touch externally.

18. Functions of Several Variables.—A function of several variables differs from a function of one variable in that the argument of the function consists of a set of variables, or is a variable point in a Cn 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 Hn of Cn with the possible exception of isolated points, and the rule of calculation is that the value of the function in any assigned part of the domain of the argument is that value which is assumed at the point by an assigned analytical expression. The limit of a function at a point a is defined in the same way as in the case of a function of one variable.

We take a positive fraction 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 x1, x2, and both are unrestricted, the domain of the argument is represented by a plane, and the values of the function are correlated with the points of the plane. The function has a limit at a point a, if we can mark out on the plane a region containing the point a within it, and such that the difference of the values of the function which correspond to any two points of the region (neither of the points being a) can be made as small as we please in absolute value by contracting all the linear dimensions of the region sufficiently. When the domain of the argument of a function of n variables extends to an infinite distance, there is a “limit at an infinite distance” if, after any number 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 x1 and x2 may have a limit at (x1 = 0, x2 = 0) if we first diminish x1 without limit, keeping x2 constant, and afterwards diminish x2 without limit. Expressed in geometrical language, this process amounts to approaching the origin along the axis of x2. The definitions of superior and inferior limits, and of maxima and minima, and the explanations of what is meant by saying that a function of several variables becomes infinite, or tends to become infinite, at a point, are almost identical verbally with the corresponding definitions and explanations in the case of a function of one variable (§ 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−1 (y/x)}. When y has any particular value this function is a continuous function of x, and, when x has any particular value this function is a continuous function of y; but the function of x and y is discontinuous at (x = 0, y = 0).