VOLUME VI, SLICE VII

Columbus to Condottiere

Articles in This Slice:

COLUMBUS (city of Georgia, U.S.A.) COMO (city of Italy) COLUMBUS (city of Indiana, U.S.A.) COMO (lake of Italy) COLUMBUS (city of Mississippi, U.S.A.) COMONFORT, IGNACIO COLUMBUS (city of Ohio, U.S.A.) COMORIN, CAPE COLUMELLA, LUCIUS JUNIUS MODERATUS COMORO ISLANDS COLUMN COMPANION COLURE COMPANY COLUTHUS COMPARATIVE ANATOMY COLVILLE, JOHN COMPARETTI, DOMENICO COLVIN, JOHN RUSSELL COMPASS COLVIN, SIDNEY COMPASS PLANT COLWYN BAY COMPAYRE, JULES GABRIEL COLZA OIL COMPENSATION COMA COMPIÈGNE COMA BERENICES COMPLEMENT COMACCHIO COMPLUVIUM COMANA (city of Cappadocia) COMPOSITAE COMANA (city of Pontus) COMPOSITE ORDER COMANCHES COMPOSITION COMAYAGUA COMPOUND COMB COMPOUND PIER COMBACONUM COMPRADOR COMBE, ANDREW COMPRESSION COMBE, GEORGE COMPROMISE COMBE, WILLIAM COMPROMISE MEASURES OF 1850 COMBE (closed-in valley) COMPSA COMBERMERE, STAPLETON COTTON COMPTON, HENRY COMBES, ÉMILE COMPTROLLER COMBINATION COMPURGATION COMBINATORIAL ANALYSIS COMTE, AUGUSTE COMBUSTION COMUS COMEDY COMYN, JOHN COMENIUS, JOHANN AMOS CONACRE COMET CONANT, THOMAS JEFFERSON COMET-SEEKER CONATION COMILLA CONCA, SEBASTIANO COMINES CONCARNEAU COMITIA CONCEPCIÓN (province of Chile) COMITY CONCEPCIÓN (city of Chile) COMMA CONCEPCIÓN (town of Paraguay) COMMANDEER CONCEPT COMMANDER CONCEPTUALISM COMMANDERY CONCERT COMMANDO CONCERTINA COMMEMORATION CONCERTO COMMENDATION CONCH COMMENTARII CONCHOID COMMENTRY CONCIERGE COMMERCE (trade) CONCINI, CONCINO COMMERCE (card-game) CONCLAVE COMMERCIAL COURT CONCORD (Massachusetts, U.S.A) COMMERCIAL LAW CONCORD (North Carolina, U.S.A.) COMMERCIAL TREATIES CONCORD (New Hampshire, U.S.A.) COMMERCY CONCORD, BOOK OF COMMERS CONCORDANCE COMMINES, PHILIPPE DE CONCORDAT COMMISSARIAT CONCORDIA (Roman goddess) COMMISSARY CONCORDIA (town of Venetia) COMMISSION CONCRETE (solidity) COMMISSIONAIRE CONCRETE (building material) COMMISSIONER CONCRETION COMMITMENT CONCUBINAGE COMMITTEE CONDÉ, PRINCES OF COMMODIANUS CONDÉ, LOUIS DE BOURBON COMMODORE CONDÉ, LOUIS II. DE BOURBON COMMODUS, LUCIUS AELIUS AURELIUS CONDÉ (villages of France) COMMON LAW CONDE, JOSÉ ANTONIO COMMON LODGING-HOUSE CONDENSATION OF GASES COMMON ORDER, BOOK OF CONDENSER COMMONPLACE CONDER, CHARLES COMMON PLEAS, COURT OF CONDILLAC, ÉTIENNE BONNOT DE COMMONS CONDITION COMMONWEALTH CONDITIONAL FEE COMMUNE CONDITIONAL LIMITATION COMMUNE, MEDIEVAL CONDOM COMMUNISM CONDOR COMMUTATION CONDORCET, CARITAT COMNENUS CONDOTTIERE

COLUMBUS, a city and the county-seat of Muscogee county, Georgia, U.S.A., on the E. bank and at the head of navigation of the Chattahoochee river, about 100 m. S.S.W. of Atlanta. Pop. (1890) 17,303; (1900) 17,614, of whom 7267 were negroes; (1910, census) 20,554. There is also a considerable suburban population. Columbus is served by the Southern, the Central of Georgia, and the Seaboard Air Line railways, and three steamboat lines afford communication with Apalachicola, Florida. The city has a public library. A fall in the river of 115 ft. within a mile of the city furnishes a valuable water-power, which has been utilized for public and private enterprises. The most important industry is the manufacture of cotton goods; there are also cotton compresses, iron works, flour and woollen mills, wood-working establishments, &c. The value of the city's factory products increased from $5,061,485 in 1900 to $7,079,702 in 1905, or 39.9%; of the total value in 1905, $2,759,081, or 39%, was the value of the cotton goods manufactured. There are many large factories just outside the city limits. Columbus was one of the first cities in the United States to maintain, at public expense, a system of trade schools. It has a large wholesale and retail trade. The city was founded in 1827 and was incorporated in 1828. In the latter year Mirabeau Buonaparte Lamar (1798-1859) established here the Columbus _Independent_, a State's-Rights newspaper. For the first twenty years the city's leading industry was trade in cotton. As this trade was diverted by the railways to Savannah, the water-power was developed and manufactories were established. During the Civil War the city ranked next to Richmond in the manufacture of supplies for the Confederate army. On the 16th of April 1865 it was captured by a Union force under General James Harrison Wilson (b. 1837); 1200 Confederates were taken prisoners; large quantities of arms and stores were seized, and the principal manufactories and much other property were destroyed.

COLUMBUS, a city and the county-seat of Bartholomew county, Indiana, U.S.A., situated on the E. fork of White river, a little S. of the centre of the state. Pop. (1890) 6719; (1900) 8130, of whom 313 were foreign-born and 224 were of negro descent (1910 census) 8813. In 1900 the centre of population of the United States was 5 m. S.E. of Columbus. The city is served by the Cleveland, Cincinnati, Chicago & St Louis, and the Pittsburg, Cincinnati, Chicago & St Louis railways, and is connected with Indianapolis and with Louisville, Ky., by an electric interurban line. Columbus is situated in a fine farming region, and has extensive tanneries, threshing-machine and traction and automobile engine works, structural iron works, tool and machine shops, canneries and furniture factories. In 1905 the value of the city's factory product was $2,983,160, being 28.4% more than in 1900. The water-supply system and electric-lighting plant are owned and operated by the city.

COLUMBUS, a city and the county-seat of Lowndes county, Mississippi, U.S.A., on the E. bank of the Tombigbee river, at the head of steam navigation, 150. m. S.E. of Memphis, Tennessee. Pop. (1890) 4559; (1900) 6484 (3366 negroes); (1910) 8988. It is served by the Mobile & Ohio and the Southern railways, and by passenger and freight steamboat lines. It has cotton and knitting mills, cotton-seed oil factories, machine shops, and wagon, stove, plough and fertilizer factories; and is a market and jobbing centre for a fertile agricultural region. It has a public library, and is the seat of the Mississippi Industrial Institute and College (1885) for women, the first state college for women--the successor of the Columbus Female Institute (1848)--of Franklin Academy (1821), and of the Union Academy (1873) for negroes. The site was first settled about 1818; the city was incorporated in 1821, and in 1830 it became the county-seat of the newly formed Lowndes county. During the Civil War the legislature met here in 1863 and 1865, and in the former year Governor Charles Clark (1810-1877) was inaugurated here.

COLUMBUS, a city, a port of entry, the capital of Ohio, U.S.A., and the county-seat of Franklin county, at the confluence of the Scioto and Olentangy rivers, near the geographical centre of the state, 120 m. N.E. of Cincinnati, and 138 m. S.S.W. of Cleveland. Pop. (1890) 88,150; (1900) 125,560, of whom 12,328 were foreign-born and 8201 were negroes; (1910) 181,511. Columbus is an important railway centre and is served by the Cleveland, Cincinnati, Chicago & St. Louis, the Pittsburg, Cincinnati, Chicago & St Louis (Pennsylvania system), the Baltimore & Ohio, the Ohio Central, the Norfolk & Western, the Hocking Valley, and the Cleveland, Akron & Columbus (Pennsylvania system) railways, and by nine interurban electric lines. It occupies a land area of about 17 sq. m., the principal portion being along the east side of the Scioto in the midst of an extensive plain. High Street, the principal business thoroughfare, is 100 ft. wide, and Broad Street, on which are many of the finest residences, is 120 ft. wide, has four rows of trees, a roadway for heavy vehicles in the middle, and a driveway for carriages on either side.

The principal building is the state capitol (completed in 1857) in a square of ten acres at the intersection of High and Broad streets. It is built in the simple Doric style, of grey limestone taken from a quarry owned by the state, near the city; is 304 ft. long and 184 ft. wide, and has a rotunda 158 ft. high, on the walls of which are the original painting, by William Henry Powell (1823-1879), of O. H. Perry's victory on Lake Erie, and portraits of most of the governors of Ohio. Other prominent structures are the U.S. government and the judiciary buildings, the latter connected with the capitol by a stone terrace, the city hall, the county court house, the union station, the board of trade, the soldiers' memorial hall (with a seating capacity of about 4500), and several office buildings. The city is a favourite meeting-place for conventions. Among the state institutions in Columbus are the university (see below), the penitentiary, a state hospital for the insane, the state school for the blind, and the state institutions for the education of the deaf and dumb and for feeble-minded youth. In the capitol grounds are monuments to the memory of Ulysses S. Grant, Rutherford B. Hayes, James A. Garfield, William T. Sherman, Philip H. Sheridan, Salmon P. Chase, and Edwin M. Stanton, and a beautiful memorial arch (with sculpture by H. A. M'Neil) to William McKinley.

The city has several parks, including the Franklin of 90 acres, the Goodale of 44 acres, and the Schiller of 24 acres, besides the Olentangy, a well-equipped amusement resort on the banks of the river from which it is named, the Indianola, another amusement resort, and the United States military post and recruiting station, which occupies 80 acres laid out like a park. The state fair grounds of 115 acres adjoin the city, and there is also a beautiful cemetery of 220 acres.

The Ohio State University (non-sectarian and co-educational), opened as the Ohio Agricultural and Mechanical College in 1873, and reorganized under its present name in 1878, is 3 m. north of the capitol. It includes colleges of arts, philosophy and science, of education (for teachers), of engineering, of law, of pharmacy, of agriculture and domestic science, and of veterinary medicine. It occupies a campus of 110 acres, has an adjoining farm of 325 acres, and 18 buildings devoted to instruction, 2 dormitories, and a library containing (1906) 67,709 volumes, besides excellent museums of geology, zoology, botany and archaeology and history, the last being owned jointly by the university and by the state archaeological and historical society. In 1908 the faculty numbered 175, and the students 2277. The institution owed its origin to federal land grants; it is maintained by the state, the United States, and by small fees paid by the students; tuition is free in all colleges except the college of law. The government of the university is vested in a board of trustees appointed by the governor of the state for a term of seven years. The first president of the institution (from 1873 to 1881) was the distinguished geologist, Edward Orton (1829-1899), who was professor of geology from 1873 to 1899.

Other institutions of learning are the Capital University and Evangelical Lutheran Theological Seminary (Theological Seminary opened in 1830; college opened as an academy in 1850), with buildings just east of the city limits; Starling Ohio Medical College, a law school, a dental school and an art institute. Besides the university library, there is the Ohio state library occupying a room in the capitol and containing in 1908 126,000 volumes, including a "travelling library" of about 36,000 volumes, from which various organizations in different parts of the state may borrow books; the law library of the supreme court of Ohio, containing complete sets of English, Scottish, Irish, Canadian, United States and state reports, statutes and digests; the public school library of about 68,000 volumes, and the public library (of about 55,000), which is housed in a marble and granite building completed in 1906.

Columbus is near the Ohio coal and iron-fields, and has an extensive trade in coal, but its largest industrial interests are in manufactures, among which the more important are foundry and machine-shop products (1905 value, $6,259,579); boots and shoes (1905 value, $5,425,087, being more than one-sixtieth of the total product value of the boot and shoe industry in the United States, and being an increase from $359,000 in 1890); patent medicines and compounds (1905 value, $3,214,096); carriages and wagons (1905 value, $2,197,960); malt liquors (1905 value, $2,133,955); iron and steel; regalia and society emblems; steam-railway cars, construction and repairing; and oleo-margarine. In 1905 the city's factory products were valued at $40,435,531, an increase of 16.4% in five years. Immediately outside the city limits in 1905 were various large and important manufactories, including railway shops, foundries, slaughter-houses, ice factories and brick-yards. In Columbus there is a large market for imported horses. Several large quarries also are adjacent to the city.

The waterworks are owned by the municipality. In 1904-1905 the city built on the Scioto river a concrete storage dam, having a capacity of 5,000,000,000 gallons, and in 1908 it completed the construction of enormous works for filtering and softening the water-supply, and of works for purifying the flow of sewage--the two costing nearly $5,000,000. The filtering works include 6 lime saturators, 2 mixing or softening tanks, 6 settling basins, 10 mechanical filters and 2 clear-water reservoirs. A large municipal electric-lighting plant was completed in 1908.

The first permanent settlement within the present limits of the city was established in 1797 on the west bank of the Scioto, was named Franklinton, and in 1803 was made the county-seat. In 1810 four citizens of Franklinton formed an association to secure the location of the capital on the higher ground of the east bank; in 1812 they were successful and the place was laid out while still a forest. Four years later, when the legislature held its first session here, the settlement was incorporated as the Borough of Columbus. In 1824 the county-seat was removed here from Franklinton; in 1831 the Columbus branch of the Ohio Canal was completed; in 1834 the borough was made a city; by the close of the same decade the National Road extending from Wheeling to Indianapolis and passing through Columbus was completed; in 1871 most of Franklinton, which was never incorporated, was annexed, and several other annexations followed.

See J. H. Studer, _Columbus, Ohio; its History and Resources_ (Columbus, 1873); A. E. Lee, _History of the City of Columbus, Ohio_ (New York, 1892).

COLUMELLA, LUCIUS JUNIUS MODERATUS, of Gades, writer on agriculture, contemporary of Seneca the philosopher, flourished about the middle of the 1st century A.D. His extant works treat, with great fulness and in a diffuse but not inelegant style which well represents the silver age, of the cultivation of all kinds of corn and garden vegetables, trees, flowers, the vine, the olive and other fruits, and of the rearing of cattle, birds, fishes and bees. They consist of the twelve books of the _De re rustica_ (the tenth, which treats of gardening, being in dactylic hexameters in imitation of Virgil), and of a book _De arboribus_, the second book of an earlier and less elaborate work on the same subject.

The best complete edition is by J. G. Schneider (1794). Of a new edition by K. J. Lundström, the tenth book appeared in 1902 and _De arboribus_ in 1897. There are English translations by R. Bradley (1725), and anonymous (1745); and treatises, _De Columellae vita et scriptis_, by V. Barberet (1887), and G. R. Becher (1897), a compact dissertation with notes and references to authorities.

COLUMN (Lat. _columna_), in architecture, a vertical support consisting of capital, shaft and base, used to carry a horizontal beam or an arch. The earliest example in wood (2684 B.C.) was that found at Kahun in Egypt by Professor Flinders Petrie, which was fluted and stood on a raised base, and in stone the octagonal shafts of the early temple at Deir-el-Bahri (c. 2850). In the tombs at Beni Hasan (2723 B.C.) are columns of two kinds, the octagonal or polygonal shaft, and the reed or lotus column, the horizontal section of which is a quatrefoil. This became later the favourite type, but it was made circular on plan. In all these examples the column rests on a stone base. (See also CAPITAL and ORDER.)

The column was employed in Assyria in small structures only, such as pavilions or porticoes. In Persia the column, employed to carry timber superstructures only, was very lofty, being sometimes 12 diameters high; the shaft was fluted, the number of flutes varying from 30 to 52.

The earliest example of the Greek column is that represented in the temple fresco at Cnossus (c. 1600 B.C.), of which portions have been found. The columns were in cypress wood raised on a stone base and tapered downwards.[1] The same, though to a less degree, is found in the stone semi-detached columns which flank the doorway of the Tomb of Agamemnon at Mycenae; the shafts of these columns were carved with the chevron design.

The earliest Greek columns in stone as isolated features are those of the Temple of Apollo at Syracuse (early 7th century B.C.) the shafts of which were monoliths, but as a rule the Greek columns were all built of drums, sometimes as many as ten or twelve. There was no base to the Doric column, but the shafts were fluted, 20 flutes being the usual number. In the Archaic Temple of Diana at Ephesus there were 52 flutes. In the later examples of the Ionic order the shaft had 24 flutes. In the Roman temples the shafts were very often monoliths.

Columns were occasionally used as supports for figures or other features. The Naxian column at Delphi of the Ionic order carried a sphinx. The Romans employed columns in various ways: the Trajan and the Antonine columns carried figures of the two emperors; the columna rostrata (260 B.C.) in the Forum was decorated with the beaks of ships and was a votive column, the miliaria column marked the centre of Rome from which all distances were measured. In the same way the column in the Place Vendôme in Paris carries a statue of Napoleon I.; the monument of the Fire of London, a finial with flames sculptured on it; the duke of York's column (London), a statue of the duke of York.

With the exception of the Cretan and Mycenaean, all the shafts of the classic orders tapered from the bottom upwards, and about one-third up the column had an increment, known as the _entasis_, to correct an optical illusion which makes tapering shafts look concave; the proportions of diameter to height varied with the order employed. Thus, broadly speaking, a Roman Doric column will be eight, a Roman Ionic nine, a Corinthian ten diameters in height. Except in rare cases, the columns of the Romanesque and Gothic styles were of equal diameter at top and bottom, and had no definite dimensions as regards diameter and height. They were also grouped together round piers which are known as clustered piers. When of exceptional size, as in Gloucester and Durham cathedrals, Waltham Abbey and Tewkesbury, they are generally called "pillars," which was apparently the medieval term for column. The word _columna_, employed by Vitruvius, was introduced into England by the Italian writers of the Revival.

In the Renaissance period columns were frequently banded, the bands being concentric with the column as in France, and occasionally richly carved as in Philibert De L'Orme's work at the Tuileries. In England Inigo Jones introduced similar features, but with square blocks sometimes rusticated, a custom lately revived in England, but of which there are few examples either in Italy or Spain.

The word "column" is used, by analogy with architecture, for any upright body or mass, in chemistry, anatomy, typography, &c. (R. P. S.)

FOOTNOTE:

[1] The tree-trunk used as a column was inverted to retain the sap; hence the shape.

COLURE (from Gr. [Greek: kolos], shortened, and [Greek: oura], tail), in astronomy, either of the two principal meridians of the celestial sphere, one of which passes through the poles and the two solstices, the other through the poles and the two equinoxes; hence designated as _solstitial colure_ and _equinoxial colure_, respectively.

COLUTHUS, or COLLUTHUS, of Lycopolis in the Egyptian Thebaid, Greek epic poet, flourished during the reign of Anastasius I. (491-518). According to Suidas, he was the author of _Calydoniaca_ (probably an account of the Calydonian boar hunt), _Persica_ (an account of the Persian wars), and _Encomia_ (laudatory poems). These are all lost, but his poem in some 400 hexameters on _The Rape of Helen_ ([Greek: Harpagê Helenês]) is still extant, having been discovered by Cardinal Bessarion in Calabria. The poem is dull and tasteless, devoid of imagination, a poor imitation of Homer, and has little to recommend it except its harmonious versification, based upon the technical rules of Nonnus. It related the history of Paris and Helen from the wedding of Peleus and Thetis down to the elopement and arrival at Troy.

The best editions are by Van Lennep (1747), G. F. Schäfer (1825), E. Abel (1880).

COLVILLE, JOHN (c. 1540-1605), Scottish divine and author, was the son of Robert Colville of Cleish, in the county of Kinross. Educated at St Andrews University, he became a Presbyterian minister, but occupied himself chiefly with political intrigue, sending secret information to the English government concerning Scottish affairs. He joined the party of the earl of Gowrie, and took part in the Raid of Ruthven in 1582. In 1587 he for a short time occupied a seat on the judicial bench, and was commissioner for Stirling in the Scottish parliament. In December 1591 he was implicated in the earl of Bothwell's attack on Holyrood Palace, and was outlawed with the earl. He retired abroad, and is said to have joined the Roman Church. He died in Paris in 1605. Colville was the author of several works, including an _Oratio Funebris_ on Queen Elizabeth, and some political and religious controversial essays. He is said to be the author also of _The Historie and Life of King James the Sext_ (edited by T. Thompson for the Bannatyne Club, Edinburgh, 1825).

Colville's _Original Letters_, 1582-1603, published by the Bannatyne Club in 1858, contains a biographical memoir by the editor, David Laing.

COLVIN, JOHN RUSSELL (1807-1857), lieutenant-governor of the North-West Provinces of India during the mutiny of 1857, belonged to an Anglo-Indian family of Scottish descent, and was born in Calcutta on the 29th of May 1807. Passing through Haileybury he entered the service of the East India Company in 1826. In 1836 he became private secretary to Lord Auckland, and his influence over the viceroy has been held partly responsible for the first Afghan war of 1837; but it has since been shown that Lord Auckland's policy was dictated by the secret committee of the company at home. In 1853 Mr Colvin was appointed lieutenant-governor of the North-West Provinces by Lord Dalhousie. On the outbreak of the mutiny in 1857 he had with him at Agra only a weak British regiment and a native battery, too small a force to make head against the mutineers; and a proclamation which he issued to the natives was censured at the time for its clemency, but it followed the same lines as those adopted by Sir Henry Lawrence and subsequently followed by Lord Canning. Exhausted by anxiety and misrepresentation he died on the 9th of September, his death shortly preceding the fall of Delhi.

His son, SIR AUCKLAND COLVIN (1838-1908), followed him in a distinguished career in the same service, from 1858 to 1879. He was comptroller-general in Egypt (1880 to 1882), and financial adviser to the khedive (1883 to 1887), and from 1883 till 1892 was back again in India, first as financial member of council, and then, from 1887, as lieutenant-governor of the North-West Provinces and Oudh. He was created K.C.M.G. in 1881, and K.C.S.I. in 1892, when he retired. He published _The Making of Modern Egypt_ in 1906, and a biography of his father, in the "Rulers of India" series, in 1895. He died at Surbiton on the 24th of March 1908.

COLVIN, SIDNEY (1845- ), English literary and art critic, was born at Norwood, London, on the 18th of June 1845. A scholar of Trinity College, Cambridge, he became a fellow of his college in 1868. In 1873 he was Slade professor of fine art, and was appointed in the next year to the directorship of the Fitzwilliam Museum. In 1884 he removed to London on his appointment as keeper of prints and drawings in the British Museum. His chief publications are lives of Landor (1881) and Keats (1887), in the English Men of Letters series; the Edinburgh edition of R. L. Stevenson's works (1894-1897); editions of the letters of Keats (1887), and of the _Vailima Letters_ (1899), which R. L. Stevenson chiefly addressed to him; _A Florentine Picture-Chronicle_ (1898), and _Early History of Engraving in England_ (1905). But in the field both of art and of literature, Mr Colvin's fine taste, wide knowledge and high ideals made his authority and influence extend far beyond his published work.

COLWYN BAY, a watering-place of Denbighshire, N. Wales, on the Irish Sea, 40½ m. from Chester by the London & North-Western railway. Pop. of urban district of Colwyn Bay and Colwyn (1901) 8689. Colwyn Bay has become a favourite bathing-place, being near to, and cheaper than, the fashionable Llandudno, and being a centre for picturesque excursions. Near it is Llaneilian village, famous for its "cursing well" (St Eilian's, perhaps Aelianus'). The stream Colwyn joins the Gwynnant. The name Colwyn is that of lords of Ardudwy; a Lord Colwyn of Ardudwy, in the 10th century, is believed to have repaired Harlech castle, and is considered the founder of one of the fifteen tribes of North Wales. Nant Colwyn is on the road from Carnarvon to Beddgelert, beyond Llyn y gader (gadair), "chair pool," and what tourists have fancifully called Pitt's head, a roadside rock resembling, or thought to resemble, the great statesman's profile. Near this is Llyn y dywarchen (sod pool), with a floating island.

COLZA OIL, a non-drying oil obtained from the seeds of _Brassica campestris_, var. _oleifera_, a variety of the plant which produces Swedish turnips. Colza is extensively cultivated in France, Belgium, Holland and Germany; and, especially in the first-named country, the expression of the oil is an important industry. In commerce colza is classed with rape oil, to which both in source and properties it is very closely allied. It is a comparatively inodorous oil of a yellow colour, having a specific gravity varying from 0.912 to 0.920. The cake left after expression of the oil is a valuable feeding substance for cattle. Colza oil is extensively used as a lubricant for machinery, and for burning in lamps.

COMA (Gr. [Greek: kôma], from [Greek: koiman], to put to sleep), a deep sleep; the term is, however, used in medicine to imply something more than its Greek origin denotes, namely, a complete and prolonged loss of consciousness from which a patient cannot be roused. There are various degrees of coma: in the slighter forms the patient can be partially roused only to relapse again into a state of insensibility; in the deeper states, the patient cannot be roused at all, and such are met with in apoplexy, already described. Coma may arise abruptly in a patient who has presented no pre-existent indication of such a state occurring. Such a condition is called _primary coma_, and may result from the following causes:--(1) concussion, compression or laceration of the brain from head injuries, especially fracture of the skull; (2) from alcoholic and narcotic poisoning; (3) from cerebral haemorrhage, embolism and thrombosis, such being the causes of apoplexy. _Secondary coma_ may arise as a complication in the following diseases:--diabetes, uraemia, general paralysis, meningitis, cerebral tumour and acute yellow atrophy of the liver; in such diseases it is anticipated, for it is a frequent cause of the fatal termination. The depth of insensibility to stimulus is a measure of the gravity of the symptom; thus the conjunctival reflex and even the spinal reflexes may be abolished, the only sign of life being the respiration and heart-beat, the muscles of the limbs being sometimes perfectly flaccid. A characteristic change in the respiration, known as Cheyne-Stokes breathing occurs prior to death in some cases; it indicates that the respiratory centre in the medulla is becoming exhausted, and is stimulated to action only when the venosity of the blood has increased sufficiently to excite it. The breathing consequently loses its natural rhythm, and each successive breath becomes deeper until a maximum is reached; it then diminishes in depth by successive steps until it dies away completely. The condition of apnoea, or cessation of breathing, follows, and as soon as the venosity of the blood again affords sufficient stimulus, the signs of air-hunger commence; this altered rhythm continues until the respiratory centre becomes exhausted and death ensues.

_Coma Vigil_ is a state of unconsciousness met with in the algide stage of cholera and some other exhausting diseases. The patient's eyes remain open, and he may be in a state of low muttering delirium; he is entirely insensible to his surroundings, and neither knows nor can indicate his wants.

There is a distinct word "coma" (Gr. [Greek: komê], hair), which is used in astronomy for the envelope of a comet, and in botany for a tuft.

COMA BERENICES ("BERENICE'S HAIR"), in astronomy, a constellation of the northern hemisphere; it was first mentioned by Callimachus, and Eratosthenes (3rd century B.C.), but is not included in the 48 asterisms of Ptolemy. It is said to have been named by Conon, in order to console Berenice, queen of Ptolemy Euergetes, for the loss of a lock of her hair, which had been stolen from a temple to Venus. This constellation is sometimes, but wrongly, attributed to Tycho Brahe. The most interesting member of this group is _24 Comae_, a fine, wide double star, consisting of an orange star of magnitude 5½, and a blue star, magnitude 7.

COMACCHIO, a town of Emilia, Italy, in the province of Ferrara, 30 m. E.S.E. by road from the town of Ferrara, on the level of the sea, in the centre of the lagoon of Valli di Comacchio, just N. of the present mouth of the Reno. Pop. (1901) 7944 (town), 10,745 (commune). It is built on no less than thirteen different islets, joined by bridges, and its industries are the fishery, which belongs to the commune, and the salt-works. The seaport of Magnavacca lies 4 m. to the east. Comacchio appears as a city in the 6th century, and, owing to its position in the centre of the lagoons, was an important fortress. It was included in the "donation of Pippin"; it was taken by the Venetians in 854, but afterwards came under the government of the archbishops of Ravenna; in 1299 it came under the dominion of the house of Este. In 1508 it became Venetian, but in 1597 was claimed by Clement VIII. as a vacant fief.

COMANA, a city of Cappadocia [frequently called CHRYSE or AUREA, i.e. the golden, to distinguish it from Comana in Pontus; mod. _Shahr_], celebrated in ancient times as the place where the rites of M[=a]-Enyo, a variety of the great west Asian Nature-goddess, were celebrated with much solemnity. The service was carried on in a sumptuous temple with great magnificence by many thousands of _hieroduli_ (temple-servants). To defray expenses, large estates had been set apart, which yielded a more than royal revenue. The city, a mere apanage of the temple, was governed immediately by the chief priest, who was always a member of the reigning Cappadocian family, and took rank next to the king. The number of persons engaged in the service of the temple, even in Strabo's time, was upwards of 6000, and among these, to judge by the names common on local tombstones, were many of Persian race. Under Caracalla, Comana became a Roman colony, and it received honours from later emperors down to the official recognition of Christianity. The site lies at Shahr, a village in the Anti-Taurus on the upper course of the Sarus (Sihun), mainly Armenian, but surrounded by new settlements of Avshar Turkomans and Circassians. The place has derived importance both in antiquity and now from its position at the eastern end of the main pass of the western Anti-Taurus range, the Kuru Chai, through which passed the road from Caesarea-Mazaca (mod. _Kaisarieh_) to Melitene (Malatia), converted by Septimius Severus into the chief military road to the eastern frontier of the empire. The extant remains at Shahr include a theatre on the left bank of the river, a fine Roman doorway and many inscriptions; but the exact site of the great temple has not been satisfactorily identified. There are many traces of Severus' road, including a bridge at Kemer, and an immense number of milestones, some in their original positions, others in cemeteries.

See P. H. H. Massy in _Geog. Journ._ (Sept. 1905); E. Chantre, _Mission en Cappadocie_ (1898). (D. G. H.)

COMANA (mod. _Gumenek_), an ancient city of Pontus, said to have been colonized from Comana in Cappadocia. It stood on the river Iris (Tozanli Su or Yeshil Irmak), and from its central position was a favourite emporium of Armenian and other merchants. The moon-goddess was worshipped in the city with a pomp and ceremony in all respects analogous to those employed in the Cappadocian city. The slaves attached to the temple alone numbered not less than 6000. St John Chrysostom died there on the way to Constantinople from his exile at Cocysus in the Anti-Taurus. Remains of Comana are still to be seen near a village called Gumenek on the Tozanli Su, 7 m. from Tokat, but they are of the slightest description. There is a mound; and a few inscriptions are built into a bridge, which here spans the river, carrying the road from Niksar to Tokat. (D. G. H.)

COMANCHES, a tribe of North American Indians of Shoshonean stock, so called by the Spaniards, but known to the French as Padoucas, an adaptation of their Sioux name, and among themselves _nimenim_ (people). They number some 1400, attached to the Kiowa agency, Oklahoma. When first met by Europeans, they occupied the regions between the upper waters of the Brazos and Colorado on the one hand, and the Arkansas and Missouri on the other. Until their final surrender in 1875 the Comanches were the terror of the Mexican and Texan frontiers, and were always famed for their bravery. They were brought to nominal submission in 1783 by the Spanish general Anza, who killed thirty of their chiefs. During the 19th century they were always raiding and fighting, but in 1867, to the number of 2500, they agreed to go on a reservation. In 1872 a portion of the tribe, the Quanhada or Staked Plain Comanches, had again to be reduced by military measures.

COMAYAGUA, the capital of the department of Comayagua in central Honduras, on the right bank of the river Ulua, and on the interoceanic railway from Puerto Cortes to Fonseca Bay. Pop. (1900) about 8000. Comayagua occupies part of a fertile valley, enclosed by mountain ranges. Under Spanish rule it was a city of considerable size and beauty, and in 1827 its inhabitants numbered more than 18,000. A fine cathedral, dating from 1715, is the chief monument of its former prosperity, for most of the handsome public buildings erected in the colonial period have fallen into disrepair. The present city chiefly consists of low adobe houses and cane huts, tenanted by Indians. The university founded in 1678 has ceased to exist, but there is a school of jurisprudence. In the neighbourhood are many ancient Indian ruins (see CENTRAL AMERICA: _ARCHAEOLOGY_).

Founded in 1540 by Alonzo Caceres, who had been instructed by the Spanish government to find a site for a city midway between the two oceans, Valladolid la Nueva, as the town was first named, soon became the capital of Honduras. It received the privileges of a city in 1557, and was made an episcopal see in 1561. Its decline dates from 1827, when it was burned by revolutionaries; and in 1854 its population had dwindled to 2000. It afterwards suffered through war and rebellion, notably in 1872 and 1873, when it was besieged by the Guatemalans. In 1880 Tegucigalpa (q.v.), a city 37 m. east-south-east, superseded it as the capital of Honduras.

COMB (a word common in various forms to Teut. languages, cf. Ger. _Kamm_, the Indo-Europ. origin of which is seen in [Greek: gomphos], a peg or pin, and Sanskrit, _gambhas_, a tooth), a toothed article of the toilet used for cleaning and arranging the hair, and also for holding it in place after it has been arranged; the word is also applied, from resemblance in form or in use, to various appliances employed for dressing wool and other fibrous substances, to the indented fleshy crest of a cock, and to the ridged series of cells of wax filled with honey in a beehive. Hair combs are of great antiquity, and specimens made of wood, bone and horn have been found in Swiss lake-dwellings. Among the Greeks and Romans they were made of boxwood, and in Egypt also of ivory. For modern combs the same materials are used, together with others such as tortoise-shell, metal, india-rubber and celluloid. There are two chief methods of manufacture. A plate of the selected material is taken of the size and thickness required for the comb, and on one side of it, occasionally on both sides, a series of fine slits are cut with a circular saw. This method involves the loss of the material cut out between the teeth. The second method, known as "twinning" or "parting," avoids this loss and is also more rapid. The plate of material is rather wider than before, and is formed into two combs simultaneously, by the aid of a twinning machine. Two pairs of chisels, the cutting edges of which are as long as the teeth are required to be and are set at an angle converging towards the sides of the plate, are brought down alternately in such a way that the wedges removed from one comb form the teeth of the other, and that when the cutting is complete the plate presents the appearance of two combs with their teeth exactly inosculating or dovetailing into each other. In india-rubber combs the teeth are moulded to shape and the whole hardened by vulcanization.

COMBACONUM, or KUMBAKONAM, a city of British India, in the Tanjore district of Madras, in the delta of the Cauvery, on the South Indian railway, 194 m. from Madras. Pop. (1901) 59,623, showing an increase of 10% in the decade. It is a large town with wide and airy streets, and is adorned with pagodas, gateways and other buildings of considerable pretension. The great _gopuram_, or gate-pyramid, is one of the most imposing buildings of the kind, rising in twelve stories to a height of upwards of 100 ft., and ornamented with a profusion of figures of men and animals formed in stucco. One of the water-tanks in the town is popularly reputed to be filled with water admitted from the Ganges every twelve years by a subterranean passage 1200 m. long; and it consequently forms a centre of attraction for large numbers of devotees. The city is historically interesting as the capital of the Chola race, one of the oldest Hindu dynasties of which any traces remain, and from which the whole coast of Coromandel, or more properly Cholamandal, derives its name. It contains a government college. Brass and other metal wares, silk and cotton cloth and sugar are among the manufactures.

COMBE, ANDREW (1797-1847), Scottish physiologist, was born in Edinburgh on the 27th of October 1797, and was a younger brother of George Combe. He served an apprenticeship in a surgery, and in 1817 passed at Surgeons' Hall. He proceeded to Paris to complete his medical studies, and whilst there he investigated phrenology on anatomical principles. He became convinced of the truth of the new science, and, as he acquired much skill in the dissection of the brain, he subsequently gave additional interest to the lectures of his brother George, by his practical demonstrations of the convolutions. He returned to Edinburgh in 1819 with the intention of beginning practice; but being attacked by the first symptoms of pulmonary disease, he was obliged to seek health in the south of France and in Italy during the two following winters. He began to practise in 1823, and by careful adherence to the laws of health he was enabled to fulfil the duties of his profession for nine years. During that period he assisted in editing the _Phrenological Journal_ and contributed a number of articles to it, defended phrenology before the Royal Medical Society of Edinburgh, published his _Observations on Mental Derangement_ (1831), and prepared the greater portion of his _Principles of Physiology Applied to Health and Education_, which was issued in 1834, and immediately obtained extensive public favour. In 1836 he was appointed physician to Leopold I., king of the Belgians, and removed to Brussels, but he speedily found the climate unsuitable and returned to Edinburgh, where he resumed his practice. In 1836 he published his _Physiology of Digestion_, and in 1838 he was appointed one of the physicians extraordinary to the queen in Scotland. Two years later he completed his _Physiological and Moral Management of Infancy_, which he believed to be his best work and it was his last. His latter years were mostly occupied in seeking at various health resorts some alleviation of his disease; he spent two winters in Madeira, and tried a voyage to the United States, but was compelled to return within a few weeks of the date of his landing at New York. He died at Gorgie, near Edinburgh, on the 9th of August 1847.

His biography, written by George Combe, was published in 1850.

COMBE, GEORGE (1788-1858), Scottish phrenologist, elder brother of the above, was born in Edinburgh on the 21st of October 1788. After attending Edinburgh high school and university he entered a lawyer's office in 1804, and in 1812 began to practise on his own account. In 1815 the _Edinburgh Review_ contained an article on the system of "craniology" of F. J. Gall and K. Spurzheim, which was denounced as "a piece of thorough quackery from beginning to end." Combe laughed like others at the absurdities of this so-called new theory of the brain, and thought that it must be finally exploded after such an exposure; and when Spurzheim delivered lectures in Edinburgh, in refutation of the statements of his critic, Combe considered the subject unworthy of serious attention. He was, however, invited to a friend's house where he saw Spurzheim dissect the brain, and he was so far impressed by the demonstration that he attended the second course of lectures. Investigating the subject for himself, he became satisfied that the fundamental principles of phrenology were true--namely "that the brain is the organ of mind; that the brain is an aggregate of several parts, each subserving a distinct mental faculty; and that the size of the cerebral organ is, _caeteris paribus_, an index of power or energy of function." In 1817 his first essay on phrenology was published in the _Scots Magazine_; and a series of papers on the same subject appeared soon afterwards in the _Literary and Statistical Magazine_; these were collected and published in 1819 in book form as _Essays on Phrenology_, which in later editions became _A System of Phrenology_. In 1820 he helped to found the Phrenological Society, which in 1823 began to publish a _Phrenological Journal_. By his lectures and writings he attracted public attention to the subject on the continent of Europe and in America, as well as at home; and a long discussion with Sir William Hamilton in 1827-1828 excited general interest.

His most popular work, _The Constitution of Man_, was published in 1828, and in some quarters brought upon him denunciations as a materialist and atheist. From that time he saw everything by the light of phrenology. He gave time, labour and money to help forward the education of the poorer classes; he established the first infant school in Edinburgh; and he originated a series of evening lectures on chemistry, physiology, history and moral philosophy. He studied the criminal classes, and tried to solve the problem how to reform as well as to punish them; and he strove to introduce into lunatic asylums a humane system of treatment. In 1836 he offered himself as a candidate for the chair of logic at Edinburgh, but was rejected in favour of Sir William Hamilton. In 1838 he visited America and spent about two years lecturing on phrenology, education and the treatment of the criminal classes. On his return in 1840 he published his _Moral Philosophy_, and in the following year his _Notes on the United States of North America_. In 1842 he delivered, in German, a course of twenty-two lectures on phrenology in the university of Heidelberg, and he travelled much in Europe, inquiring into the management of schools, prisons and asylums. The commercial crisis of 1855 elicited his remarkable pamphlet on _The Currency Question_ (1858). The culmination of the religious thought and experience of his life is contained in his work _On the Relation between Science and Religion_, first publicly issued in 1857. He was engaged in revising the ninth edition of the _Constitution of Man_ when he died at Moor Park, Farnham, on the 14th of August 1858. He married in 1833 Cecilia Siddons, a daughter of the great actress.

COMBE, WILLIAM (1741-1823), English writer, the creator of "Dr Syntax," was born at Bristol in 1741. The circumstances of his birth and parentage are somewhat doubtful, and it is questioned whether his father was a rich Bristol merchant, or a certain William Alexander, a London alderman, who died in 1762. He was educated at Eton, where he was contemporary with Charles James Fox, the 2nd Baron Lyttelton and William Beckford. Alexander bequeathed him some £2000--a little fortune that soon disappeared in a course of splendid extravagance, which gained him the nickname of Count Combe; and after a chequered career as private soldier, cook and waiter, he finally settled in London (about 1771), as a law student and bookseller's hack. In 1776 he made his first success in London with _The Diaboliad_, a satire full of bitter personalities. Four years afterwards (1780) his debts brought him into the King's Bench; and much of his subsequent life was spent in prison. His spurious _Letters of the Late Lord Lyttelton_[1] (1780) imposed on many of his contemporaries, and a writer in the _Quarterly Review_, so late as 1851, regarded these letters as authentic, basing upon them a claim that Lyttelton was "Junius." An early acquaintance with Lawrence Sterne resulted in his _Letters supposed to have been written by Yorick and Eliza_ (1779). Periodical literature of all sorts--pamphlets, satires, burlesques, "two thousand columns for the papers," "two hundred biographies"--filled up the next years, and about 1789 Combe was receiving £200 yearly from Pitt, as a pamphleteer. Six volumes of a _Devil on Two Sticks in England_ won for him the title of "the English le Sage"; in 1794-1796 he wrote the text for Boydell's _History of the River Thames_; in 1803 he began to write for _The Times_. In 1809-1811 he wrote for Ackermann's _Political Magazine_ the famous _Tour of Dr Syntax in search of the Picturesque_ (descriptive and moralizing verse of a somewhat doggerel type), which, owing greatly to Thomas Rowlandson's designs, had an immense success. It was published separately in 1812 and was followed by two similar _Tours_, "in search of Consolation," and "in search of a Wife," the first Mrs Syntax having died at the end of the first _Tour_. Then came _Six Poems_ in illustration of drawings by Princess Elizabeth (1813), _The English Dance of Death_ (1815-1816), _The Dance of Life_ (1816-1817), _The Adventures of Johnny Quae Genus_ (1822)--all written for Rowlandson's caricatures; together with _Histories_ of Oxford and Cambridge, and of Westminster Abbey for Ackermann; _Picturesque Tours_ along the Rhine and other rivers, _Histories of Madeira_, _Antiquities of York_, texts for _Turner's Southern Coast Views_, and contributions innumerable to the _Literary Repository_. In his later years, notwithstanding a by no means unsullied character, Combe was courted for the sake of his charming conversation and inexhaustible stock of anecdote. He died in London on the 19th of June 1823.

Brief obituary memoirs of Combe appeared in Ackermann's _Literary Repository_ and in the _Gentleman's Magazine_ for August 1823; and in May 1859 a list of his works, drawn up by his own hand, was printed in the latter periodical. See also _Diary of H. Crabb Robinson_, _Notes and Queries for 1869_.

FOOTNOTE:

[1] Thomas, 2nd Baron Lyttelton (1744-1779), commonly known as the "wicked Lord Lyttelton," was famous for his abilities and his libertinism, also for the mystery attached to his death, of which it was alleged he was warned in a dream three days before the event.

COMBE, or COOMB, a term particularly in use in south-western England for a short closed-in valley, either on the side of a down or running up from the sea. It appears in place-names as a termination, e.g. Wiveliscombe, Ilfracombe, and as a prefix, e.g. Combemartin. The etymology of the word is obscure, but "hollow" seems a common meaning to similar forms in many languages. In English "combe" or "cumb" is an obsolete word for a "hollow vessel," and the like meaning attached to Teutonic forms _kumm_ and _kumme_. The Welsh _cwm_, in place-names, means hollow or valley, with which may be compared _cum_ in many Scots place-names. The Greek [Greek: kumbê] also means a hollow vessel, and there is a French dialect word _combe_ meaning a little valley.

COMBERMERE, STAPLETON COTTON, 1ST VISCOUNT (1773-1865), British field-marshal and colonel of the 1st Life Guards, was the second son of Sir Robert Salusbury Cotton of Combermere Abbey, Cheshire, and was born on the 14th of November 1773, at Llewenny Hall in Denbighshire. He was educated at Westminster School, and when only sixteen obtained a second lieutenancy in the 23rd regiment (Royal Welsh Fusiliers). A few years afterwards (1793) he became by purchase captain in the 6th Dragoon Guards, and he served in this regiment during the campaigns of the duke of York in Flanders. While yet in his twentieth year, he joined the 25th Light Dragoons (subsequently 22nd) as lieutenant-colonel, and, while in attendance with his regiment on George III. at Weymouth, he became a great favourite of the king. In 1796 he went with his regiment to India, taking part _en route_ in the operations in Cape Colony (July-August 1796), and in 1799 served in the war with Tippoo Sahib, and at the storming of Seringapatam. Soon after this, having become heir to the family baronetcy, he was, at his father's desire, exchanged into a regiment at home, the 16th Light Dragoons. He was stationed in Ireland during Emmett's insurrection, became colonel in 1800, and major-general five years later. From 1806 to 1814 he was M.P. for Newark. In 1808 he was sent to the seat of war in Portugal, where he shortly rose to the position of commander of Wellington's cavalry, and it was here that he most displayed that courage and judgment which won for him his fame as a cavalry officer. He succeeded to the baronetcy in 1809, but continued his military career. His share in the battle of Salamanca (22nd of July 1812) was especially marked, and he received the personal thanks of Wellington. The day after, he was accidentally wounded. He was now a lieutenant-general in the British army and a K.B., and on the conclusion of peace (1814) was raised to the peerage under the style of Baron Combermere. He was not present at Waterloo, the command, which he expected, and bitterly regretted not receiving, having been given to Lord Uxbridge. When the latter was wounded Cotton was sent for to take over his command, and he remained in France until the reduction of the allied army of occupation. In 1817 he was appointed governor of Barbadoes and commander of the West Indian forces. From 1822 to 1825 he commanded in Ireland. His career of active service was concluded in India (1826), where he besieged and took Bhurtpore--a fort which twenty-two years previously had defied the genius of Lake and was deemed impregnable. For this service he was created Viscount Combermere. A long period of peace and honour still remained to him at home. In 1834 he was sworn a privy councillor, and in 1852 he succeeded Wellington as constable of the Tower and lord lieutenant of the Tower Hamlets. In 1855 he was made a field-marshal and G.C.B. He died at Clifton on the 21st of February 1865. An equestrian statue in bronze, the work of Baron Marochetti, was raised in his honour at Chester by the inhabitants of Cheshire. Combermere was succeeded by his only son, Wellington Henry (1818-1891), and the viscountcy is still held by his descendants.

See Viscountess Combermere and Captain W. W. Knollys, _The Combermere Correspondence_ (London, 1866).

COMBES, [JUSTIN LOUIS] ÉMILE (1835- ), French statesman, was born at Roquecourbe in the department of the Tarn. He studied for the priesthood, but abandoned the idea before ordination, and took the diploma of doctor of letters (1860), then he studied medicine, taking his degree in 1867, and setting up in practice at Pons in Charente-Inférieure. In 1881 he presented himself as a political candidate for Saintes, but was defeated. In 1885 he was elected to the senate by the department of Charente-Inférieure. He sat in the Democratic left, and was elected vice-president in 1893 and 1894. The reports which he drew up upon educational questions drew attention to him, and on the 3rd of November 1895 he entered the Bourgeois cabinet as minister of public instruction, resigning with his colleagues on the 21st of April following. He actively supported the Waldeck-Rousseau ministry, and upon its retirement in 1903 he was himself charged with the formation of a cabinet. In this he took the portfolio of the Interior, and the main energy of the government was devoted to the struggle with clericalism. The parties of the Left in the chamber, united upon this question in the _Bloc republicain_, supported Combes in his application of the law of 1901 on the religious associations, and voted the new bill on the congregations (1904), and under his guidance France took the first definite steps toward the separation of church and state. He was opposed with extreme violence by all the Conservative parties, who regarded the secularization of the schools as a persecution of religion. But his stubborn enforcement of the law won him the applause of the people, who called him familiarly _le petit père_. Finally the defection of the Radical and Socialist groups induced him to resign on the 17th of January 1905, although he had not met an adverse vote in the Chamber. His policy was still carried on; and when the law of the separation of church and state was passed, all the leaders of the Radical parties entertained him at a noteworthy banquet in which they openly recognized him as the real originator of the movement.

COMBINATION (Lat. _combinare_, to combine), a term meaning an association or union of persons for the furtherance of a common object, historically associated with agreements amongst workmen for the purpose of raising their wages. Such a combination was for a long time expressly prohibited by statute. See TRADE UNIONS; also CONSPIRACY and STRIKES AND LOCK OUTS.

COMBINATORIAL ANALYSIS.

Historical Introduction.

The Combinatorial Analysis, as it was understood up to the end of the 18th century, was of limited scope and restricted application. P. Nicholson, in his _Essays on the Combinatorial Analysis_, published in 1818, states that "the Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things that has not been enumerated." Writers on the subject seemed to recognize fully that it was in need of cultivation, that it was of much service in facilitating algebraical operations of all kinds, and that it was the fundamental method of investigation in the theory of Probabilities. Some idea of its scope may be gathered from a statement of the parts of algebra to which it was commonly applied, viz., the expansion of a multinomial, the product of two or more multinomials, the quotient of one multinomial by another, the reversion and conversion of series, the theory of indeterminate equations, &c. Some of the elementary theorems and various particular problems appear in the works of the earliest algebraists, but the true pioneer of modern researches seems to have been Abraham Demoivre, who first published in _Phil. Trans._ (1697) the law of the general coefficient in the expansion of the series a + bx + cx² + dx³ + ... raised to any power. (See also _Miscellanea Analytica_, bk. iv. chap. ii. prob. iv.) His work on Probabilities would naturally lead him to consider questions of this nature. An important work at the time it was published was the _De Partitione Numerorum_ of Leonhard Euler, in which the consideration of the reciprocal of the product (1 - xz) (1 - x²z) (1 - x³z) ... establishes a fundamental connexion between arithmetic and algebra, arithmetical addition being made to depend upon algebraical multiplication, and a close bond is secured between the theories of discontinuous and continuous quantities. (Cf. Numbers, Partition of.) The multiplication of the two powers x^a, x^b, viz. x^a + x^b = x^(a+b), showed Euler that he could convert arithmetical addition into algebraical multiplication, and in the paper referred to he gives the complete formal solution of the main problems of the partition of numbers. He did not obtain general expressions for the coefficients which arose in the expansion of his generating functions, but he gave the actual values to a high order of the coefficients which arise from the generating functions corresponding to various conditions of partitionment. Other writers who have contributed to the solution of special problems are James Bernoulli, Ruggiero Guiseppe Boscovich, Karl Friedrich Hindenburg (1741-1808), William Emerson (1701-1782), Robert Woodhouse (1773-1827), Thomas Simpson and Peter Barlow. Problems of combination were generally undertaken as they became necessary for the advancement of some particular part of mathematical science: it was not recognized that the theory of combinations is in reality a science by itself, well worth studying for its own sake irrespective of applications to other parts of analysis. There was a total absence of orderly development, and until the first third of the 19th century had passed, Euler's classical paper remained alike the chief result and the only scientific method of combinatorial analysis.

In 1846 Karl G. J. Jacobi studied the partitions of numbers by means of certain identities involving infinite series that are met with in the theory of elliptic functions. The method employed is essentially that of Euler. Interest in England was aroused, in the first instance, by Augustus De Morgan in 1846, who, in a letter to Henry Warburton, suggested that combinatorial analysis stood in great need of development, and alluded to the theory of partitions. Warburton, to some extent under the guidance of De Morgan, prosecuted researches by the aid of a new instrument, viz. the theory of finite differences. This was a distinct advance, and he was able to obtain expressions for the coefficients in partition series in some of the simplest cases (_Trans. Camb. Phil. Soc._, 1849). This paper inspired a valuable paper by Sir John Herschel (_Phil. Trans._ 1850), who, by introducing the idea and notation of the circulating function, was able to present results in advance of those of Warburton. The new idea involved a calculus of the imaginary roots of unity. Shortly afterwards, in 1855, the subject was attacked simultaneously by Arthur Cayley and James Joseph Sylvester, and their combined efforts resulted in the practical solution of the problem that we have to-day. The former added the idea of the prime circulator, and the latter applied Cauchy's theory of residues to the subject, and invented the arithmetical entity termed a denumerant. The next distinct advance was made by Sylvester, Fabian Franklin, William Pitt Durfee and others, about the year 1882 (_Amer. Journ. Math._ vol. v.) by the employment of a graphical method. The results obtained were not only valuable in themselves, but also threw considerable light upon the theory of algebraic series. So far it will be seen that researches had for their object the discussion of the partition of numbers. Other branches of combinatorial analysis were, from any general point of view, absolutely neglected. In 1888 P. A. MacMahon investigated the general problem of distribution, of which the partition of a number is a particular case. He introduced the method of symmetric functions and the method of differential operators, applying both methods to the two important subdivisions, the theory of composition and the theory of partition. He introduced the notion of the separation of a partition, and extended all the results so as to include multipartite as well as unipartite numbers. He showed how to introduce zero and negative numbers, unipartite and multipartite, into the general theory; he extended Sylvester's graphical method to three dimensions; and finally, 1898, he invented the "Partition Analysis" and applied it to the solution of novel questions in arithmetic and algebra. An important paper by G. B. Mathews, which reduces the problem of compound partition to that of simple partition, should also be noticed. This is the problem which was known to Euler and his contemporaries as "The Problem of the Virgins," or "the Rule of Ceres"; it is only now, nearly 200 years later, that it has been solved.

Fundamental problem.

The most important problem of combinatorial analysis is connected with the distribution of objects into classes. A number n may be regarded as enumerating n similar objects; it is then said to be unipartite. On the other hand, if the objects be not all similar they cannot be effectively enumerated by a single integer; we require a succession of integers. If the objects be p in number of one kind, q of a second kind, r of a third, &c., the enumeration is given by the succession pqr... which is termed a multipartite number, and written,

______ pqr...,

where p + q + r + ... = n. If the order of magnitude of the numbers p, q, r, ... is immaterial, it is usual to write them in descending order of magnitude, and the succession may then be termed a partition of the number n, and is written (pqr...). The succession of integers thus has a twofold signification: (i.) as a multipartite number it may enumerate objects of different kinds; (ii.) it may be viewed as a partitionment into separate parts of a unipartite number. We may say either that the objects are represented by the multipartite number

______ pqr...,

or that they are defined by the partition (pqr...) of the unipartite number n. Similarly the classes into which they are distributed may be m in number all similar; or they may be p1 of one kind, q1 of a second, r1 of a third, &c., where p1 + q1 + r1 + ... = m. We may thus denote the classes either by the multipartite numbers

_________ p1q1r1...,

or by the partition (p1q1r1...) of the unipartite number m. The distributions to be considered are such that any number of objects may be in any one class subject to the restriction that no class is empty. Two cases arise. If the order of the objects in a particular class is immaterial, the class is termed a _parcel_; if the order is material, the class is termed a _group_. The distribution into parcels is alone considered here, and the main problem is the enumeration of the distributions of objects defined by the partition (pqr...) of the number n into parcels defined by the partition (p1q1r1...) of the number m. (See "Symmetric Functions and the Theory of Distributions," _Proc. London Mathematical Society_, vol. xix.) Three particular cases are of great importance. Case I. is the "one-to-one distribution," in which the number of parcels is equal to the number of objects, and one object is distributed in each parcel. Case II. is that in which the parcels are all different, being defined by the partition (1111...), conveniently written (1^m); this is the theory of the compositions of unipartite and multipartite numbers. Case III. is that in which the parcels are all similar, being defined by the partition (m); this is the theory of the partitions of unipartite and multipartite numbers. Previous to discussing these in detail, it is necessary to describe the method of symmetric functions which will be largely utilized.

The distribution function.

Let [alpha], [beta], [gamma], ... be the roots of the equation

x^n - a1x^(n-1) + a2x^(n-2) - ... = 0.

The symmetric function [Sigma][alpha]^p[beta]^q[gamma]^r..., where p + q + r + ... = n is, in the partition notation, written (pqr...). Let

A_[(pqr...), (p1q1r1...)]

denote the number of ways of distributing the n objects defined by the partition (pqr...) into the m parcels defined by the partition (p1q1r1...). The expression

[Sigma]A_[(pqr...), (p1q1r1...)]·(pqr...),

where the numbers p1, q1, r1 ... are fixed and assumed to be in descending order of magnitude, the summation being for every partition (pqr...) of the number n, is defined to be the distribution function of the objects defined by (pqr...) into the parcels defined by (p1q1r1...). It gives a complete enumeration of n objects of whatever species into parcels of the given species.

Case I.

1. _One-to-One Distribution. Parcels m in number (i.e. m = n)._--Let hs be the homogeneous product-sum of degree s of the quantities [alpha], [beta], [gamma], ... so that

(1 - [alpha]x. 1 - [beta]x. 1 - [gamma]x. ...)^-1 = 1 + h1x + h2x² + h3x³ + ...

h1 = [Sigma][alpha] = (1) h2 = [Sigma][alpha]² + [Sigma][alpha][beta] = (2) + (1²) h3 = [Sigma][alpha]³ + [Sigma][alpha]²[beta] + [Sigma][alpha][beta][gamma] = (3) + (21) + (1³).

Form the product h_(p1)h_(q1)h_(r1)...

Any term in h_(p1) may be regarded as derived from p1 objects distributed into p1 similar parcels, one object in each parcel, since the order of occurrence of the letters [alpha], [beta], [gamma], ... in any term is immaterial. Moreover, every selection of p1 letters from the letters in [alpha]^p[beta]^q[gamma]^r ... will occur in some term of h_(p1), every further selection of q1 letters will occur in some term of h_(q1), and so on. Therefore in the product h_(p1)h_(q1)h_(r1) ... the term [alpha]^p[beta]^q[gamma]^r ..., and therefore also the symmetric function (pqr ...), will occur as many times as it is possible to distribute objects defined by (pqr ...) into parcels defined by (p1q1r1 ...) one object in each parcel. Hence

[Sigma]A_[(pqr...), (p1q1r1...)]·(pqr ...) = h_(p1)h_(q1)h_(r1)....

This theorem is of algebraic importance; for consider the simple particular case of the distribution of objects (43) into parcels (52), and represent objects and parcels by small and capital letters respectively. One distribution is shown by the scheme

A A A A A B B a a a a b b b

wherein an object denoted by a small letter is placed in a parcel denoted by the capital letter immediately above it. We may interchange small and capital letters and derive from it a distribution of objects (52) into parcels (43); viz.:--

A A A A B B B a a a a a b b.

The process is clearly of general application, and establishes a one-to-one correspondence between the distribution of objects (pqr ...) into parcels (p1q1r1 ...) and the distribution of objects (p1q1r1 ...) into parcels (pqr ...). It is in fact, in Case I., an intuitive observation that we may either consider an object placed in or attached to a parcel, or a parcel placed in or attached to an object. Analytically we have

_Theorem._--"The coefficient of symmetric function (pqr ...) in the development of the product h_(p1)h_(q1)h_(r1) ... is equal to the coefficient of symmetric function (p1q1r1 ...) in the development of the product h_p·h_q·h_r...."

The problem of Case I. may be considered when the distributions are subject to various restrictions. If the restriction be to the effect that an aggregate of similar parcels is not to contain more than one object of a kind, we have clearly to deal with the elementary symmetric functions a1, a2, a3, ... or (1), (1²), (1³), ... in lieu of the quantities h1, h2, h3, ... The distribution function has then the value a_(p1)a_(q1)a_(r1)... or (1^p1) (1^q1) (1^r1) ..., and by interchange of object and parcel we arrive at the well-known theorem of symmetry in symmetric functions, which states that the coefficient of symmetric function (pqr ...) in the development of the product ap1aq1ar1 ... in a series of monomial symmetric functions, is equal to the coefficient of the function (p1q1r1 ...) in the similar development of the product a_p·a_q·a_r....

The general result of Case I. may be further analysed with important consequences.

Write X1 = (1)x1, X2 = (2)x2 + (1²)x1², X3 = (3)x3 + (21)x2x1 + (1³){x1}³

. . . . .

and generally

X_s = [Sigma]([lambda][mu][nu] ...)x_[lambda]x_[mu]x_[nu] ...

the summation being in regard to every partition of s. Consider the result of the multiplication--

X_p1 X_q1 X_r1 ... = [Sigma]P(x_s1)^[sigma]1 (x_s2)^[sigma]2 (x_s3)^[sigma]3 ...

To determine the nature of the symmetric function P a few definitions are necessary.

_Definition I._--Of a number n take any partition ([lambda]1[lambda]2[lambda]3 ... [lambda]s) and separate it into component partitions thus:--

([lambda]1[lambda]2) ([lambda]3[lambda]4[lambda]5) ([lambda]6) ...

in any manner. This may be termed a _separation_ of the partition, the numbers occurring in the separation being identical with those which occur in the partition. In the theory of symmetric functions the separation denotes the product of symmetric functions--

[Sigma] [alpha]^[lambda]1 [beta]^[lambda]2 [Sigma][alpha]^[lambda]3 [beta]^[lambda]4 [gamma]^[lambda]5 [Sigma][alpha]^[lambda]6 ...

The portions ([lambda]1[lambda]2), ([lambda]3[lambda]4[lambda]5), ([lambda]6)... are termed _separates_, and if [lambda]1 + [lambda]2 = p1, [lambda]3 + [lambda]4 + [lambda]5 = q1, [lambda]6 = r1... be in descending order of magnitude, the usual arrangement, the separation is said to have a _species_ denoted by the partition (p1q1r1...) of the number n.

_Definition II._--If in any distribution of n objects into n parcels (one object in each parcel), we write down a number [xi], whenever we observe [xi] similar objects in similar parcels we will obtain a succession of numbers [xi]1, [xi]2, [xi]3, ..., where ([xi]1, [xi]2, [xi]3 ...) is some partition of n. The distribution is then said to have a _specification_ denoted by the partition ([xi]1[xi]2[xi]3...).

Now it is clear that P consists of an aggregate of terms, each of which, to a numerical factor _près_, is a separation of the partition

( s1^{[sigma]1} s2^{[sigma]2} s3^{[sigma]3} ...)

of species (p1q1r1...). Further, P is the distribution function of objects into parcels denoted by (p1q1r1...), subject to the restriction that the distributions have each of them the specification denoted by the partition

( s1^{[sigma]1} s2^{[sigma]2} s3^{[sigma]3} ...).

Employing a more general notation we may write

X_p1^[pi]1 X_p2^[pi]2 X_p3^[pi]3 ... = [Sigma]P x_s1^[sigma]1 x_s2^[sigma]2 x_s3^[sigma]3 ...

and then P is the distribution function of objects into parcels

(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...),

the distributions being such as to have the specification

(s1^[sigma]1 s2^[sigma]2 s3^[sigma]3 ...),

Multiplying out P so as to exhibit it as a sum of monomials, we get a result--

X_p1^[pi]1 X_p2^[pi]2 X_p3^[pi]3 ... = [Sigma][Sigma][theta] ([lambda]1^l1 [lambda]2^l2 [lambda]3^l3) x_s1^[sigma]1 x_s2^[sigma]2 x_s3^[sigma]3 ...

indicating that for distributions of specification

(s1^[sigma]1 s2^[sigma]2 s3^[sigma]3 ...)

there are [theta] ways of distributing n objects denoted by

([lambda]1^l1 [lambda]2^l2 [lambda]3^l3 ...)

amongst n parcels denoted by

(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...),

one object in each parcel. Now observe that as before we may interchange parcel and object, and that this operation leaves the specification of the distribution unchanged. Hence the number of distributions must be the same, and if

X_p1^[pi]1 X_p2^[pi]2 X_p3^[pi]3 ... = = ... + [theta]([lambda]1^l1 [lambda]2^l2 [lambda]3^l3) x_s1^[sigma]1 x_s2^[sigma]2 x_s3^[sigma]3 ... + ...

then also

X_[lambda]1^l1 X_[lambda]2^l2 X_[lambda]3^l3 ... = = ... + [theta](p1^[pi]1 p2^[pi]2 p3^[pi]3) x_s1^[sigma]1 x_s2^[sigma]2 x_s3^[sigma]3 ... + ...

This extensive theorem of algebraic reciprocity includes many known theorems of symmetry in the theory of Symmetric Functions.

The whole of the theory has been extended to include symmetric functions symbolized by partitions which contain as well zero and negative parts.

Case II.

2. _The Compositions of Multipartite Numbers. Parcels denoted by (I^m)._--There are here no similarities between the parcels.

Let ([pi]1 [pi]2 [pi]3) be a partition of m.

(p1^[pi]1 p2^[pi]2 p3^[pi]3) a partition of n.

Of the whole number of distributions of the n objects, there will be a certain number such that n1 parcels each contain p1 objects, and in general [pi]s parcels each contain ps objects, where s = 1, 2, 3, ... Consider the product h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ... which can be permuted in m! / ([pi]1![pi]2![pi]3! ...) ways. For each of these ways h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ... will be a distribution function for distributions of the specified type. Hence, regarding all the permutations, the distribution function is

m! ------------------------ h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ... [pi]1! [pi]2! [pi]3! ...

and regarding, as well, all the partitions of n into exactly m parts, the desired distribution function is

m! [Sigma] ------------------------ h_p1^[pi]1 h_p2^[pi]2 h_p3^[pi]3 ... [pi]1! [pi]2! [pi]3! ... [ [Sigma]_[pi] = ([Sigma]_[pi])p = n ],

that is, it is the coefficient of x^n in (h1x + h2x² + h3x³ + ... )^m. The value of A_{(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...), (1^m)} is the coefficient of (p1^[pi]1 p2^[pi]2 p3^[pi]3 ...)x^n in the development of the above expression, and is easily shown to have the value

/p1 + m - 1\^[pi]1 /p2 + m - 1\^[pi]2 /p3 + m - 1\^[pi]3 \ p1 / \ p2 / \ p3 / ...

- /m\ /p1 + m - 2\^[pi]1 /p2 + m - 2\^[pi]2 /p3 + m - 2\^[pi]3 \1/ \ p1 / \ p2 / \ p3 / ...

- /m\ /p1 + m - 3\^[pi]1 /p2 + m - 3\^[pi]2 /p3 + m - 3\^[pi]3 \1/ \ p1 / \ p2 / \ p3 / ...

- ... to m terms.

Observe that when p1 = p2 = p3 = ... = [pi]1 = [pi]1 = [pi]1 ... = 1 this expression reduces to the mth divided differences of 0^n. The expression gives the compositions of the multipartite number ______________________________ p1^[pi]1 p2^[pi]2 p3^[pi]3 ...

into m parts. Summing the distribution function from m = 1 to w = [oo] and putting x = 1, as we may without detriment, we find that the totality of the compositions is given by

h1 + h2 + h3 + ... ---------------------- which may be given the form 1 - h1 - h2 - h3 + ...

a1 - a2 + a3 - ... -------------------------. 1 - 2(a1 - a2 + a3 - ...)

Adding ½ we bring this to the still more convenient form

1 ½ -------------------------. 1 - 2(a1 - a2 + a3 - ...)

Let F(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...) denote the total number of compositions of the multipartite /{p1^[pi]1 p2^[pi]2 p3^[pi]3}.... Then ½{1/1 - 2[alpha]} = ½ + [Sigma]F(p)[alpha]^p, and thence F(p) = 2^(p-1).

1 Again ½ --------------------------------------- = 1 - 2([alpha] + [beta] - [alpha][beta])

= ½ + [Sigma]F(p)[alpha]^p1 [beta]^p2,

and expanding the left-hand side we easily find

(p1 + p2)! (p1 + p2 - 1)! F(p1p2) = 2^(p1+p2-1) ---------- - 2^(p1+p2-2) --------------------- 0! p1! p2! 1!(p1 - 1)! (p2 - 1)!

(p1 + p2 - 2)! + 2^(p1+p2-3) --------------------- - .... 2!(p1 - 2)! (p2 - 2)!

We have found that the number of compositions of the multipartite /(p1p2p3 ... ps) is equal to the coefficient of symmetric function (p1p2p3...ps) _or_ of the single term [alpha]1^p1 [alpha]2^p2 [alpha]3^p3 ... [alpha]s^ps in the development according to ascending powers of the algebraic fraction

1 ½ · ----------------------------------------------------------------------------------------------------. 1 - 2([Sigma][a]1 - [Sigma][a]1 [a]2 + [Sigma][a]1 [a]2 [a]3) - ... + (-)^(s+1)[a]1 [a]2 [a]3...[a]s

This result can be thrown into another suggestive form, for it can be proved that this portion of the expanded fraction

1 ½ · -------------------------------------------------------------------------------------------------------------------, {1 - t1(2[a]1 + [a]2 + ... + [a]3)} {1 - t2(2[a]1 + 2[a]2 + ... + [a]s)} ... {1 - t_s(2[a]1 + 2[a]2 + ... + 2[a]s)}

which is composed entirely of powers of

t1[alpha]1, t2[alpha]2, t3[alpha]3, ... t_s[alpha]_s

has the expression

1 ½ · -----------------------------------------------------------------------------------------------------------------, 1 - 2([Sigma]t1[a]1 - [Sigma]t1t2[a]1[a]2 + [Sigma]t1t2[a]1[a]2[a]3 - ... + (-)^(s+1) t1t2...t_s[a]1[a]2...[a]_s)

and therefore the coefficient of [alpha]1^p1 [alpha]2^p2...[alpha]s^ps in the latter fraction, when t1, t2, &c., are put equal to unity, is equal to the coefficient of the same term in the product

½ (2[a]1 + [a]2 + ... + [a]s)^p1 (2[a]1 + 2[a]2 + ... +[a]s)^p2 ... (2[a]1 + 2[a]2 + ... + 2[a]s)^ps.

This result gives a direct connexion between the number of compositions and the permutations of the letters in the product [alpha]1^p1 [alpha]2^p2...[alpha]s^ps. Selecting any permutation, suppose that the letter a_r occurs q_r times in the last p_r + p_(r+1) + ... + p_s places of the permutation; the coefficient in question may be represented by ½[Sigma] 2^(q1+q2+...+qs), the summation being for every permutation, and since q1 = p1 this may be written

2p1^(-1)[Sigma] 2^(q2+q3+...+qs).

_Ex. Gr._--For the bipartite /22, p1 = p2 = 2, and we have the following scheme:--

[a]1 [a]1 | [a]2 [a]2 q2 = 2 [a]1 [a]2 | [a]1 [a]2 = 1 [a]1 [a]2 | [a]2 [a]1 = 1 [a]2 [a]1 | [a]1 [a]2 = 1 [a]2 [a]1 | [a]2 [a]1 = 1 [a]2 [a]2 | [a]1 [a]1 = 0

Hence F(22) = 2(2² + 2 + 2 + 2 + 2 + 2°) = 26.

We may regard the fraction

1 -------------------------------------------------------------------------------------------------------------------, ½ · {1 - t1(2[a]1 + [a]2 + ... + [a]s)} {1 - t2(2[a]1 + 2[a]2 + ... + [a]s)} ... {1 - t_s(2[a]1 + 2[a]2 + ... + 2[a]s)}

as a redundant generating function, the enumeration of the compositions being given by the coefficient of

(t1[alpha]1)^p1 (t2[alpha]2)^p2 ... (t_s[alpha]_s )^ps.

The transformation of the pure generating function into a factorized redundant form supplies the key to the solution of a large number of questions in the theory of ordinary permutations, as will be seen later.

The theory of permutations.

[The transformation of the last section involves a comprehensive theory of Permutations, which it is convenient to discuss shortly here.

If X1, X2, X3, ... Xn be linear functions given by the matricular relation

(X1, X2, X3, ... Xn) = (a11 a12 ... a1n)(x1, x2, ... xn) |a21 a22 ... a2n| | . . ... . | | . . ... . | |an1 an2 ... ann|

that portion of the algebraic fraction,

1 ---------------------------------, (1 - s1X1)(1 - s2X2)...(1 - snXn)

which is a function of the products s1x1, s2x2, s3x3, ... snxn only is

1 -------------------------------------------------------- |(1 - a11s1x1)(1 - a22s2x2)(1 - a33s3x3)(1 - ann·sn·xn)|

where the denominator is in a symbolic form and denotes on expansion

1 - [Sigma]|a11|s1x1 + [Sigma]|a11a22|s1s2x1x2 - ... + (-)^n|a11a22a33...ann|s1s2 ... sn·x1x2...xn,

where |a11|, |a11a22|, ... |a11a22,...ann| denote the several co-axial minors of the determinant

|a11a22...ann|

of the matrix. (For the proof of this theorem see MacMahon, "A certain Class of Generating Functions in the Theory of Numbers," _Phil. Trans. R. S._ vol. clxxxv. A, 1894). It follows that the coefficient of

x1^[xi]1 x2^[xi]2 ... xn^[xi]n

in the product

(a11x1 + a12x2 + ... + a1n·xn )^[xi]¹ (a21x1 + a22x2 + ... + + a2n·xn)^[xi]²...(an1x1 + an2x2 + ... + ann·xn)^[xi]n

is equal to the coefficient of the same term in the expansion ascending-wise of the fraction

1 --------------------------------------------------------------------------. 1 - [Sigma]|a11|x1 + [Sigma]|a11a22|x1x2 - ... + (-)^n|a11a22...|x1x2...xn

If the elements of the determinant be all of them equal to unity, we obtain the functions which enumerate the unrestricted permutations of the letters in

x1^[xi]1 x2^[xi]2 ... xn^[xi]n,

viz. (x1 + x2 + ... - xn)^{[xi]1 + [xi]2 + ... + [xi]n}

1 and ------------------------. 1 - (x1 + x2 + ... + xn)

Suppose that we wish to find the generating function for the enumeration of those permutations of the letters in x1^[xi]1 x2^[xi]2...x3^[xi]n which are such that no letter xs is in a position originally occupied by an x3 for all values of s. This is a generalization of the "Problème des rencontres" or of "derangements." We have merely to put

a11 = a22 = a33 = ... = ann = 0

and the remaining elements equal to unity. The generating product is

(x2 + x3 + ... + xn)^[xi]1 (x1 + x3 + ... + xn)^[xi]2 ... (x1 + x2 + ... + x_n-1)^[xi]n,

and to obtain the condensed form we have to evaluate the co-axial minors of the invertebrate determinant--

| 0 1 1 ... 1 | | 1 0 1 ... 1 | | 1 1 0 ... 1 | | . . . ... . | | 1 1 1 ... 0 |

The minors of the 1st, 2nd, 3rd ... nth orders have respectively the values

0 -1 +2 ... (-)^(n-1)(n - 1),

therefore the generating function is

1 --------------------------------------------------------------------------------------; 1 - [Sigma]x1x2 - 2[Sigma]x1x2x3 - ... - s[Sigma]x1x2...x_s+1 - ... - (n - 1)x1x2...xn

or writing

(x - x1)(x - x2)...(x - xn) = x^n - a1x^(n-1) + a2x^(n-2) - ...,

this is

1 ------------------------------------- 1 - a2 - 2a3 - 3a4 - ... - (n - 1)a_n

Again, consider the general problem of "derangements." We have to find the number of permutations such that exactly _m_ of the letters are in places they originally occupied. We have the particular redundant product

(ax1 + x2 + ... + xn)^[xi]¹ (x1 + ax2 + ... + xn)^[xi]² ... (x1 + x2 + ... + ax_n)^[xi]n,

in which the sought number is the coefficient of a^m x1^[xi]¹ x2^[xi]²...xn^[xi]n. The true generating function is derived from the determinant

| a 1 1 1 . . . | | 1 a 1 1 . . . | | 1 1 a 1 . . . | | 1 1 1 a . . . | | . . . . | | . . . . |

and has the form

1 ------------------------------------------------------------------------------------------. 1 - a[Sigma]x1 + (a - 1)(a + 1)[Sigma]x1x2 - ... + (-)^n(a - 1)^(n-1)(a + n - 1)x1x2... xn

It is clear that a large class of problems in permutations can be solved in a similar manner, viz. by giving special values to the elements of the determinant of the matrix. The redundant product leads uniquely to the real generating function, but the latter has generally more than one representation as a redundant product, in the cases in which it is representable at all. For the existence of a redundant form, the coefficients of x1, x2, ... x1x2 ... in the denominator of the real generating function must satisfy 2^n - n² + n - 2 conditions, and assuming this to be the case, a redundant form can be constructed which involves n - 1 undetermined quantities. We are thus able to pass from any particular redundant generating function to one equivalent to it, but involving n - 1 undetermined quantities. Assuming these quantities at pleasure we obtain a number of different algebraic products, each of which may have its own meaning in arithmetic, and thus the number of arithmetical correspondences obtainable is subject to no finite limit (cf. MacMahon, _loc. cit._ pp. 125 et seq.)]

Case III.

3. _The Theory of Partitions. Parcels defined by (m)._--When an ordinary unipartite number n is broken up into other numbers, and the order of occurrence of the numbers is immaterial, the collection of numbers is termed a partition of the number n. It is usual to arrange the numbers comprised in the collection, termed the parts of the partition, in descending order of magnitude, and to indicate repetitions of the same part by the use of exponents. Thus (32111), a partition of 8, is written (321³). Euler's pioneering work in the subject rests on the observation that the algebraic multiplication

x^a × x^b × x^c × ... = x^(a+b+c+...)

is equivalent to the arithmetical addition of the exponents a, b, c, ... He showed that the number of ways of composing n with p integers drawn from the series a, b, c, ..., repeated or not, is equal to the coefficient of [zeta]^p·x^n in the ascending expansion of the fraction

1 ------------------------------------------------, 1 - [zeta]x^a. 1 - [zeta]x^b. 1 - [zeta]x^c. ...

which he termed the generating function of the partitions in question.

If the partitions are to be composed of p, or fewer parts, it is merely necessary to multiply this fraction by 1/(1 - [zeta]). Similarly, if the parts are to be unrepeated, the generating function is the algebraic product

(1 + [zeta]x^a)(1 + [zeta]x^b)(1 + [zeta]x^c)...;

if each part may occur at most twice,

(1 + [zeta]x^a + [zeta]²x^2a)(1 + [zeta]x^b + [zeta]²x^2b) (1 + [zeta]x^c + [zeta]²x^2c)...;

and generally if each part may occur at most k - 1 times it is

1 - [zeta]^k·x^ka 1 - [zeta]^k·x^kb 1 - [zeta]^k·x^kc ----------------- · ----------------- · ----------------- · ... 1 - [zeta]x^a 1 - [zeta]x^b 1 - [zeta]x^c

It is thus easy to form generating functions for the partitions of numbers into parts subject to various restrictions. If there be no restriction in regard to the numbers of the parts, the generating function is

1 ------------------------------ 1 - x^a. 1 - x^b. 1 - x^c. ...

and the problems of finding the partitions of a number n, and of determining their number, are the same as those of solving and enumerating the solutions of the indeterminate equation in positive integers

ax + by + cz + ... = n.

Euler considered also the question of enumerating the solutions of the indeterminate simultaneous equation in positive integers

ax + by + cz + ... = n a'x + b'y + c'z + ... = n' a"x + b"y + c"z + ... = n"

which was called by him and those of his time the "Problem of the Virgins." The enumeration is given by the coefficient of x^n·y^n'·z^n" ... in the expansion of the fraction

1 ---------------------------------------------------------------------- (1 - x^a·y^b·z^c...)(1 - x^a'·y^b'·z^c'...)(1 - x^a"·y^b"·z^c"...) ...

which enumerates the partitions of the multipartite number /nn'n"... into the parts

/abc..., /a'b'c'..., /a"b"c"..., ...

Sylvester has determined an analytical expression for the coefficient of x^n in the expansion of

1 ------------------------------ (1 - x^a)(1 - x^b)...(1 - x^i)

To explain this we have two lemmas:--

_Lemma 1._--The coefficient of x^-1, i.e., after Cauchy, the residue in the ascending expansion of (1 - e^x)^-i, is -1. For when i is unity, it is obviously the case, and

(1 - e^x)^-i-1 = (1 - e^x)^-i + e^x(1 - e^x)^-i-1

d 1 = (1 - e^x)^-i + -- (1 - e^x)^-i·--. dx i

d 1 Here the residue of -- (1 - e^x)^-i·-- is zero, and therefore the residue dx i of (1 - e^x)^-i is unchanged when i is increased by unity, and is therefore always -1 for all values of i.

_Lemma 2._--The constant term in any proper algebraical fraction developed in ascending powers of its variable is the same as the residue, with changed sign, of the sum of the fractions obtained by substituting in the given fraction, in lieu of the variable, its exponential multiplied in succession by each of its values (zero excepted, if there be such), which makes the given fraction infinite. For write the proper algebraical fraction

c_{[lambda],[mu]} [gamma]_[lambda] F(x) = [Sigma][Sigma]-------------------- + [Sigma]----------------. (a_[mu] - x)[lambda] x^[lambda]

c_{[lambda],[mu]} The constant term is [Sigma][Sigma]-----------------. a_[mu]^[lambda]

Let a_[nu] be a value of x which makes the fraction infinite. The residue of

c_{[lambda],[mu]} [gamma]_[lambda] [Sigma][Sigma][Sigma]------------------------------ + [Sigma]----------------------------- (a_[mu] - a_[nu]·e^x)^[lambda] a_[nu]^[lambda]·e^{[lambda]x}

is equal to the residue of

c_{[lambda],[mu]} [Sigma][Sigma][Sigma]------------------------------, (a_[mu] - a_[nu]·e^x)^[lambda]

and when [nu] = [mu], the residue vanishes, so that we have to consider

c_{[lambda],[mu]} [Sigma][Sigma]----------------------------------, a_[mu]^[lambda]·(1 - e^x)^[lambda]

and the residue of this is, by the first lemma,

c_{[lambda],[mu]} - [Sigma][Sigma]-----------------, a_[mu]^[lambda]

which proves the lemma.

1 f(x) Take F(x) = --------------------------------- = ----, since the sought x^n(1 - x^a)(1 - x^b)...(1 - x^l) x^n

number is its constant term.

Let [rho] be a root of unity which makes f(x) infinite when substituted for x. The function of which we have to take the residue is

[Sigma][rho]^-n·e^nx·f([rho]e^-x)

[rho]^-n·e^nx = [Sigma]------------------------------------------------------------. (1 - [rho]^a·e^-ax)(1 - [rho]^b·e^-bx)...(1 - [rho]^l·e^-lx)

We may divide the calculation up into sections by considering separately that portion of the summation which involves the primitive qth roots of unity, q being a divisor of one of the numbers a, b, ... l. Thus the qth _wave_ is

[rho]_q^-n·e^nx [Sigma]-------------------------------------------------------------------- , (1 - [rho]_q^a·e^-ax)(1 - [rho]_q^b·e^-bx)...(1 - [rho]_q^l·e^-lx)

which, putting 1/[rho]_q for [rho]_q and [nu] = ½(a + b + ... + l), may be written

[rho]_q^[nu]·e^[nu]x [Sigma]------------------------------------------------------------------------------------------------------------------------, ([rho]_q^½a·e^½ax - [rho]_q^-½a·e^-½ax)([rho]_q^½b·e^½bx - [rho]_q^-½b·e^-½bx)...([rho]_q^½l·e^½lx - [rho]_q^-½l·e^-½lx)

and the calculation in simple cases is practicable.

Thus Sylvester finds for the coefficient of x^n in

1 --------------------- 1 - x. 1 - x². 1 - x³

[nu]² 7 1 1 the expression ---- - -- - --(-)[nu] + --([rho]_3^[nu] + [rho]_3^-[nu]), 12 72 8 9

where [nu] = n + 3.

Sylvester's graphical method.

Sylvester, Franklin, Durfee, G. S. Ely and others have evolved a constructive theory of partitions, the object of which is the contemplation of the partitions themselves, and the evolution of their properties from a study of their inherent characters. It is concerned for the most part with the partition of a number into parts drawn from the natural series of numbers 1, 2, 3.... Any partition, say (521) of the number 8, is represented by nodes placed in order at the points of a rectangular lattice,

o---o---o---o---o------ | | | | | | | | | | o---o---+---+---+------ | | | | | | | | | | o---+---+---+---+------ | | | | | | | | | | | | | | |

when the partition is given by the enumeration of the nodes by lines. If we enumerate by columns we obtain another partition of 8, viz. (321³), which is termed the conjugate of the former. The fact or conjugacy was first pointed out by Norman Macleod Ferrers. If the original partition is one of a number n in i parts, of which the largest is j, the conjugate is one into j parts, of which the largest is i, and we obtain the theorem:--"The number of partitions of any number into [i parts]/[i parts or fewer], and having the largest part [equal to j]/[equal or less than j], remains the same when the numbers i and j are interchanged."

The study of this representation on a lattice (termed by Sylvester the "graph") yields many theorems similar to that just given, and, moreover, throws considerable light upon the expansion of algebraic series.

The theorem of reciprocity just established shows that the number of partitions of n into; parts or fewer, is the same as the number of ways of composing n with the integers 1, 2, 3, ... j. Hence we can

1 expand ----------------------------------------- in ascending powers of 1 - a. 1 - ax. 1 - ax². 1 - ax³...ad inf.

a; for the coefficient of a^j·x^n in the expansion is the number of ways of composing n with j or fewer parts, and this we have seen in the coefficients of x^n in the ascending expansion of

1 -----------------------. 1 - x. 1 - x²...1 - x^j

Therefore

1 a a² -------------------------- = 1 + ----- + ------------- + ... 1 - a. 1 - ax. 1 - ax².... 1 - x 1 - x. 1 - x²

a^j + ----------------------- + .... 1 - x. 1 - x²...1 - x^j

The coefficient of a^j·x^n in the expansion of

1 ------------------------------------ 1 - a. 1 - ax. 1 - ax². ... 1 - ax^i

denotes the number of ways of composing n with j or fewer parts, none of which are greater than i. The expansion is known to be

1 - x^(j+1). 1 - x^(j+2). ... 1 - x^(j+i) [Sigma]-----------------------------------------a^j. 1 - x. 1 - x². ... 1 - x^i

It has been established by the constructive method by F. Franklin (_Amer. Jour. of Math._ v. 254), and shows that the generating function for the partitions in question is

1 - x^(j+1). 1 - x^(j+2). ... 1 - x^(j+i) -----------------------------------------, 1 - x. 1 - x². ... 1 - x^i

which, observe, is unaltered by interchange of i and j.

Franklin has also similarly established the identity of Euler

j=-[oo] (1 - x)(1 - x²)(1 - x³)...ad inf. = [Sigma](-)jx^{½(3j²+j)}, j=+[oo]

known as the "pentagonal number theorem," which on interpretation shows that the number of ways of partitioning n into an even number of unrepeated parts is equal to that into an uneven number, except when n has the pentagonal form ½(3j² + j), j positive or negative, when the difference between the numbers of the partitions is (-)^j.

+----------+ |· · · ·| · · · · · |· · · ·| · · |· · · ·| · |· · · ·| +----------+ · · · . . . . .

To illustrate an important dissection of the graph we will consider those graphs which read the same by columns as by lines; these are called self-conjugate. Such a graph may be obviously dissected into a square, containing say [theta]² nodes, and into two graphs, one lateral and one subjacent, the latter being the conjugate of the former. The former graph is limited to contain not more than [theta] parts, but is subject to no other condition. Hence the number of self-conjugate partitions of n which are associated with a square of [theta]² nodes is clearly equal to the number of partitions of ½(n = [theta]²) into [theta] or few parts, i.e. it is the coefficient of x^{½(n-[theta]²)} in

1 -----------------------------------------, 1 - x. 1 - x². 1 - x³. ... 1 - x^[theta].

x^[theta]² or of x^n in --------------------------------------------. 1 - x². 1 - x^4. 1 - x^6. ... 1 - x^2[theta]

and the whole generating function is

[theta]=[oo] x^[theta]² 1 + [Sigma] --------------------------------------------. [theta]=1 1 - x². 1 - x^4. 1 - x^6. ... 1 - x^2[theta]

Now the graph is also composed of [theta] angles of nodes, each angle containing an uneven number of nodes; hence the partition is transformable into one containing [theta] unequal uneven numbers. In the case depicted this partition is (17, 9, 5, 1). Hence the number of the partitions based upon a square of [theta]² nodes is the coefficient of a^[theta]·x^n in the product (1 + ax)(1 + ax³)(1 + ax^5)...(1 + ax^{2s-1}), and thence the coefficient of a^[theta] in this product is

x^[theta]² --------------------------------------------, and we have the expansion 1 - x². 1 - x^4. 1 - x^6. ... 1 - x^2[theta]

(1 + ax)(1 + ax³)(1 + ax^5)...ad inf.

x x^4 x^9 = 1 + ------ a + --------------- a² + ---------------------- a³ + ... 1 - x² 1 - x². 1 - x^4 1 - x². 1 - x^4. - x^6

Again, if we restrict the part magnitude to i, the largest angle of nodes contains at most 2i - 1 nodes, and based upon a square of [theta]² nodes we have partitions enumerated by the coefficient of a^[theta]·x^n in the product (1 + ax)(1 + ax³)(1 + ax^5)...(1 + ax^{2i-1}); moreover the same number enumerates the partition of ½(n - [theta]²) into [theta] or fewer parts, of which the largest part is equal to or less than i -[theta], and is thus given by the coefficient of x^{½(n-[theta]²)} in the expansion of

1 - x^{i-[t]+1}. 1 - x^{i-[t]+2}. 1 - x^{i-[t]+3}. ... 1 - x^i --------------------------------------------------------------, 1 - x. 1 - x². 1 - x³. ... 1 - x^[t] ([t] = [theta]) or of x^n in

1 - x^{2i-2[t]+2}. 1 - x^{2i-2[t]+4}. ... 1 - x^2i -------------------------------------------------- x[t]²; 1 - x². 1 - x^4. 1 - x^6. ... 1 - x^[t]

hence the expansion

(1 + ax)(1 + ax³)(1 + ax^5)...(1 + ax^{2i-1})

[t]=i 1 - x^{2i-2[t]+2}. 1 - x^{2i-2[t]+4}. ... 1 + x^2i = 1 + [Sigma] -------------------------------------------------- x^[t]²·a^[t]. [t]=1 1 - x². 1 - x^4. 1 - x^6. ... 1 - x^2[t]

Extension to three dimensions.

There is no difficulty in extending the graphical method to three dimensions, and we have then a theory of a special kind of partition of multipartite numbers. Of such kind is the partition

_________ _________ _________ (a1a2a3...), (b1b2b3...), (c1c2c3..., ...)

of the multipartite number _______________________________________________________________ (a1 + b1 + c1 + ..., a2 + b2 + c2 + ..., a3 + b3 + c3 + ..., ...)

if a1 >= a2 >= a3 >= ...; b1 >= b2 >= b3 >= ..., ... a3 >= b3 >= c3 >= ...,

for then the graphs of the parts /a1a2a3..., /b1b2b3..., ... are superposable, and we have what we may term a _regular_ graph in three dimensions. Thus the partition (/643, /632, /411) of the multipartite /(16, 8, 6) leads to the graph

0+------------------------------------ x | | ((·)) ((·)) ((·)) ((·)) (·) (·) | | ((·)) (·) (·) · | | ((·)) (·) · | y

and every such graph is readable in six ways, the axis of z being perpendicular to the plane of the paper.

_Ex. Gr._ ___ ___ ___ Plane parallel to xy, direction Ox reads (643,632,411) ______ ______ ______ " " xy, " Oy " (333211,332111,311100) ___ ___ ___ ___ ___ ___ " " yz, " Oy " (333,331,321,211,110,110) ___ ___ ___ ___ ___ ___ " " yz, " Oz " (333,322,321,310,200,200) ______ ______ ______ " " zx, " Oz " (333322,322100,321000) ___ ___ ___ " " zx, " Ox " (664,431,321)

the partitions having reference to the multipartite numbers /16, 8, 6, 976422, /13, 11, 6, which are brought into relation through the medium of the graph. The graph in question is more conveniently represented by a numbered diagram, viz.--

3 3 3 3 2 2 3 2 2 1 3 2 1

and then we may evidently regard it as a unipartite partition on the points of a lattice,

0 +-----+-----+-----+-----+------- x | | | | | | | | | | +-----+-----+-----+-----+------- | | | | | | | | | | +-----+-----+-----+-----+------- | | | | | | | | | | +-----+-----+-----+-----+------- | | | | | y

the descending order of magnitude of part being maintained along _every_ line of route which proceeds from the origin in the positive directions of the axes.

This brings in view the modern notion of a partition, which has enormously enlarged the scope of the theory. We consider any number of points _in plano_ or _in solido_ connected (or not) by lines in pairs in any desired manner and fix upon any condition, such as is implied by the symbols >=, >, =, <=, <>, as affecting any pair of points so connected. Thus in ordinary unipartite partition we have to solve in integers such a system as

[a]1 >= [a]2 >= [a]3 >= ... [a]n

[a]1 + [a]2 + [a]3 + ... + [a]n = n, ([a] = [alpha])

the points being in a straight line. In the simplest example of the three-dimensional graph we have to solve the system

[a]1 >= [a]2 v = [a]1 + [a]2 + [a]3 + [a]4 = n, = v [a]3 >= [a]4

and a system for the general lattice constructed upon the same principle. The system has been discussed by MacMahon, _Phil. Trans._ vol. clxxxvii. A, 1896, pp. 619-673, with the conclusion that if the numbers of nodes along the axes of x, y, z be limited not to exceed the numbers m, n, l respectively, then writing for brevity 1 - x^s = (s), the generating function is given by the product of the factors

+----------------------------------------------x | | (l + 1) (l + 2) (l + m) | ------- . ------- ... ------- | (1) (2) (m) | | (l + 2) (l + 3) (l + m + 1) | ------- . ------- ... ----------- | (2) (3) (m + 1) | . . ... . | . . ... . | . . ... . | (l + n) (l + n + 1) (l + m + n - 1) | ------- . ----------- ... --------------- | (n) (n + 1) (m + n - 1) y

one factor appearing at each point of the lattice.

In general, partition problems present themselves which depend upon the solution of a number of simultaneous relations in integers of the form

[lambda]_1·[alpha]_1 + [lambda]_2·[alpha]_2 + [lambda]_3·[alpha]_3 + ... >= 0,

the coefficients [lambda] being given positive or negative integers, and in some cases the generating function has been determined in a form which exhibits the fundamental solutions of the problems from which all other solutions are derivable by addition. (See MacMahon, _Phil. Trans._ vol. cxcii. (1899), pp. 351-401; and _Trans. Camb. Phil. Soc._ vol. xviii. (1899), pp. 12-34.)

Method of symmetric functions.

The number of distributions of n objects (p1p2p3 ...) into parcels (m) is the coefficient of b^m(p1p2p3 ...)x^n in the development of the fraction

1 ----------------------------------------------------------------------- (1 - b[alpha]x. 1 - b[beta]x. 1 - b[gamma]x ... ) × (1 - b[alpha]²x². 1 - b[alpha][beta]x². 1 - b[beta]²x² ... ) × (1 - b[alpha]³x³. 1 - b[alpha]²[beta]x³. 1 - b[alpha][beta][gamma]x³ ...)

. . . . . .

and if we write the expansion of that portion which involves products of the letters [alpha], [beta], [gamma], ... of degree r in the form

1 + h_r1·bx^r + h_r2·b²x^2r + ...,

we may write the development

r=[oo] [Pi] (1 + h_r1·bx^r + h_r2·b²x^2r + ...), r=1

and picking out the coefficient of b^m x^n we find

[Sigma] h_[tau]1·h_[tau]2·h_[tau]3 ..., t1 t2 t3

where [Sigma][tau] = m, [Sigma][tau]t = n.

The quantities h are symmetric functions of the quantities [alpha], [beta], [gamma], ... which in simple cases can be calculated without difficulty, and then the distribution function can be formed.

_Ex. Gr._--Required the enumeration of the partitions of all multipartite numbers (p1p2p2 ...) into exactly two parts. We find

h2² = h4 - h3h1 + (h2)²

h3² = h6 - h5h1 + h4h2

h4² = h8 - h7h1 + h6h2 + h5h3 + (h4)²,

and paying attention to the fact that in the expression of h_r2 the term (h_r)² is absent when r is uneven, the law is clear. The generating function is

h2x² + h2h1x³ + (h4 + h2²)x^4 + (h4h1 + h3h2)x^5 + (h6 + 2h4h2)x^6 + (h6h1 + h6h2 + h4h3)x^7 + (h8 + 2h6h2 + h4²)x8 + ...

Taking h4 + h2² = h4 + {(2) + (1²)}²

= 2(4) + 3(31) + 4(2²) + 5(21²) + 7(1^4),

the term 5(21²) indicates that objects such as a, a, b, c can be partitioned in five ways into two parts. These are a|a, b, c; b|a; a, c; c|a, a, b; a, a|b, c; a, b|a, c. The function h_{r^s} has been studied. (See MacMahon, _Proc. Lond. Math. Soc._ vol. xix.) Putting x equal to unity, the function may be written (h2 + h4 + h6 + ...)(1 + h1 + h2 + h3 + h4 + ...), a convenient formula.

Method of differential operators.

The method of differential operators, of wide application to problems of combinatorial analysis, has for its leading idea the designing of a function and of a differential operator, so that when the operator is performed upon the function a number is reached which enumerates the solutions of the given problem. Generally speaking, the problems considered are such as are connected with lattices, or as it is possible to connect with lattices.

To take the simplest possible example, consider the problem of finding the number of permutations of n different letters. The function is here x^n, and the operator (d/dx)^n = [delta]_x^n, yielding [delta]_x^n·x^n = n! the number which enumerates the permutations. In fact--

[delta]_x·x^n = [delta]_x. x. x. x. x. x. ...,

and differentiating we obtain a sum of n terms by striking out an x from the product in all possible ways. Fixing upon any one of these terms, say x. [x]. x. x. ..., we again operate with [delta]_x by striking out an x in all possible ways, and one of the terms so reached is x. [x]. x. [x]. x. .... Fixing upon this term, and again operating and continuing the process, we finally arrive at one solution of the problem, which (taking say n = 4) may be said to be in correspondence with the operator diagram-- ([x] = striken-out x)

or say +-------+-------+-------+-------+ +-------+-------+-------+-------+ | | [d]_x | | | | | 1 | | | +-------+-------+-------+-------+ +-------+-------+-------+-------+ | | | | [d]_x | | | | | 1 | +-------+-------+-------+-------+ +-------+-------+-------+-------+ | | | [d]_x | | | | | 1 | | +-------+-------+-------+-------+ +-------+-------+-------+-------+ | [d]_x | | | | | 1 | | | | +-------+-------+-------+-------+ +-------+-------+-------+-------+ ([d] = [delta])

the number in each row of cempartments denoting an operation of [delta]_x. Hence the permutation problem is equivalent to that of placing n units in the compartments of a square lattice of order n in such manner that each row and each column contains a single unit. Observe that the method not only enumerates, but also gives a process by which each solution is actually formed. The same problem is that of placing n rooks upon a chess-board of n² compartments, so that no rook can be captured by any other rook.

Regarding these elementary remarks as introductory, we proceed to give some typical examples of the method. Take a lattice of m columns and n rows, and consider the problem of placing units in the compartments in such wise that the sth column shall contain [lambda]_s units (s = 1, 2, 3, ... m), and the tth row p1 units (t = l, 2, 3, ... n).

Writing

1 + a1x + a2x² + ... + ... = (1 + a1x)(1 + a2x)(1 + a3x) ...

1 and D_p = --([d]_[a]1 + [a]1[d]_[a]2 + [a]2[d]_[a]3 + ...)^p, p! ([d] = [delta], [a] = [alpha])

the multiplication being symbolic, so that D_p is an operator of order p, the function is

a_[lambda]1·a_[lambda]2·a_[lambda]3...a_[lambda]m,

and the operator D_p1·D_p2·D_p3...D_pn. The number D_p1·D_p2...D_pn·a_[lambda]1·a_[lambda]2·a_[lambda]3...a_[lambda]m enumerates the solutions. For the mode of operation of D_p upon a product reference must be made to the section on "Differential Operators" in the article ALGEBRAIC FORMS. Writing

a_[l]1·a_[l]2...a_[l]m = ... + [Delta][Sigma][a]1^p1·[a]2^p2...[a]n^pn + ...,

or, in partition notation,

(1^[l]1)(1^[l]2)...(1^[l]m) = ... + A(p1p2...pn) ... + D_p1·D_p2...D_pn·(1^[l]1)(1^[l]2)...(1^[l]m) = A, ([l] = [lambda])

and the law by which the operation is performed upon the product shows that the solutions of the given problem are enumerated by the number A, and that the process of operation actually represents each solution.

_Ex. Gr._--Take [lambda]1 = 3, [lambda]2 = 2, [lambda]4 = 1,

p1 = 2, p2 = 2, p3 = 1, p4 = 1,

D2²D1²·a3a2a1 = 8,

and the process yields the eight diagrams:--

+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | 1 | 1 | | | 1 | 1 | | | | 1 | 1 | | 1 | 1 | | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | 1 | 1 | | | 1 | 1 | | | 1 | 1 | | | | 1 | 1 | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | 1 | | | | | | 1 | | 1 | | | | 1 | | | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | | | 1 | | 1 | | | | 1 | | | | 1 | | | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+

+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | 1 | | 1 | | 1 | | 1 | | 1 | 1 | | | 1 | 1 | | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | 1 | 1 | | | 1 | 1 | | | 1 | | 1 | | 1 | | 1 | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | 1 | | | | | 1 | | | 1 | | | | | 1 | | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ | | 1 | | | 1 | | | | | 1 | | | 1 | | | +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+

viz. every solution of the problem. Observe that transposition of the diagrams furnishes a proof of the simplest of the laws of symmetry in the theory of symmetric functions.

For the next example we have a similar problem, but no restriction is placed upon the magnitude of the numbers which may appear in the compartments. The function is now h_[lambda]1·h_[lambda]2...h_[lambda]m, h_[lambda]m being the homogeneous product sum of the quantities a, of order [lambda]. The operator is as before

D_p1·D_p2...D_pn,

and the solutions are enumerated by

D_p1·D_p2...D_pn·h_[lambda]1·h_[lambda]2...h_[lambda]m.

Putting as before [lambda]1 = 2, [lambda]2 = 2, [lambda]1 = 1, p1 = 2, P2 = 2, p3 = 1, p4 = 1, the reader will have no difficulty in constructing the diagrams of the eighteen solutions.

The next and last example of a multitude that might be given shows the extraordinary power of the method by solving the famous problem of the "Latin Square," which for hundreds of years had proved beyond the powers of mathematicians. The problem consists in placing n letters a, b, c, ... n in the compartments of a square lattice of n² compartments, no compartment being empty, so that no letter occurs twice either in the same row or in the same column. The function is here

{[Sigma][a]1^(2^n-1)·[a]2^(2^n-2)...([a]_n-1)²·[a]n}^n,

and the operator D_n^{2^(n-1)}, the enumeration being given by

D_n^{2^(n-1)}·{[Sigma][a]1^(2^n-1)·[a]2^(2^n-2)...([a]_n-1)²·[a]n}^n, ([a] = [alpha])

See _Trans. Camb. Phil. Soc._ vol. xvi. pt. iv. pp. 262-290.

AUTHORITIES.--P. A. MacMahon, "Combinatory Analysis: A Review of the Present State of Knowledge," _Proc. Lond. Math. Soc._ vol. xxviii. (London, 1897). Here will be found a bibliography of the Theory of Partitions. Whitworth, _Choice and Chance_; Édouard Lucas, _Théorie des nombres_ (Paris, 1891); Arthur Cayley, _Collected Mathematical Papers_ (Cambridge, 1898), ii. 419; iii. 36, 37; iv. 166-170; v. 62-65, 617; vii. 575; ix. 480-483; x. 16, 38, 611; xi. 61, 62, 357-364, 589-591; xii. 217-219, 273-274; xiii. 47, 93-113, 269; Sylvester, _Amer. Jour, of Math._ v. 119 251; MacMahon, _Proc. Lond. Math. Soc._ xix. 228 et seq.; _Phil. Trans._ clxxxiv. 835-901; clxxxv. 111-160; clxxxvii. 619-673; cxcii. 351-401; _Trans. Camb. Phil. Soc._ xvi. 262-290. (P. A. M.)

COMBUSTION (from the Lat. _comburere_, to burn up), in chemistry, the process of burning or, more scientifically, the oxidation of a substance, generally with the production of flame and the evolution of heat. The term is more customarily given to productions of flame such as we have in the burning of oils, gas, fuel, &c., but it is conveniently extended to other cases of oxidation, such as are met with when metals are heated for a long time in air or oxygen. The term "spontaneous combustion" is used when a substance smoulders or inflames apparently without the intervention of any external heat or light; in such cases, as, for example, in heaps of cotton-waste soaked in oil, the oxidation has proceeded slowly, but steadily, for some time, until the heat evolved has raised the mass to the temperature of ignition.

The explanation of the phenomena of combustion was attempted at very early times, and the early theories were generally bound up in the explanation of the nature of fire or flame. The idea that some extraneous substance is essential to the process is of ancient date; Clement of Alexandria (c. 3rd century A.D.) held that some "air" was necessary, and the same view was accepted during the middle ages, when it had been also found that the products of combustion weighed more than the original combustible, a fact which pointed to the conclusion that some substance had combined with the combustible during the process. This theory was supported by the French physician Jean Ray, who showed also that in the cases of tin and lead there was a limit to the increase in weight. Robert Boyle, who made many researches on the origin and nature of fire, regarded the increase as due to the fixation of the particles of fire. Ideas identical with the modern ones were expressed by John Mayow in his _Tractatus quinque medico-physici_ (1674), but his death in 1679 undoubtedly accounts for the neglect of his suggestions by his contemporaries. Mayow perceived the similarity of the processes of respiration and combustion, and showed that one constituent of the atmosphere, which he termed _spiritus nitro-aereus_, was essential to combustion and life, and that the second constituent, which he termed _spiritus nitri acidi_, inhibited combustion and life. At the beginning of the 18th century a new theory of combustion was promulgated by Georg Ernst Stahl. This theory regarded combustibility as due to a principle named phlogiston (from the Gr. [Greek: phlogistos], burnt), which was present in all combustible bodies in an amount proportional to their degree of combustibility; for instance, coal was regarded as practically pure phlogiston. On this theory, all substances which could be burnt were composed of phlogiston and some other substance, and the operation of burning was simply equivalent to the liberation of the phlogiston. The Stahlian theory, originally a theory of combustion, came to be a general theory of chemical reactions, since it provided simple explanations of the ordinary chemical processes (when regarded qualitatively) and permitted generalizations which largely stimulated its acceptance. Its inherent defect--that the products of combustion were invariably heavier than the original substance instead of less as the theory demanded--was ignored, and until late in the 18th century it dominated chemical thought. Its overthrow was effected by Lavoisier, who showed that combustion was simply an oxidation, the oxygen of the atmosphere (which was isolated at about this time by K. W. Scheele and J. Priestley) combining with the substance burnt.

COMEDY, the general term applied to a type of drama the chief object of which, according to modern notions, is to amuse. It is contrasted on the one hand with tragedy and on the other with farce, burlesque, &c. As compared with tragedy it is distinguished by having a happy ending (this being considered for a long time the essential difference), by quaint situations, and by lightness of dialogue and character-drawing. As compared with farce it abstains from crude and boisterous jesting, and is marked by some subtlety of dialogue and plot. It is, however, difficult to draw a hard and fast line of demarcation, there being a distinct tendency to combine the characteristics of farce with those of true comedy. This is perhaps more especially the case in the so-called "musical comedy," which became popular in Great Britain and America in the later 19th century, where true comedy is frequently subservient to broad farce and spectacular effects.

The word "comedy" is derived from the Gr. [Greek: kômôidia], which is a compound either of [Greek: kômos] (revel) and [Greek: aoidos] (singer; [Greek: aeidein], [Greek: aidein], to sing), or of [Greek: kômê] (village) and [Greek: aoidos]: it is possible that [Greek: kômos] itself is derived from [Greek: komê], and originally meant a village revel. The word comes into modern usage through the Lat. _comoedia_ and Ital. _commedia_. It has passed through various shades of meaning. In the middle ages it meant simply a story with a happy ending. Thus some of Chaucer's Tales are called comedies, and in this sense Dante used the term in the title of his poem, _La Commedia_ (cf. his _Epistola_ X., in which he speaks of the comic style as "loquutio vulgaris, in qua et mulierculae communicant"; again "comoedia vero remisse et humiliter"; "differt a tragoedia per hoc, quod t. in principio est admirabilis et quieta, in fine sive exitu est foetida et horribilis"). Subsequently the term is applied to mystery plays with a happy ending. The modern usage combines this sense with that in which Renaissance scholars applied it to the ancient comedies.

The adjective "comic" (Gr. [Greek: kômikos]), which strictly means that which relates to comedy, is in modern usage generally confined to the sense of "laughter-provoking": it is distinguished from "humorous" or "witty" inasmuch as it is applied to an incident or remark which provokes spontaneous laughter without a special mental effort. The phenomena connected with laughter and that which provokes it, the comic, have been carefully investigated by psychologists, in contrast with other phenomena connected with the emotions. It is very generally agreed that the predominating characteristics are incongruity or contrast in the object, and shock or emotional seizure on the part of the subject. It has also been held that the feeling of superiority is an essential, if not the essential, factor: thus Hobbes speaks of laughter as a "sudden glory." Physiological explanations have been given by Kant, Spencer and Darwin. Modern investigators have paid much attention to the origin both of laughter and of smiling, babies being watched from infancy and the date of their first smile being carefully recorded. For an admirable analysis and account of the theories see James Sully, _On Laughter_ (1902), who deals generally with the development of the "play instinct" and its emotional expression.

See DRAMA; also HUMOUR; CARICATURE; PLAY, &c.

COMENIUS (or KOMENSKY), JOHANN AMOS (1592-1671), a famous writer on education, and the last bishop of the old church of the Moravian and Bohemian Brethren, was born at Comna, or, according to another account, at Niwnitz, in Moravia, of poor parents belonging to the sect of the Moravian Brethren. Having studied at Herborn and Heidelberg, and travelled in Holland and England, he became rector of a school at Prerau, and after that pastor and rector of a school at Fulnek. In 1621 the Spanish invasion and persecution of the Protestants robbed him of all he possessed, and drove him into Poland. Soon after he was made bishop of the church of the Brethren. He supported himself by teaching Latin at Lissa, and it was here that he published his _Pansophiae prodromus_ (1630), a work on education, and his _Janua linguarum reserata_ (1631), the latter of which gained for him a widespread reputation, being produced in twelve European languages, and also in Arabic, Persian and Turkish. He subsequently published several other works of a similar kind, as the _Eruditionis scholasticae janua_ and the _Janua linguarum trilinguis_. His method of teaching languages, which he seems to have been the first to adopt, consisted in giving, in parallel columns, sentences conveying useful information, in the vernacular and the languages intended to be taught (i.e. in Comenius's works, Latin and sometimes Greek). In some of his books, as the _Orbis sensualium pictus_ (1658), pictures are added; this work is, indeed, the first children's picture-book. In 1638 Comenius was requested by the government of Sweden to draw up a scheme for the management of the schools of that country; and a few years after he was invited to join the commission that the English parliament then intended to appoint, in order to reform the system of education. He visited England in 1641, but the disturbed state of politics prevented the appointment of the commission, and Comenius passed over to Sweden in August 1642. The great Swedish minister, Oxenstjerna, obtained for him a pension, and a commission to furnish a plan for regulating the Swedish schools according to his own method. Devoting himself to the elaboration of his scheme, Comenius settled first at Elbing, and then at Lissa; but, at the burning of the latter city by the Poles, he lost nearly all his manuscripts, and he finally removed to Amsterdam, where he died in 1671.

As an educationist, Comenius holds a prominent place in history. He was disgusted at the pedantic teaching of his own day, and he insisted that the teaching of words and things must go together. Languages should be taught, like the mother tongue, by conversation on ordinary topics; pictures, object lessons, should be used; teaching should go hand in hand with a happy life. In his course he included singing, economy, politics, world-history, geography, and the arts and handicrafts. He was one of the first to advocate teaching science in schools.

As a theologian, Comenius was greatly influenced by Boehme. In his _Synopsis physicae ad lumen divinum reformatae_ he gives a physical theory of his own, said to be taken from the book of Genesis. He was also famous for his prophecies and the support he gave to visionaries. In his _Lux in tenebris_ he published the visions of Kotterus, Dabricius and Christina Poniatovia. Attempting to interpret the book of Revelation, he promised the millennium in 1672, and guaranteed miraculous assistance to those who would undertake the destruction of the Pope and the house of Austria, even venturing to prophesy that Cromwell, Gustavus Adolphus, and Rakoczy, prince of Transylvania, would perform the task. He also wrote to Louis XIV., informing him that the empire of the world should be his reward if he would overthrow the enemies of God.

Comenius also wrote against the Socinians, and published three historical works--_Ratio disciplinae ordinisque in unitate fratrum Bohemorum_, which was republished with remarks by Buddaeus, _Historia persecutionum ecclesiae Bohemicae_ (1648), and _Martyrologium Bohemicum_. See Raumer's _Geschichte der Pädogogik_, and Carpzov's _Religionsuntersuchung der böhmischen und mährischen Brüder_.

COMET (Gr. [Greek: komêtês], long-haired), in astronomy, one of a class of seemingly nebulous bodies, moving under the influence of the sun's attraction in very eccentric orbits. A comet is visible only in a small arc of its orbit near perihelion, differing but slightly from the arc of a parabola. An obvious but not sharp classification of comets is into bright comets visible to the naked eye, and telescopic comets which can be seen only with a telescope. The telescopic class is much the more numerous of the two, only from 20 to 30 bright comets usually appearing in any one century, while several telescopic comets, frequently 6 or 8, are generally observed in the course of a year.

A bright comet consists of (1) a star-like nucleus; (2) a nebulous haze, called the _coma_, surrounding this nucleus, the latter fading into the haze by insensible gradations; (3) a tail or luminous stream flowing from the coma in a direction opposite to that of the sun. The nuclei and comae of different comets exhibit few peculiarities to the unaided vision except in respect to brightness; but the tails of comets differ widely, both in brightness and in extent. They range from a barely visible brush or feather of light to a phenomenon extending over a considerable arc of the heavens, which, comparatively bright near the head of the comet, becomes gradually fainter and more diffuse towards its end, fading out by gradations so insensible that a precise length cannot be assigned to it. When a telescopic comet is first discovered the nucleus is frequently invisible, the object presenting the appearance of a faint nebulous haze, scarcely distinguishable in aspect from a nebula. When the nucleus appears it may at first be only a comparatively faint condensation, and may or may not develop into a point of light as the comet approaches the sun. A tail also is generally not seen at great distances from the sun, but gradually develops as the comet approaches perihelion, to fade away again as the comet recedes from the sun.

A few comets are known to revolve in orbits with a regular period, while, in the case of others, no evidence is afforded by observation that the orbit deviates from a parabola. Were the orbit a parabola or hyperbola the comet would never return (see ORBIT). Periodicity may be recognized in two ways: observations during the apparition may show that the motion is in an elliptic and not in a parabolic orbit; or a comet may have been observed at more than one return. In the latter case the comet is recognized as distinctly periodic, and therefore a member of the solar system. The shortest periods range between 3 and 10 years. The majority of comets which have been observed are shown by observation to be periodic; the period is usually very long, being sometimes measured by centuries, but generally by thousands of years. It is conceivable that a comet might revolve in a hyperbolic orbit. Although there are several of these bodies observations on which indicate such an orbit, the deviation from the parabolic form has not in any case been so well marked as to be fully established. Circumstances lead to the classification of newly appearing comets as _expected_ and _unexpected_. An expected comet is a periodic one of which the return is looked for at a determinate time and in a certain region of the heavens. When this is not the case the comet is an unexpected one.

_Physical Constitution of Comets._--The subject of the physical constitution of these bodies is one as to the details of which much uncertainty still exists. The considerations on which conclusions in this field rest are very various, and can best be set forth by beginning with what we may consider to be the best established facts.

We must regard it as well established that comets are not, like planets and satellites, permanent in mass, but are continuously losing minute portions of the matter which belongs to them, through a progressive dissipation--at least when they are in the neighbourhood of the sun. When near perihelion the matter of a comet is seen to be undergoing a process in the nature of evaporation, successive envelopes of vapour rising from the nucleus to form the coma, and then gradually repelled from the sun to form the tail. If this process went on indefinitely every comet would, in the course of ages, be entirely dissipated. This result has actually happened in the case of some known comets, the best established example of which is that of Biela, in which the process of disintegration was clearly followed. As the amount of matter lost by a comet at any one return cannot be estimated, and may be very small, it is impossible to set any limit to the period during which its life may continue. It is still an unsettled question whether, in every case, the evaporation will ultimately cease, leaving a residuum as permanent as any other mass of matter.

The next question in logical order is one of great difficulty. It is whether the nucleus of a comet is an opaque solid body, a cluster of such bodies, or a mass of particles of extreme tenuity. Some light is thrown on this and other questions by the spectroscope. This instrument shows in the spectrum of nearly every comet three bright bands, recognized as those of hydrocarbons. The obvious conclusion is that the light forming these bands is not reflected sunlight, but light radiated by the gaseous hydrocarbons. Since a gas at so great a distance from the sun cannot be heated to incandescence, the question arises how incandescence is excited. The generalizations of recent years growing out of the phenomena of radioactivity make it highly probable that the source is to be found in some form of electrical excitation, produced by electrons or other corpuscles thrown out by the sun. The resemblance of the cometary spectrum to the spectrum of hydrocarbons in the Geissler tube lends great plausibility to this view. It is remarkable that the great comet of 1882 also showed the bright lines of sodium with such intensity that they were observed in daylight by R. Copeland and W. O. Lohse. In addition to these gaseous spectra, all but the fainter comets show a continuous spectrum, crossed by the Fraunhofer lines, which is doubtless due to reflected sunlight. It happens that, since the spectroscope has been perfected, no comet of great brilliancy has been favourably situated for observation. Until the opportunity is offered, the conclusions to be derived from spectroscopic observation cannot be further extended.

PLATE I.

PLATE II.

In the telescope the nucleus of a bright comet appears as an opaque mass, one or more seconds in diameter, the absolute dimensions comparing with those of the satellites of the planets, sometimes, indeed, equal to our moon. But the actual results of micrometric measures are found to differ very widely. In the case of Donati's comet of 1858 the nucleus seemed to grow smaller as perihelion was approached. This is evidently due to the fact that the coma immediately around the nucleus was so bright as apparently to form a part of it at considerable distances from the sun. G. P. Bond estimated the diameter of the actual nucleus at 500