Dn2 n-1 (Σ α12 n-1 α22 n-2 ... α2n-1 αn)n.
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. φλογιστός, 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. κωμῳδία, which is a compound either of κῶμος (revel) and ἀοιδός (singer; ἀείδειν, ᾄδειν, to sing), or of κώμη (village) and ἀοιδός: it is possible that κῶμος itself is derived from κώμη, 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.