function is here xn, and the operator (ddx)n = δn x ,
yielding δn x xn = n!
the number which enumerates the permutations. In fact—
δxxn = δx . 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 δx 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—
the number in each row of compartments denoting an operation of δx. 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 λs units (s = 1, 2, 3, ... m), and the tth row pt units (t = 1, 2, 3, ... n).
Writing
and Dp = 1p! (δα1 + α1δα2 + α2δα3 + ...)p, the multiplication being symbolic, so that Dp is an operator of order p, the function is
and the operator Dp1Dp2Dp3 ... Dpn. The number Dp1Dp2 ... Dpnaλ1aλ2aλ3 ... aλm enumerates the solutions. For the mode of operation of Dp upon a product reference must be made to the section on “Differential Operators” in the article Algebraic Forms. Writing
... ΑΣαp11 αp22 ... αpnn + ...,
or, in partition notation,
... + Α(p1p2 ... pn) ... + Dp1Dp2 ... Dpn (1λ1) (1λ2) ... (1λm) = Α
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
|
λ1 = 3, λ2 = 2, λ3 = 1, |
and the process yields the eight diagrams:—
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 hλ1hλ2 ... hλm, hλm being the homogeneous product sum of the quantities a, of order λ. The operator is as before
Dp1Dp2 ... Dpm,
and the solutions are enumerated by
hλ1hλ2 ... hλm.
Putting as before λ1 = 3, λ2 = 2, λ3 = 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
α2n−1 αn)n,
and the operator Dn2n−1, the enumeration being given by
(Σα2n−11 α2n−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
