306 On the Action of Solids in liberating Gas from Solution. It was stated by De Luc* that water purged of air cannot be boiled; and Donny † gives an experiment in which water, heated in an oil-bath many degrees above its boiling-point, at length suddenly bursts into steam with the force of an explosion. The water, it is said, cannot boil unless air be present in the liquid, because, according to De Luc's theory, or, as it is now called, the theory of Clausius, air is required for the steam to expand into. There is a well-known experiment by Grove, in which water covered with a layer of oil was repeatedly boiled, and it was found impossible to get rid of the dissolved air. I repeated this experiment some years ago, and found that air was carried down by the oil itself. As soon as the water was fairly boiling, the oil was broken up into globules, and one or more bubbles of air attached to the oil was carried down into the liquid. On removing the lamp, the oil rose to the surface with a ring of air-bubbles beneath. But supposing all these facts to be accurately represented, and that a liquid, at or near the boiling-point, is constituted like soda- or Seltzerwater, then I cannot admit that a solid, such as a glass rod, introduced into a boiling liquid (water for example), becomes covered with bubbles of steam by virtue of the air carried down by the rod. If the rod be unclean (that is, contaminated with a greasy film), the steam-bubbles cover it precisely after the manner of the gas-bubbles, because there is adhesion between the steam-bubbles and the film and not between the water and the film, and hence there is a separation. A chemically clean glass rod has no such action, not because the act of cleaning it deprives it of its adhering air, but because there is perfect adhesion between a vaporous supersaturated solution and a chemically clean surface. It is perhaps a fortunate thing for science (the object of which, some people suppose, is the discovery of truth) that men are so enamoured of their own theories that they defend them with the strongest dialectical weapons that they can furbish up; and it is out of the battle of rival theorists that truth finally emerges. So that while each man pursues science with perfect honesty and sincerity, but nevertheless for his own glory, Nature, like one of her own stars, ohne Hast aber ohne Rast, *"Quand on a préalablement purgé l'eau de tout l'air qu'elle contenoit, elle ne peut plus bouiller; et la raison en est que les vapeurs ne peuvent se former qu'à des surfaces libres."-Recherches &c., Geneva, 1772. For authorities on some interesting points in connexion with boiling, see "Historical Notes" &c., Phil. Mag. for March 1869. Mém. de l'Acad. Roy. de Bruxelles, vol. xvii. 1845. In a paper contained in the Phil. Mag. for October 1868, a precise meaning is given to the terms "clean and "unclean." See also Phil. Mag. for April 1873 and November 1874. moves silently on, and at length asserts herself, with perfect indifference to the fame of her votaries, and the final verdict is in favour of truth. Many a fact wanders about the world as a scientific waif, not finding rest in any sufficient theory, until at length it falls into its proper place and becomes associated with a host of other facts, outcasts like itself, or interlopers in theoretical lodging-houses; but now, in its right place, it performs valuable and unexpected work. The motions of camphor &c. on water were known during upwards of a century and a half before they found their true resting-place in the surface-tension of liquids; and it is quite possible that the varied phenomena connected with supersaturation may, at some future time, be embraced by some general law, or be held together by some satisfactory theory, when the labours of those who have been working on the subject from the commencement of this century to the present time will have been forgotten, or referred to by the curious in journals that will then have become old, with a wondering smile that men could have been so blind to the obvious teaching of the facts. Highgate, N. March 9, 1875. XXXIII. Note on Partitions. By J. W. L. GLAISHER, M.A.* DENG ENOTING by P(a, b, c...q) the number of ways of forming x by addition of the elements a, b, c...q, each element being repeatable any number of times, I propose to consider the value of P(1, 3, 5...), viz. the number of ways of partitioning x into parts all of which shall be uneven. To fix the ideas, suppose x=10, and consider any one partition, say 1+1+3+5; write this in the form 1, 1, 1, 1 the top line consisting of units only (as many as the partition contains parts), and the second line containing only even numbers; the other partitions of 10 into four parts, viz. 1+1+1+7, 1+3+3+3 are to be written so that the number of partitions of 10 into four uneven parts is equal to the number of partitions of 10-4 into even parts not exceeding four in number. It is at once evident that this process is general, and that the number of partitions of 2x into 2r uneven parts is equal to the number of ways of partitioning * Communicated by the Author. 2x-2r into even parts, subject to the condition that the number of these latter (even) parts must not exceed 2r (the number of units in the top line). Thus, writing for the moment P'n(a, b, c,...) to represent the number of ways of partitioning x into the parts a, b, c..., no more than n of such parts appearing in any one partition, we see that P(1, 3, 5...)2x=1+ P'2 (2, 4, 6...) (2x −2) 2 +P(2, 4, 6...) (2x−4) ... + P'2-(2, 4, 6...) 2, the first term in the latter expression corresponding to the single partition of 2x as the sum of 2x units. Now obviously the number of ways of partitioning 2n into the parts 2, 4, 6... is equal to the number of ways of partitioning n into the parts 1, 2, 3..., so that P(1, 3, 5...)2x=1+ P'q(1, 2, 3 ...) (x − 1) + P1⁄4 (1, 2, 3 ...) (x−2) ... + P'2x-2 (1, 2, 3 ...) 1, which, transformed by means of the well-known theorem P'2 (1, 2, 3 ...) x = P(1, 2, 3 ... becomes B...n)x, (1) P(1, 3, 5 ...)2x=1+P(1, 2) (x− 1) + P(1, 2, 3, 4) (x −2)... By a similar method we easily arrive at the expression for the decomposition of an uneven number, viz. P(1, 3, 5 ...) (2x+1)=2+ P(1, 2, 3) (x − 1) + P(1, 2, 3, 4, 5)(x-2)... + P(1, 2, 3 ... 2x-1)1, (3) formulæ expressing the value of P(1, 3, 5...) in terms of partitions into the elements 1,2; 1, 2, 3, &c., and which have thus been obtained by general reasoning without analysis: they can, of course, be derived more directly by means of the identity + 1−x2.1−x1.1-x6 + &c. The proposition (1) that has been used (viz. that the number of partitions of x into parts not exceeding n in number is equal to the number of partitions of a into the parts 1, 2, 3 ...n) admits of almost intuitive proof by Mr. Ferrers's method of breaking up and arranging the parts in a partition, so that when read as lines we obtain the number of partitions into parts not exceeding n in number, and when read as columns the number of partitions into 1, 2, 3 ... n: sce Phil. Mag. S. 4. vol. v. p. 201 (1853). An application of the same principle affords another transformation of some interest for P(1, 3, 5...). It was proved by Euler that P(1, 3, 5...) is equal to the number of partitions of z into the parts 1, 2, 3... in which each part only appears once in each partition, or, say, is equal to P(1, 2, 3...) without repetitions. Now, applying Mr. Ferrers's method to the latter, we see that a partition without repetitions corresponds to a partition without omissions. For example, writing 1+2+3+5 as 1 1, 1 1, 1, 1 1, 1, 1, 1, 1, and adding the columns, we obtain 1+1+2+3+4; and it is clear that though any part may appear any number of times, on part can appear unless all the lesser parts appear also. We thus see that P(1, 3, 5...)x= P(1, 2, 3...)x without omissions; and this latter quantity is equal to 1+P(1, 2) (x − 1 −2) + P(1, 2, 3) (x—1—2—3) +P(1, 2, 3, 4) (x−1—2—3—4) + &c. ; (4) for the second term represents the number of partitions of x into parts in which both 1 and 2 must occur and no higher part can occur; the third term represents the number of partitions in which 1, 2, 3 must, and no higher part can, occur, &c. By use of the well-known theorem P(1, 2, 3 ... n) (x —n) + P(1, 2, 3 ...n-1)x = P(1, 2, 3 ... n)x (which can be readily established by general reasoning), the last equality can be transformed into +P(1, 2, 3, 4) (x−1−2−3) + &c., . which can be at once identified with the formula (5) That Mr. Ferrers's method should, as it were, afford a demonstration of this identity is what we should expect, as it was in effect remarked by Mr. Sylvester (Brit. Assoc. Report, 1871, p. 25, Sect. Proc.) that it established Euler's more general The value of P(1, 3, 5...)x given by (4) is in fact that found by Euler (Opera minora collecta, vol. i. p. 93). The class of partitions without omissions is one which has not, I believe, been particularly noticed before; and it is curious that it should be identical with the partitionment into uneven numbers. There is also another mode of transforming this latter class of partitions, which can be best made clear by an example. Thus, consider the partition 1+3+3+7, and arrange the parts as in the following scheme : the 1 occupying the left-hand lower corner, the three squares surrounding it being occupied by 2's, as there are two 3's in the partition, the next five squares being left vacant as there is no 5, and the next seven squares being filled by 1's as there is one 7. Then dividing the square into similar belts to those which represented the different parts in its formation, only beginning at the opposite (viz. the upper right-hand) corner, we have the parts 1, 1 + 1, 1 + 2 + 1,1+2+1+2+1, that is 1, 2, 4, 7, which are always thus given in order of magnitude, and are subject to the following laws of formation, viz. :—that any number of parts (except the first) may be equal; but that, taking no account of these repetitions, i. e. regarding, for example, a+b+b+c simply as a+b+c, then the parts a, b, c, d, &c. are such that b=2a+a, c=b+a+B, d=c+B+y, &c. (so that the second part must be at least the double of the first). The same decomposition may also be derived without a diagram by observing that, for example, a+3b+7d=d+ (2d) + (2d+b) + (2d+2b+a); but the mode of formation is too complicated to render the transformation one of much interest. The number of transformations of P(1, 3, 5...)x, however, is noteworthy, as we have seen the equivalence of the numbers of (i) partitions into the uneven elements 1, 3, 5..., repetitions not excluded; (ii) partitions into the elements 1,2,3... without repetitions; (iii) partitions into the parts |