NA TURE 



i}i 



THURSDAY, NOVEMBER 23, 1893. 



WATSON'S KINETIC THEORY OF GASES. 

 A Treatise on the Kinetic Theory of Gases. By Henry 

 William Watson, D.Sc, F.R.S. Second Edition. 

 (Oxford: at the Clarendon Press, 1893.) 



THE rather pointed reference to myself, which Dr. 

 Watson makes at the end of this new edition of 

 his work, seems to call for an answer. Had this call 

 come some five or six years ago, when the questions once 

 again at issue were debated in a somewhat lively way, I 

 should have had little difficulty in rising to it ; — but I 

 have in the interval been so busy with questions of a 

 totally different nature that I am taken at a disadvantage, 

 especially as I cannot at present find time to read up 

 again the discussions of that period. I remember enough 

 about them, however, to^make the very positive assertion 

 that the questions then raised turned on points of logic, 

 relevancy, and consistency, much more than upon physical 

 ideas or mathematical processes ; and a perusal of Dr. 

 Watson's volume shows me that he has reproduced from 

 Boltzmann and others much of what I then objected to. 1 

 believe that I gave, in 1886 {Trans. R S.E. vol. xxxiii.), 

 the first (and possibly even now the sole) thoroughly 

 legitimate, and at least approximately complete, 

 demonstration of what is known as Clerk-Maxwell's 

 Theorem, relating to the ultimate partition of energy 

 between or among two or more sets of hard, smooth, and 

 perfectly elastic spherical particles. And I then pointed 

 out, in considerable detail, the logical deficiencies or con- 

 tradictions which vitiated Maxwell's own proof of 1859, 

 as well as those involved in the mode of demonstration 

 which he subsequently adopted from Boltzmann. Dr. 

 Boltzmann entered, at the time, on an elaborate defence 

 of his position ; but he did not, in my opinion, satis- 

 factorily dispose of the objections I had raised. Of 

 course I am fully aware how very much easier it is for 

 one to discover flaws in another man's logic than in his 

 own, and how unprepared he usually is to acknowledge 

 his own defects of logic even when they are pointed out 

 to him. But the only attacks which, so far as I know, 

 have been made on my investigation, were easily shown 

 to be due to misconception of some of the terms or pro- 

 cesses employed. 



Dr. Watson's little work has been for many years the 

 recognized text-book on the kinetic theory of gases : — 

 and there can be no doubt that, considered in the fierce 

 light of the Examination Hall, it is well adapted to the 

 wants alike of actual Moderators and of would-be 

 Wranglers. It is so framed as to be easily dissected 

 into compact and thoroughly self-contained pieces of 

 book-work: — from "easy" up to " rather stiff" : — and, 

 were these to be answered at all nearly in the words and 

 formula of the text, few Examiners would venture to 

 refuse full marks. From this point of view nothing 

 more could be desired ; for incorrect historical notices, 

 such as the ascription of the origin of the theory to J. 

 Bernouilli (properly D. Bernoulli) instead of R. Hooke. 

 will injure no man's place in the Tripos. The purely 

 mathematical part, mainly a series of exercises in the 

 transformation (by functional determinants) of differen- 

 tial elements from one system of variables to another, 



NO. T256, VOL. 49] 



though elegant enough, presents an aspect of sameness. 

 Toiijoiirs perdrix .' To this point we will recur. 



Considered as a scientific treatise, however, and as 

 practically the only one in Britain which deals at all 

 fully with the subject, the work is not quite so deserving 

 of commendation. Much of course has, in all cases, to 

 be allowed for the almost necessary defects of a book 

 which deals in any way with questions of probability. 

 It has been the good fortune of but a very few, even 

 among the most gifted of mathematicians, to be able to 

 thread their way in safety through the countless traps 

 and pitfalls which lurk unnoticed, often undiscoverable 

 till they have done their worst, in every part of every 

 region of this fascinating domain : — not, as in other sub- 

 jects, in the partially explored nooks and crannies alone. 

 But probability is only one application of logic : — and, 

 in the passages we most object to, it is in general 

 ordinary logic which we think is somewhat lightly 

 treated. We do not require to go far in search of an 

 example. 



At the very commencement of the work, while dealing 

 with Maxwell's well-known result for the permanent dis- 

 tribution of velocities among a- number of equal, smooth, 

 spherical particles. Dr. Watson says : — 



" We assume that in the permanent state the distribu- 

 tion of the spheres throughout the space occupied by 

 them is homogeneous in all respects ; that is to say, on 

 an average of any long time there are the same number 

 of spheres in a given volume wherever that volume may 

 be situated, and the law of distribution of velocities is 

 the same throughout that volume as in the whole region 

 under consideration." 



On this statement we would remark that it is rather 

 vague and incomplete : — for surely it is meant that the 

 distribution is isotropic as well as homogeneous ; and 

 the word "long" has absolutely no meaning until the 

 time-unit is assigned. Dr. Watson then proceeds to in- 

 vestigate the circumstances of an individual (but typical) 

 collision. Here, however, logic steps in, and says : — 

 " Halt ! You have already assumed all that you need 

 learn from collisions, so far at least as conf:erns the solu- 

 tion of the problem before you." In fact the assumption, 

 read as above, leads at once to Maxwell's Law, by the 

 very process which its discoverer first employed ; a pro- 

 cess depending on principles freely used throughout the 

 text of this book. When Dr. Watson has found the 

 state of motion (F), of two spheres after collision, in 

 terms of the state (E) before it, he proceeds thus : — 



" For permanence of distribution ... it is sufficient 

 that the number of collisions of pairs of spheres in state 

 E during the time dt should be equal to the number of 

 collisions of pairs in the state F during the same time." 



This leads, of course, to Maxwell's result. But it is 

 not hypercritical to ask whether the above-mentioned re- 

 quirement is not merely" sufficient" but miuh more than 

 sufficient : — so exacting, in fact, as to be absolutely un- 

 attainable. Note the consequences of it. From the very 

 nature of the data, the whole motion in the present case 

 is strictly reversible, so as exactly to retrace its entire his- 

 tory. But, if we were to reverse it, we should still have 

 the "permanent" state: — i.e. one which could never 

 have been otherwise than as it is ! This principle of 

 reversion underlies a great part of the theory ; and a 

 mere reference to it would, in many of the later pages of 



E 



