Kinetic Theory of Gases. 721 



A large part of his criticism, however, centres round the 

 assumption which he has called assumption A. He suggests 

 that I make this assumption unawares, that I prove it to be 

 both true and untrue, and so on. Now the /whole point of 

 my former paper as regards assumption A wns that it is not 

 a genuine assumption at all, but that it merely amounts to a 

 hcence to misapply the calculus of probabilities. It is, if I 

 was right, as illogical to base a kinetic theory on this assump- 

 tion, coupled with the laws of dynamics, as it would be to 

 base a system of dynamics on the assumption that there is no 

 causation in nature, coupling this assumption with the funda- 

 mental laws of dynamics. For the laws of dynamics imply 

 causation with no greater certainty (I am still stating my own 

 case) than they imply the negation of assumption A. 



To establish my contention that assumption A is illogical, I 

 attempted to show, in §§ 2 and 3 of my paper, that, when 

 made, it leads to inconsistent results, and it is, I think, on the 

 strength of this that Mr. Biirbury charges me with the incon- 

 sistency of showing assumption A to be both true and untrue. 

 If by making assumption A, I can prove this assumption to be 

 both true and untrue, surely my contention that assumption A 

 is illogical is sufficiently proved ? 



To estabKsh my further contention that assumption A is 

 unnecessary, I next attempt to develop the theory of gases 

 without its help. Mr. Burbury now charges me with having 

 made the assumption unawares. I wish that he had sub- 

 stantiated his charge by pointing to the definite stage in my 

 argument at which he thinks the assumption is implied. Let 

 me, however, recapitulate the steps of this argument freed 

 from analytical details. At any instant 6 N independent 

 (in the usual algebraic sense) variables are required to define 

 the state of the gas. The ranges of these 6 N variables I 

 propose to represent in a space of 6 N dimensions. Each 

 point of this space, then, represents a gas. I take a census of 

 all the points in this space, and prove that the laws of equi- 

 partition &c. hold for all these points except an inappreciable 

 minority lying on systems of singular points, lines, surfaces, 

 &c. And now let us confine our attention to any particular 

 gas in which the molecules are moving in accordance with 

 the laws of dynamics. The changes taking place in the gas 

 are represented by the motion of a point in my generalized 

 space. Except when this point lies on one of the singular 

 lines, surfaces, &c. just mentioned, equipartition of energy 

 will hold in the gas. It follows from Liouville's Theorem, as 

 I have shown, that there is no tendency of the representative 

 points to crowd on to these singular lines, surfaces, &c. ; and 



