252 PROFESSOR TAIT ON THE 



" special state," and can like it be assumed to hold with sufficient accuracy to 

 be made the basis of calculation. The mere fact that a "steady" state, say of 

 diffusion, can be realized experimentally is a sufficient warrant for this assump- 

 tion ; and there seems to be no reason for supposing that the irregularities of 

 distribution of the translatory velocity among the particles of a group should be 

 more serious for the higher than for the lower speeds, or vice versa. For each 

 particle is sometimes a quick, sometimes a slow, moving one : — and exchanges 

 these states many thousand times per second. All that is really required by 

 considerations of this kind is allowed for by our way of looking at the mean 

 free paths for different speeds. 



I may take this opportunity of answering an objection which has been 

 raised in correspondence by Professor Newcomb, and by Messrs Watson and 

 Burbury, to a passage in § 3 of the First Part of this paper.* The words 

 objected to are put in Italics : — 



"But the argument above shows, further, that this density must be ex- 

 pressible in the form 



whatever rectangular axes be chosen, passing through the origin." 



The statement itself is not objected to, but it is alleged that it does not 

 follow from the premises assumed. 



This part of my paper was introduced when I revised it for press, some 

 months after it was read ; the date of revision, not of reading, being put at the 

 head. It was written mainly for the purpose of stringing together what had 

 been a set of detached fragments, and was in consequence not so fully detailed 

 as they were. I made some general statements as to the complete verification 

 of these preliminary propositions which was to be obtained from the more 

 complex investigations to which they led ; thus showing that I attached com- 

 paratively little weight to such introductory matters. If necessary, a detailed 

 proof can be given on the lines of § 21 of the paper. The " argument " in 

 question, however, may be given as below. It is really involved in the 

 italicised words of the following passage of § 1 : — " in place of the hopeless 

 question of the behaviour of innumerable absolutely isolated individuals, the 

 comparatively simple statistical question of the average behaviour of the various 

 groups of a community." 



Suppose two ideal planes, parallel to x = 0, to move with common speed, x, 

 through the gas. The portion of gas between them will consist of two quite 

 distinct classes of particles : — the greatly more numerous class being mere 



* In the Phil. May., for April 1887, the same objection is raised by Prof. Boltzmann ; who has 

 appended it to the English translation of his paper presently to be referred to. But he goes farther 

 than the other objectors, and accuses me of reasoning in a circle. 



