for the Theoretical Proof of Avogadro's Law. 323 



square bracket. The arithmetic mean of the four expressions 

 thus obtained is to be substituted in equation (24) for the 

 transformed term, The term 



1<£>{X, t) . [3>(X', t)®(a: + X-X, *)-<S>(X, t)&(a 9 t) 



is to be treated in exactly the same way. 



The remaining terms have now only to be once transformed 

 by means of equation (27), so that the factors —l<f>(x', t) and 

 — l&{x-\-X— x, t) appear instead of the factors l<j)(x, t) and 

 Z<E> (X, t) ; and again the arithmetic mean of the original and 

 transformed expressions is to be taken. In this way again 



— - proves to be a sum of terms of which each vanishes sepa- 



rately if -j- vanishes, ifr and M? may be equal to zero. As soon 



as x vanishes for no definite set of variables, that is only so 

 soon as the molecules of the first kind impinge freely upon 

 those of the second kind, these terms only vanish separately if 

 Maxwell's distribution of velocities holds amongst the mole- 

 cules of the first kind as well as those of the- second kind. 



The foregoing considerations serve also to completely 

 establish the still more general proposition which Mr. Burbury 

 has stated in the place already referred to. In fact, let us 

 assume that the first kind of gas consists, as above, of very 

 many molecules (molecules A) which do not impinge amongst 

 themselves ; but that the second kind, on the other hand, 

 consists but of a single molecule B which comes into collision 

 with the molecules of the first sort. Let the time of its free 

 motion be great in comparison with the time of a collision. 

 Let the whole be enclosed in a vessel R with rigid elastic 

 walls. Let R denote also the volume of the vessel. There 

 must at length ensue a stationary condition in which the 

 molecules of the first kind are, on the average, uniformly dis- 

 tributed through the vessel, and in which any direction is as 

 probable as another for their velocity. In this condition, let 

 there be in unit volume Amrv^f^vjdv molecules whose velo- 

 cities lie between the limits (6). The molecule B will of 

 course continually change its velocity ; but if we take a very 

 long time after the commencement of the stationary condition, 

 then during that time its velocity , on the average, will, with 

 equal probability _, have assumed all possible directions in 

 space; and the probability that its velocity will lie between 

 the limits (7) will be expressed as some function of V, which 

 we will denote by 



4ttV 2 RF 1 (Y)^V. 



Let us now imagine a very great number of similar vessels 



Z2 



