in the Kinetic Theory of Gases. 303 



negative unless F P r f p < = F P< /> for every two directions of the 

 diameter through C ; that is unless, for given V, all direc- 

 tions of the relative velocity are equally probable. Now in 

 stationary motion H must be invariable with the time. There- 

 fore the motion is not stationary, or the distribution of 

 velocities is not unaltered by encounters, unless the condition 

 be satisfied. So (b) is proved. 



7. We will next assume that the two molecules M and m 

 act on each other with finite forces. Then in the infinitely 

 short time dt the relative velocity is by their mutual action 

 during encounter turned through some small angle ; and 

 generally also altered in magnitude, in such manner as that 



Mm 



*M+^ p2 + % " C ° nstant ' 



% being the potential of the mutual action. 



It is not necessary to restrict the number of m molecules 

 which may simultaneously be in encounter with, and affect 

 the motion of, a given M. But by the superposition of small 

 motions, we may regard the total time-variation of the velocity 

 of M as the sum, or resultant, of the time-variation due to the 

 action of all the m molecules separately, each acting for the 

 time dt. We can then prove that for any given class of M's 

 with any class of m's, as the result of their mutual action, 



JTT 



-j— is negative unless the condition is satisfied, in which 



case it is zero. 



The mutual action is assumed to depend on the relative 

 positions, not on the velocities, of M and m. 



We assume also the distribution to be uniform in space, so 

 that the number per unit of volume of any given class of 

 molecules is independent of the position. 



Let, then, at any instant the distance between M and m be 

 between q and q + dq. 



Let the common velocity be OC = V, and let the velocity 

 of MbeOP(B 2 dBdS). 



The velocity of m or Op is then determined by these con- 

 ditions. In order completely to define the motion we require 

 one other coordinate of position. Let it be 6. Consequently 

 the number of pairs M and m satisfying these conditions at 

 any instant is F P / P B 2 dR d$ dq d6. Call this the first state. 



By mutual action V is not affected. The motion under 

 the influence of the mutual action is determinate, and after 

 time at the same variables shall be denoted by accented letters. 

 Then q', B/, S 7 , 6' are functions of q, B, S, 6, known if the 



Y2 



