FOUNDATIONS OF THE KINETIC THEORY OF OASES. 79 



the number of particles of one kind is not overwhelmingly greater than that of 

 the other kind. 



[This is one of the essential points which seem to be wholly ignored by 

 Boltzmann and his commentators. There is no proof given by them that one 

 system, while regulating by its internal collisions the distribution of energy 

 among its own members, can also by impacts regulate the distribution of 

 energy among the members of another system, when these are not free to 

 collide with one another. In fact, if (to take an extreme case) the particles of 

 one system were so small, in comparison with the average distance between 

 any two contiguous ones, that they practically had no mutual collisions, they 

 would behave towards the particles of another system much as Le Sage 

 supposed his ultra-mundane corpuscles to behave towards particles of gross 

 matter. Thus they would merely alter the apparent amount of the molecular 

 forces between the particles of a gas. And it is specially to be noted that this 

 is a question of effective diameters merely, and not of masses : — so that those 

 particles which are virtually free from the self-regulating power of mutual 

 collisions, and therefore form a disturbing element, may be much more massive 

 than the others.] 



19. With these assumptions we may proceed as follows: — Let P and Q be the 

 masses of particles from the two systems respectively ; and when they impinge, 

 let u, v be their velocity-components measured towards the same parts along 

 the line of centres at impact. If these velocities become, after impact, u', v' 

 respectively, we have at once 



P(u'-u)=-|^(u-v)=-Q(v'-v); 

 an immediate consequence of which is 



p (u '2_ u2)= __J|Q_(p u 2_Q v 2_ ( p_Q )uv ) = _Q (v '2_ v 2 ) . 



Hence, denoting by a bar the average value of a quantity, we see that trans- 

 ference of energy between the systems must cease when 



P[r 2 -Qv2-(P-Q)u^ = 0, (1), 



and the question is reduced to finding these averages. 



[I thought at first that uv might be assumed to vanish, and that u 2 and v 2 

 might each be taken as one-third of the mean square speed in its system. 

 This set of suppositions would lead to Maxwell's Theorem at once. But 

 it is clear that, when two particles have each a given speed, they are more 

 likely to collide when they are moving towards opposite parts than when 

 towards the same parts. Hence uv must be an essentially negative quantity, 

 and therefore Pu 2 necessarily less than Qv 2 , if P be greater than Q. Thus it 



