346 Prof. Tait on the Foundations of 



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 col- 

 lisions, they would behave towards the particles of another 

 system much as Le Sage supposed his ultra-mundane cor- 

 puscles 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.] 



3. With these assumptions we may proceed as follows: — Let 

 P and Q be the masses of particles from the two systems respec- 

 tively ; 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 / ' , y f respectively, we have at once 



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

 an immediate consequence of which is 



P(u /2_ U 2 )= „ 4PQ .p u2 _ Qv2 _ (P _Q )UV ) 



[ + ^ = _Q( V ' 2 -V 2 ). 



Hence, denoting by a bar the average value of a quantity, we 

 see that transference of energy between the systems must 

 cease when 



Pu 2 -Qv 2 -(P-Q)uv = 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? 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 velocity, 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 seemed as if the 

 greater masses would have on the average less energy than 

 the smaller. 



4. But these first impressions are entirely dissipated when 

 we proceed to calculate the average values. For it is found 



