J 



for the Theoretical Proof of Avogadro's Law. 311 



In the latter case V and v must be exchanged. If, there- 

 fore, the functions / and F are chosen quite arbitrarily, 

 DdZ will in general not disappear if the molecules of the 

 first and second kinds have equal energy. In other words, 

 even though the molecules of the first and second kinds have 

 equal energy, yet, if f and F are chosen arbitrarily, energy 

 may at first be transferred from molecules of the first kind 

 to molecules of the second kind, or vice versa. It would be 

 transferred to them continuously if by any external action 

 the originally chosen values of the functions / and F were 

 maintained constant. 



If, for example, with uniform distribution in space and 

 equal probability of all directions of velocity, the molecules of 

 the first kind had all the same velocity v, the second kind all 

 the same velocity Y>v, then there would be at first no 

 transference of energy to the molecules of the first kind by 

 the molecules of the second kind if expression (5) should 

 disappear. If we put the positive quantities 



m-M V 2 _ 



we have 



(l-p>=* + /* + \/ g + if + Hp 



3M^ = 2m-M + ^|(5m 2 -2mM + 2M 2 ). 



If m is only a little larger than M, say m = M(l + e), then 

 this gives nearly 



MV 2 =W./l+|); 



for m=4M, we have 



MV 2 =m^(l-21 . . .). 

 If m is very much greater than M, we have nearly 



MV 2 = fmi; 2 . 



The condition that, on the average, no energy shall be 

 communicated in the first moment to the molecules of the first 

 order, requires then that the molecules of smaller mass should 

 have greater energy than the others. But it must be observed 

 that this only holds good for the first moment ; the velocities 

 even of the molecules of the first kind immediately become 

 different amongst themselves as the result of impacts, and so 



