20 The Law of Distribution of Velocities [CH. n 



changed so long as this law remains unchanged. Hence a necessary condi- 

 tion for a steady state is given by dHfdt = 0. We proceed to evaluate 

 dH/dt in the general case. 



After an interval dt the value of /log/ corresponding to any specified 

 values of u, v, w will of course have changed into 



or, what is the same thing, into 



(ITT 

 Hence the increase in H, which may be written , - dt, will be given by 



CLL 



(14), 



' J 



and if we substitute the value of df/dt from equation (12), this becomes 

 = *////////(! + lo g/) (//' ~ff) V* 2 cos e dudvdwdu'dv'dw'da. . .(15). 



23. Equation (13) regards H as the sum of a number of contributions, 

 one from each class of molecule, a.nd in this equation class A is taken as the 

 typical class. If we had chosen class B as the typical class, we might have 

 written H in the form 



r r r 



Iv'dw' (16), 



and the increase in H, instead of being given by equation (14), would then 

 have been given by 



d du'dv'dw' .................. (17). 



To evaluate the right-hand member of this equation we need to know the 

 value of df/dt. Now equation (12) regards the change in / as the sum of a 

 number of contributions, one from every class of collision in which either of 

 the molecules either before or after impact is of class A, and the typical 

 classes of collision are taken to be classes a. and /3. In a similar way we can 

 express df'/dt as the sum of a number of contributions, one from every class 

 of collision in which either of the molecules either before or after impact is of 

 class B. The typical classes of collision may again be taken to be classes a 

 and (3, and if this is done we obtain for df'/dt an expression similar to that 

 given for df/dt in equation (12), except that accented symbols replace un- 

 accented, and vice-versa. In fact molecules of classes A and B exchange 

 roles. 



