FOUNDATIONS OF THE KINETIC THEORY OF GASES. 93 



process of variation. For this purpose we must calculate the changes which 

 take place in the momentum, and in the number of particles, in a layer; or, 

 rather, we must inquire into the nature of the processes which, by balancing- 

 one another's effects, leave these quantities unchanged. 



Recur to § 29, and suppose the particles to be subject to a potential, U, 

 which depends on x only. Then the whole momentum passing per unit of 

 time perpendicularly across unit surface of any plane parallel to yz is 



J Vn Jr=Th> 



where n (the number of particles per cubic unit), and h (which involves the 

 mean-square speed), are functions of x. 



At a parallel plane, distant a. from the first in the direction of x positive, 

 the corresponding value is 



But the difference must be sufficient to neutralise, in the layer between these 

 planes, the momentum which is due to the external potential, i.e., 



Hence 



dx 



1 p d n _ p d\J 



2 dxh dx 



or 



,d\J _ 1 dn 1 dh 

 dx n dx h dx 



-2h^..=± -_— (1) . 



Again, the number of particles which, in unit of time, leave the plane unit 

 towards the side x positive is 



- n I vv I cosPsinft dfi= — n / vV . 

 2 <Jo >Jo 4 <so 



Hence those which leave the corresponding area at distance a are, in number, 



i( 1+a l)(^)- 



But, by our postulate of last section, they can also be numbered as 



where 



