RECENT STATISTICAL THEORIES 729 



time between impacts) is equal to first approximation to af{^ + d^, 

 7], ^)dr]d^.^'' This loss is partly balanced by an inward drift of particles 

 which are accelerated into the compartment from the one lying 

 beyond the plane ^; the balance is not perfect, for the number drifting 

 in per unit time is equal to af(^, rj, ^)dr]d^ and there is a difference 

 a{dfi'd^)dT outstanding.^* This difference must be balanced by the 

 entrances and exits of particles which undergo collisions. 



Let adl represent the number of electrons which, being initially in 

 the compartment C, suffer impacts during unit time and are thus 

 suddenly bumped out of it; and bdt the number which, being initially 

 in other compartments, suffer such impacts during unit time that 

 they suddenly turn up in C. The function g must be so chocen, 

 that the lack of balance between the electrons drifting out and the 

 electrons drifting in is just compensated by the lack of balance between 

 those bumped into the compartment and those which are bumped out: 



— -T-=b-a. (93) 



m d^ 



We must therefore evaluate (b — a) in terms of the distribution- 

 function. 



We already have a formula ready-made for the number of impacts 

 experienced per unit time by the particles of speed y; it is v/l for each 

 particle, so that: 



a = {vll)fdkdrjd^. (94) 



Since however we have also to compute h, we shall find it expedient 

 to classify these impacts according to the destinations of the particles, 

 so to speak — according to the compartments of velocity-space into 

 which they are bounced. A particle of speed v is located, in the 

 velocity-space, on a sphere of radius v centered at the origin. Collision 

 with an immovable atom changes the direction, but not the magnitude 

 of the velocity; in the velocity-space, the particle suddenly moves to 

 some other point on the same sphere. When electrons jump out of 

 the compartment C because of impacts, they land in the other com- 

 partments which with C form a spherical layer around the origin. 

 When electrons jump into C because of impacts, they come from the 

 other compartments of that same layer. We shall derive an expression 



'^ During a time dt so short that adt is small by comparison with d^, the particles 

 which initially lie between the plane {^ -\- d^ — adt) and the side (^ -|- d^ of the 

 compartment move out of it, while the particles which initially lie between the 

 side ^ of the compartment and the plane ($— adt) move into it; these two classes 

 of particles number /(| -i- d^, ??, ^)adt-dr]d^ and/(|, 77, ^) adt ■ d-i)d'g respectively. 



'* This expression figures in the equations as a net loss, but in fact has a negative 

 sign (since dfjdi, < 0) and therefore is actually a gain. 



