MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 233 



For the right hand member we observe that we have the eightfold 

 integration 



that is an integration over the collision angles and all final velocity 

 components. By a general principle of kinetic theory ^^ we can invert in 

 this integration the final and the initial quantities and write 



dfi,^ dU du = d^ dU' du (108) 



This puts us in a position to eliminate the 5-f unction by integration. 

 We find 



'''' ^ "^ ^^""^ ^hll ^(^')^^^ - ^)n(xj dn, du' (109) 



with the side condition that u, U, u', U' form a quadruple of vectors 

 in the sense discussed in Section IB for which in addition 



U' = 



If we substitute (107) into (104), denoting the dummy variable by u' 

 instead of u, then the two equations (104) and (109) take on a very 

 similar look. A proof of their identity hinges upon proving the identity 

 of the integral terms : 



j hiu') dn' J m(c - u)n(xn) dQ, 



(110) 

 = j h{n') du jj M(C V(c' - u )n(xc) do,' dC 



The form of this relation suggests the assumption that the expressions 

 are identical before integration over u; this assumption is proved by 

 the events below. The complicated function h(u) thus disappears from 

 the problem. The other such function, namely 11 (x) disappears then also; 

 for it is by assumption arbitrary, hence could be replaced by a 5-function 

 for a fixed, but arbitrary x- The two sides of (110) must therefore be 

 equal before we integrate over Xu or Xe , and the two x's are to be taken 

 equal and fixed. Defining angles as shown in the spherical diagram Fig. 

 15 we thus get (110) in the form 



I m(c -u)d€^ jl MiC')m{c - u') d<f> dC (Ilia) 



2' See Reference 4, Section 3.52. 



