MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 207 



If we substitute this into (66) we see that the K's decrease as (m/M)^ ', 

 with a possible vl slowing up the final convergence. In any case /12(c) 

 ^ comes out small compared to hi{c) which is all that is needed to make 

 equation (65) approximately correct. 



While the case of small m/M is generally known it appears to be other- 

 wise for large m/M. The intuitive basis for the solution of this case is 

 the fact observable from (52), (57) and (59) that (c) increases indefinitely 

 with m/M J but that the relative deviation from the mean decreases so 

 that the distribution function approaches a 6-f unction. The structure of 

 this limiting function may be explored, starting directly from (40), 

 because in the limit of large m/M the sphere of integration shrinks as 

 may be verified from (37). This makes it possible to replace the integral 

 in (40) by differential terms. This becomes clearer if (40) is written in 

 the form 



« ^ = J f n(x) sin X .X 1- r i. m t:? - ?^| m 



dCz 2 Jo 27r Jo [\c/ t{c') t{c)) 



Let us call the inner average ^. It exhibits the differential properties 

 discussed earlier: the curly bracket is the difference of two terms which 

 are almost identical. Hence we approximate the value of J^by expanding 

 the slowly varying terms to first order in c' — c, while the fast varying 

 h(c') will be expanded to square terms. This expansion is obviously 

 permissible for everything except the rapidly varying function h{c'). 

 For /i(c') itself no justification can be offered except success. By proceed- 

 ing to square terms in this expansion we mitigate any possible error com- 

 mitted, but it is quite possible that structural details are lost in the 

 procedure. 



The development of g is straightforward. We proceed as follows 



The expansion of the first term involves only (41) which to this order 

 reads 



C' — C^C (1 — COS x) 



m 

 This formula does not contain the azimuth w which therefore disappears 

 trivially. In the second term on the contrary we are dealing with a 

 vectorial difference involving all three polar coordinates c\ k, co of c'. 



