I 



MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 205 



(41) and (42) in the simplified form 



f'-'l+Jd-cosx) (61) 



cos K ^ COS X + ir7 sin^ X (62) 



M 



and develop the equations in powers of m/M. Starting out with the 

 simpler equation j/ = 1 we find 



/ I - cos x \ , / N dhoic) . . 



This same procedure is not adequate for the equation v = because 

 the two left hand terms in (47) cancel in zero approximation. We must 

 therefore develop the integral up to linear terms in m/M; this is a per- 

 fectly straightforward, though somewhat cumbersome, step. It leads to 

 the following equation 



The equation may be integrated by multiplication with c ; this yields 



Elimination of hi{c) from (63) and (64) gives a differential equation for 

 /io(c) which is easily solved by quadrature; the result is 



Except for the dependence on the angular law of scattering this formula 

 may be found in the literature.^ Its most important special cases are 

 obtained for r = const (Pseudo-Maxwellian distribution) and 

 T = const/c (Druyvesteyn distribution). 



The derivation of (65) should be completed by a proof that indeed 

 /12(c) is small compared to hi{c). This is not true for all values of c; on 

 the contrary, the argument below shows that near the origin where 

 c '-^ aric), hiic) is actually comparable to hi(c). For our purposes, however, 

 it is sufficient if it is true in the overwhelming majority of cases. As the 

 proof applies equally to all h,'s we will run it in this manner. Assuming 



