MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 247 



This is a consequence of the limited form of the convolution theorem 

 proved in Section III A; it probably implies also that the theorem can 

 be extended in some form to include the case of diffusion. 



It has been mentioned in the Section ID that the Nernst-Townsend 

 relation (30) applies only to ions moving in a low field. We are now in a 

 position to examine possible extensions of it to general fields. Equations 

 (144), (145) and (117) suggest the form 



Dn _ 2 X mean random energy along n (\aq\ 



mobility e 



where n stands for one of the principal directions of the diffusion tensor. 

 This formula contains equation (30) as a specialization to the low 

 field case. 



Formula (148) is one of the formulas obtained in this study of ion 

 motion in which model parameters do not appear. It is valid (a) for all 

 interactions at low field and (b) for the mean free time case at all fields. 

 It also holds dimensionally at high field for models obeying (25); this 

 may be seen from (26a) and (28a). It appears a reasonable conjecture 

 that (148) is approximately true for any law of interaction; the question 

 will be taken up again in the next section. 



Let us, in conclusion, write down the formulas resulting from (146) 

 and (147) for the two special mean free time models studied in detail 

 in Section IIIB: the polarization force and the isotropic model. The 

 necessary averages are (129), (130), (134) and (135). They yield for the 

 polarization force 



Z) = ^^-i^0.905T«./cT 

 " Mm 



1 {M + m)\M + 3.72m) . 



(149) 



^ ^ M + m 0905^^.j^y + 1 (^+7)' ^ a^(0.905r.)' (150) 



Mm 3 M^m{M + 1.908m) 



and for the case of isotropic scattering 



M + m , ^ , 1 (M + mf{M + 4m) 



^"=nJf^^'^^ + 3 MMM + 2m) "' ^^^^^ 



M + m 1 {M + mY 23 ,,.0) 



Just as in the earlier study the results for the two models do not differ 

 appreciably. 



