MOTION OF GASEOUS IONS IN STKONG ELECTRIC FIELDS 245 



rather than by generalizing the formal procedure of the Sections IIA, 

 IIB and IIIA. Such a generalization would no doubt be possible, but 

 would increase unduly the bulk of this paper. We shall operate therefore 

 directly on equation (20). To get out the integral (23) we multiply the 

 equation vectorially with c and integrate over dc. This operation makes 

 the first term vanish completely. This is obvious from symmetry for the 

 components Cx and Cy of the multiplier c. For Ct we have 



•/ 



dCz 



An integration by parts brings this in the form (18) and thus makes 

 it equal to zero. 



Temporarily, we may break the integral term of (20) into two parts, 

 using some artificial procedure to eliminate small angle collisions. The 

 first half of the integral term reads then simply 



J /.(c) 



c do, 



This is already the desired average (23). On the second half we use the 

 identity (108) to give it the form 



— Ij M{C')g{c)cU{x) dQy dC' dc 



47rT 

 We now use (7) to replace c by the expression 



m t , M ^, , M 



c = 



c' + ^^^^- C + 



M + m M -\- m M + m 



Only Y is affected by the integration over dQy which we take up first. 

 Using y as the axis of a polar coordinate system we may write 



r = Til + r-L 



For every value of x, Yi i has the fixed value y cos x- On the other hand 

 the average of yj. vanishes through integration over all azimuths. Thence 

 we may write 



-L f cn(x) d^y = -^ c' + -^ C + ^^ (c' - C) (cos x) 



47rJ ^ M + m M -\- m M -{- m 



^ m + M (cos x) / , M(l - cos x) ^/ 

 M -{- m ^ M + m 



We now multiply with M(C')g{c) and integrate over dc dC' . The inte- 

 gration of the term containing C obviously vanishes for two independent 

 reasons. The integration of the term in c', finally, yields again the average 



