MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 235 



because of (9). This yields with the angles as shown on Fig. 15 

 X = r m{c - n)de = (^^' ' exp ["- ^mc' 



,2 M^ -\- rn -{- 2Mm cos x , o/^^../ ^ + ^^ ^'os x ^_ ,1 



— amu 



{M + m)' 



+ 2^mcu 



M + m 



COS ^ 



f 



exp 



2^ 





cu sin X sin ^ cos e de 



The integral is evaluated by a formula known from the theory of Bessel 

 functions 



and yields 



v3/2 



X = —7= (8mr" exp - /3mc - /3mi^ 



,2 AT ^ + m^ + 2Mm cos x 



(112) 



(M + m)' 



, ^^ , m + M cos X , 



+ 2Bmcu Tir— cos i/^ 



M + m 



^ / 2^Mm , . .A 

 • ^0 ( ir^—; ^^ si^ X sm ^ 1 



(113) 



\M + 



m 



Passing now to the right hand side of (11 la) we may replace in the 

 first place dC by dy, because of (111b). c' and C' are then replaced by 

 the expressions 



c = c — 



C' = c 



M ^ M , 



Y + .r ■ Y 



ilf + m M -\- m 



M 



M 



Y - 



M + m ' M + m 

 With the angles defined in Fig. 15, we thus get for Y 



Y = (^)" (^)" exp [- ^Mc^ - ^mic - uTl 

 • ffffl^ dy sin e dB db d4> 

 expf- jSMt' + ^McT (cos i/^ cos ^ + sin yj/ sin ^ cos 5) 



2^ 



mM 



M -\- m 



uy (cos <? - COS ^ COS X - sin ^ sin x cos 0) J 



