MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 197 



We have now to consider simultaneously the three vectors c, c' and a 

 as well as the angles between them. These angles are defined in Fig 8. 

 We study equation (34) or (40) term by term in order to see what be- 

 comes of it upon substitution of (43). Starting with h{c') under the inte- 

 gral sign we get from Fig. 8 and the addition theorem for spherical har- 

 monics 



hie) = E hXc)lP, (cos T»P. (cos k) 



+ 2 X; ^/ 7 ^l\ P". (cos ^)K (cos k) cos M 



M=l (»' + m) I 



For this expression, the integration over co is elementary and gives 



\ ' h{c) do, = 2irY. Hc)Pv (cos t?)P. (cos k) (44) 



Further, we get for the derivative in (34) or (40) 

 #- (Z K ic)P. (cos t») 



OCz \f=0 / 



= ± ^ _!_ { (, + DP^i (cos ,» + yP^i (cos ,» 1 (45) 

 Po dc Zj/ -t- 1 



c 2i' H- 1 



Through the equations (43), (44) and (45), all terms in equation (34) 

 or (40) are developed in spherical harmonics with respect to the angle t? 

 between c and the field direction. We can therefore annul separately 

 the coefficient of each Legendre polynomial in cos t?. This gives the follow- 

 ing set of equations 



(M + mf (mi'h^p, (eos .) n(x) dc' -"^ 



= ^^ /^^^::lW __ y- 1 h^^ic)) (46) 



2v - l\ dc c I 



{y + l)a IdK^ , v_±2 ;l_(c)^ 

 '^ 2v^Z\ dc c J 



