of Stationary States of Motion, 



249 



necessary and putting for the volume element dY its value 

 ^rdzd^d^, we obtain in succession 



We 

 'bt 



BO, 



bt 



= — XX t/{Ci sin + (Co sin i/r-f- C 3 cos i/r) cos 0} 



exp t{r +^ cos 6 + k^—jiv sin sin -\/r}, 



= — XX ij { — C 2 cos yjr -f C 3 sin ^} 



exp i{r-\rjz cos + kyjr —jnr sin sin sjr}, 



with similar expressions for b^e/bt and BK<p/?)£ ; and also 



U fl = XX exp tT . jJJ { — C 2 cos -v/r -|- C 3 sin ty + K 2 sin 6 ~1 

 + (K 2 sin i/r + K 3 cos i/r) cos 0} . ijw 

 . exptj^costf + Z;^ — j'ur sin sin ^dzd^rdyjr, 



V<p= —XX exp it . jjj {OiSin 0+ (0 2 simjr 



+ C 3 cos-v/r) cos #H-K 2 cos^ — K 3 sin ^} . t/'sr 

 . exp t-f/r cos # + ^— /srsin 0sin ^dzd^rd-^r 



On the assumption which we have made the limits for 4r 

 are and 2tt ; hence we find from (11) with the usual 

 notation for Bessel Functions of* order k, 



(11) 



K 8 cosfl- 



sin 



-•* 



)Jk(/ 



«r sin 0) 



JJ 6 = 27rXX exp it . ( t J WKJ^sin^H- 



+ (C 3 + K 2 cos 0)JvtJk O'ot sin 0) V exp y> cos 6 . d-&dz, 

 L = 27rS2exp it > ff{ -/C^tf sin + ^^e^ ty^ 



sin<9) 



+ (K 3 — 2 cos 6)J'&Jk(J'&- sin 0) > exp y> cos . dmdz, 



where the summations with respect to j and h are from 

 — x> to go as beforehand we must remember that k is an 

 integer, whilst j is not necessarily so. The limits for ot and 

 z are determined by the form of the cross-section of the 

 ring-shaped region swept out by the electron in its motion ; 

 each element of charge has been implicitly supposed to 

 describe a circle with its centre on the axis of z, otherwise 

 we should have had to assume -sr and z to depend on ^ in an 

 assigned manner. The motion is not, however, assumed to 

 be uniform. 



at) 



