produced by Motion in the Electric Field. 7 



Equating the coefficients of e* p * to zero in equations (2) and 

 (3) we have 



— -: -V 2 Fi = 7- 1 ~ inside the sphere, 



Airfjb dt doe r 



with similar equations for G x and H^ ; outside the sphere we 

 have 



_! ^--^-^ + ^-^1 (A) 



/ZK V x ~dt 2 dtdx + K dxdz r> ' { J 



with similar equations for G ± and Hj. 



The form of equation (4) suggests that we should put 



i|r 1 = J3- r -. 



7 dz r 



A particular integral of (4) is then 



_cB_^_JL _coe d 2 1 

 1 p dxdz r lLp 2 dxdz r 



The complementary function is that solution of the differential 

 equation 



V^ + ^K^F^O, 



which, when considered as a function of the angular coordi- 



d 2 1 

 nates of a point, varies as r z --. — j- - ; this (see Proc. Math. Soc. 

 r ax dz r x 



vol. xv. p. 212) is 



0E 2 (i\ry J~- -, 



v y dx dz r 



where 



E 2 UXr) = ... . 3 + ,.^ , 2 + ^— > 



v ' (?Xr) d («\r) 2 2Xr 



and \ 2 = f juKf. 



Thus, outside the sphere, 



F -CE (i\r)r 3 ® * iBeP * — X ffl ^ d * X 

 1 3 e£# dz r p dx dz r Kp 2 dx dz r 



v y dydzr p dy dz r Kp J dydzr 1 



1 K ' dz 2 r uy y p dz 2 r Kp 2 dz 2 r 



where 'Ei (iXr)=€~ i? " r /iXr, and is introduced iuto the expression 



