Prof. A. Schuster's Electrical Notes. 7 



and inside the spherical surface 



Take now two similar spheres at a distance w from each 

 other along the axis of Z, the first acting as a sink, while the 

 second acts like a similar source of electricity ; and in order 

 to do away with sinks and sources, unite each point of the 

 first sphere to the corresponding point of the second by 

 means of a straight-line current parallel to the axis of Z, and 

 of such intensity that all currents now flow in closed paths. 

 If w is small, the vector-potential due to the second sphere at 



any point P will be —i A x -~iv), and the complete vector- 

 potential therefore, taking account of convection currents, will 

 be 



Al= ^ M7 £K(6^ 1 )] = ^ 



.2 — „2 



d 2 



A 



fzqiv 



fiqw 



. d fz ( a 2 ., Yl /mow 



Inside the sphere, 



An^A^O, 



-tis — n ■ 



(5 



-a 2 

 6 



fiqw 



dz dx 



d 2 

 dzdy 

 r 2 -a 2 d 2 

 d. 



b 



(})■ 



>■ • (8a) 



figw 



\ 



(8b) 



The condition f — H — — ^ H — ^ =0 is necessarily fulfilled 



dx dy dz J 



as the currents have been made to flow in closed paths. The 

 currents which are defined by equations (8a) and (86) are in 

 the outside space 



4z7T{A 



Air dz dx\r ) 



^ A2 ~^d~z dy\rP 

 _qw d 2 /1\ 



47TyL6 



But these are the current-densities which J. J. Thomson 

 takes as the basis of his investigation. The vector-potential 

 given by (8a) and (86) therefore represents the solution of the 

 problem if q measures the charge of the sphere and w the 



