Spherical Conducting S 'i hell on Dielectric Induction. 161 

 Now <J> is zero at infinity and must decrease as r increases, 



e~ lXr 



Q—I.XT 



Substituting in (14) 



„ (2n + l)be~ lXr > v 



/j—n—i— 7 t^t : ? A_ n _i 



(ji + l)a + no 

 =ir** X_„_! (15) 



if both a and b be large compared with their difference. 



Let O denote the solid angle subtended by the circuit at 

 any point ; 



V the potential due to the circuit ; 



U the normal magnetic force. 



n = 27rsin 2 «X ?T ^ I (y Fi(a)P n (0), 

 if r > c. P„ is the nth zonal harmonic. 



U_ dr' 



= 27rysm*ae«^Z^y +1 PMl> n (d) (16) 



Now in the dielectric inside the shell, 



n(Vi+l) /xr „ , . 



v=- -i HX„ + X_ n _ 1 )^, 



7i(n+l) 



X_ w _ 1 =27r > sm 2 «0y+»P^(«)P»(^), 



. ^ _ 27T7sin 2 a ip (c\ n + l TiU NT> , m n _. 



•• X - n - 1 - n(n+l) W P » P '^' ■ ' ( 17 > 



.'. from (15) 



Z.,_ 1 = -.-«, 2 ^« ■ »» (£)•*' P»P. W . . (18) 



But in the dielectric outside the shell 



n(n + l) n . 



l/c\ tt+1 - 



.-. u=e-^27r 7 sin 2 a^V^ P»P„(0). 

 Now 477yn/? n , 2 2 



