a Spherical Conducting Shell on Dielectric Induction. 157 



• • • (1) 



.-. P= — sin0.<l>, Q = cos (/>.<£, R=0. 



<I> is by symmetry independent of </>. 

 Now P satisfies the equation 



2p_. — ft __i n the conductor, 

 cr at 

 and 



d 2 F 

 V 2 P=/i'K' -j-g- in the dielectric. 



Assume 

 and let 



P varies as e lpt , 

 X 2__473vup 



X' 2 = fi'K'p 2 . 



V 2 P + X 2 P=0; 



in the dielectric we must write A/ for X. 



Q and R satisfy equations of the same form. 



(2) 



Assume 



P=*,A>')(y s -^->^ 



(3) 



These equations make -; — |- -j- + -r- =0, as should be the 

 case. ^ 



j^ is an arbitrary solid spherical harmonic of degree n. 

 Substituting in (2) we find 



dr 2 r dr r 



Since R = 0, the last equation of (3) gives 



d<j> u * 



(4) 



.*. from the first of (3). 



P=-sin<£.^(\r)^V^. 



■M't 



(5) 



