266-267] SPHERICAL BOUNDARY. 485 



where <f> n is a solid harmonic of degree n, and R n is a function of r 



only. Now 



Ua/p A \ ^7<>7? A, _L 



V"(Jtl n (b n )=\"lt n . n + 



...... (6). 



And, by the definition of a solid harmonic, we have 



Hence 



2(n+l)dR n 



(7) - 



If we substitute in (1), the terms in <j) n must satisfy the 

 equation independently, whence 



which is the differential equation in R n . 



This can be integrated by series. Thus, assuming that 



the relation between consecutive coefficients is found to be 



This gives two ascending series, one beginning with m 0, and the 

 other with m = 2n 1 ; thus 



p A fi k * r * . ^ V __ ^ 



\ 2(27i+3)" r 2.427i + 327i+5 "J 



(l 



where A, B are arbitrary constants. Hence putting <j> n = r n S n , so 

 that S n is a surface-harmonic of order ?i, the general solution of (1) 

 may be written 



kr)}r n S n ............ (9), 



