266-267] SPHERICAL BOUNDARY. 485 



where &amp;lt;f&amp;gt; n is a solid harmonic of degree n, and R n is a function of r 

 only. Now 



dy dy dz dz 



dx dy dz 



(6). 



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



Hence 



If we substitute in (1), the terms in &amp;lt;/&amp;gt; n must satisfy the 

 equation independently, whence 



which is the differential equation in R n . 



This can be integrated by series. Thus, assuming that 



E n = 2A m (kr) } 



the relation between consecutive coefficients is found to be 

 m (2n +1+ ra) A m + A m _^ = 0. 



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

 other with m = 2n 1 ; thus 



- 

 2(l-2n)2.4(l-27i)(3-2n) &quot;V 



where A, B are arbitrary constants. Hence putting &amp;lt; n = r w &amp;lt;8f n , so 

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

 may be written 



^)J r& ............ (9), 



