204 Methods for the Solution of Special Problems [OH. vm 



The first term is a function of r only, while the last two terms are inde- 

 pendent of r. Thus the equation can only be satisfied by taking 



R dr 



.(133), 



a / . Q ds\ i 8 2 >sf 



^ sm<9 ^ + o-o/i ^r = -^ ............ (134), 



2 2 



where JT is a constant. Equation (133). regarded as a differential equation 

 for R can be solved, the solution being 



(1.35), 



where J., 5 are arbitrary constants, and n(n + l) = K. After simplification 

 equation (134) becomes 



, (siue d ~} + J-a d ~+n(n + I)S = ...... (136). 



sin 080 \ 80 / sm 2 # 8</> 2 



Any solution of this equation will be denoted by S n , the solution being a 

 function of n as well as of 6 and <. The solution of Laplace's equation we 

 have obtained is now 



and by the addition of such solutions, the most general solution of Laplace's 

 equation may be reached. 



234. DEFINITION. Any solution of Laplace s equation is said to be a 

 spherical harmonic. 



A solution which is homogeneous in x, y, z of dimensions n is said to be a 

 spherical harmonic of degree n. 



A spherical harmonic of degree n must be of the form r n multiplied by 

 a function of 9 and <f>, it must therefore be of the form Ar n S n , where S n 

 is a solution of equation (136). 



Any solution S n of equation (136) is said to be a surface-harmonic of 

 degree n. 



235. THEOREM. If V is any spherical harmonic of degree n, then 

 Y/r 2n+l is a spherical harmonic of degree (n + 1). 



For V must be of the form Ar n S n , so that 



V AS n 



which is known to be a solution of Laplace's equation, and is of dimensions 

 (n + 1) in r. Conversely if V is a spherical harmonic of degree (n + 1), 

 then r* n+1 V is a spherical harmonic of degree n. 



