141, 142, 143] SPHERICAL HARMONICS. 395 



Consequently if m = ~ (2n + 1), the product r m V n is a harmonic. 

 Since V n is of degree n, and r is of degree unity in the coordinates, 

 r _ (2,1+1) y n - g Q Degree .- (^ -f. 1). Accordingly to any spherical 

 harmonic V n = r n Y n of degree n there corresponds another, 



Y n 



of degree (n -f 1). Compare this with the corresponding property 

 of circular harmonics, where the degrees of the two corresponding 

 harmonics are equal and opposite. 



142. Dirichlet's Problem for Sphere. By means of these 

 harmonics we may solve Dirichlet's problem for the sphere. If a 

 function harmonic within a sphere of radius E can be developed at 

 the surface in an infinite series of surface harmonics, 



89) F=r + r 1 + r 2 ..., 



the internal problem is solved by the series 



90) 7 = r c + Jir l + gr 1 + .... 



For each term is harmonic, and therefore the series 90), if con- 

 vergent, is harmonic. At the surface the series takes the given 

 values of V. Every term of the series 90) is less than the corre- 



sponding term of the series 89) in virtue of the factor ? therefore 



R n 



if the series 89) converges, the series 90) does as well. Since the 

 series fulfils all the conditions it is the only solution. 



We may in like manner fulfil the outer problem by the series 

 of harmonics of negative degree, which vanish at infinity. 



91) F =fr + | s r 1 +^r 2 + .., 



For the space bounded by two concentric spheres, we must use 

 the series in harmonics of positive and negative degrees. 



143. Forms of Spherical Harmonics. Before considering 

 the question of development in spherical harmonics, we will briefly 

 consider some convenient forms. Since if 



we have 



