186 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



96. 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 R can be developed 

 at the surface in an infinite series of surface harmonics, 



(6) F=F, + F 1+ F 2 , 



the internal problem is, solved by the series 



For each term is harmonic, and therefore the series (7), if con- 

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

 values of V. Every term of the series (7) is less than the corre- 

 sponding term of the series (6) in virtue of the factor r n jR n , 

 therefore if the series (6) converges, the series (7) 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. 



7? 7? 2 7? 3 



(8) v = ^Y + ^T 1 + ^T,+ .... 



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

 the series in positive and negative degrees, as will be illustrated by 

 an example in 198. 



97. Forms of Spherical Harmonics. Before considering 

 the question of development in spherical harmonics, we will 

 briefly consider some convenient forms. Since if 



we have 



and any derivative of a harmonic is itself a harmonic, so that 



is a harmonic of degree n (a + /3 + 7). Since to F = c corre- 

 sponds the harmonic F_ x = c/r, we have 



V9) o~i r\ n ,Q 27 



