233-238] Spherical Harmonics 205 



236. THEOREM. // V is any spherical harmonic of degree n, then 



fix+t+u y 



where s, t, and u are any integers, is a spherical harmonic of degree n st u. 



9 2 F 9 2 F 9 2 F 



For W + W + a^ =0 ' 



so that on differentiation s times with respect to #, t times with respect to y, 

 and u times with respect to z t 



or V 2 



which proves the theorem. 



237. THEOREM. If S m ,S n are two surf ace harmonics of different degrees 

 ra, n, then 



II S n S m dw = 0, 



where the integration is over the surface of a unit sphere. 

 In Green's Theorem ( 181), 



^_\p"_ d^ 

 put 4> = r n S n , ^f = r m S m , and take the surface to be the unit sphere. 



Then V 2 <3> = 0, V 2 "^ = 0, -? = -^ = nr n ~ l 8 n , and = mr m ~ l S m . 



on or on 



Thus the volume integral vanishes, and the equation becomes 



1 1 (nr m+n - l S n S m - mr m+n - l S n S m ) da) = 0, 

 or, since n is, by hypothesis, not equal to m, 



jjs n s m do>=o. 



Harmonics of Integral Degree. 



238. The following table of examples of harmonics of integral degrees n = 0, - 1, -2, 

 + 1, is taken from Thomson and Tait's Natural Philosophy. 



n = 0. 1, tan-^, log^f, tan-i2log r + * 





Also if F is any one of these harmonics, - , -^ , -^ are harmonics of degree - 1, so 



that r -^ , r -^ , r -~ are harmonics of degree zero. As examples of harmonics derived 

 8# ' dy ' dz 



in this way may be given 



rx ry zx 



