Prof. J. J. Sylvester on Spher ieal Harmonies. 297 



reckoned either both internal or both external. The results 

 established in these two cases are not complementary, but 

 mutually equivalent to each other, and to the theorem that the 

 integral along a spherical surface of the product of two sphe- 

 rical harmonics of unequal degrees is zero. In the third case, 

 where one point is internal and the other external, then for the 

 case of space of three dimensions the equation between i and 

 k will have to be satisfied, not by i—h but by i + h + 2j + l=s Oj 

 as previously stated in a footnote ; and for two dimensions the 

 equation would have to be satisfied, not by z = & but by 



The advantage of the method here indicated is that it is im- 

 mediately applicable to space of any number of dimensions. 

 I shall now proceed to show that it leads at once to the deter- 

 mination of the values of the surface-integral of the product 

 of any two given types of spherical harmonics of equal degrees, 

 and mutatis mutandis to the corresponding surface-integral in 

 space of any order. 



To prove that the degrees must be equal or else the inte- 

 gral will vanish, Ave have combined the two Laplacian opera- 

 tors applicable to R and R' respectively ; to find the value of 

 the integral in a series, I use either of these operators to act 

 singly on the result acquired by their use in combination. 

 For greater simplicity suppose the point-pair to be internal ; 

 then, calling 



a -f b + c = ^ = « + /3 -f y , 



the problem to be solved is in effect that of finding the value 

 of the numerical coefficient of h a k b l c . 7/ a Wv in the inte- 

 gral I. Now we know by what precedes that the value, 

 say 1^, of that part of I which is of the yu-th order in the two 

 sets h, k, I ; hf, //, V respectively is a rational function of rt 

 and s ; and we may accordingly write 



where 

 and 



s = lih r + kk' + ll r , 



When A, B, C, . . . are determined, the problem is virtually 

 solved, as we shall then know the coefficient of 



by mere binomial expansions. 

 Since 



\dk) + \dk) * \dl) ' 



