2^2 Prof, J. J. Sylvester on Spherical Harmonics. 



product of the distances of the two points from the centre; 

 and in fact, if we write d$ for an element of a spherical sur- 

 face, it is easy to iind, by direct integration, that 



d& 



JJ >/c 2 -: 



fact that the surface 1 1 -^ '„ where 



2hx + hWc 2 -2h f a; + h' 2 



for the entire surface is proportional to 



•J.KN c 2 +</hh' 



In like manner, the truth of the more general theorem rela- 

 ting to the surface-integral of the product of any two har- 

 monics of unequal degrees involves, and is involved in, the 



r/2 = (a _ h y + ( -y_ yy + ( z _ iy 



and h 2 + k 2 + I 2 and h 12 + k' 2 + Z /2 are each less or each greater 

 than the square of the radius of the sphere, is not merely a 

 function (as we see a priori from the symmetry of the sphere 

 must be the case) of the three quantities 



h 2 + P + 1 2 , I/ 2 + V 2 + V 2 , hh' + W + IV\ 



but, more definitely, is a complete function of the product of 

 two of them, viz. (A 2 + k 2 + F)(h!*<+ k/ 2 + V 2 ), and of the third. 

 In other words, the fundamental law of spherical harmonics is 

 exactly tantamount to the assertion that if each element of a 

 sphere is charged with a density proportional to the product 

 of its distances from two internal or two external points, then 

 the mass of the sphere will be a function only of the density at 

 the centre and of the angle subtended at the centre by the line 

 joining the given pair of points ; or, venturing upon an irre- 

 pressible neologism, which explains its own meaning, the Bi- 

 potential, with respect to a given uniform sphere at any point- 

 pair, is a function only of the Bipotential thereat with respect to 

 a unit particle at the centre, and of the angle subtended at the 

 centre by the line joining the two given points. Of course, 

 if this is true for the volume of the sphere, it must be true for 

 any shell of uniform thickness, or, in other words, for the surface, 

 and vice versa. In what immediately follows the volume of a 

 spherical shell is to be understood. It is, I think, very notice- 

 able that in that proof no process whatever of integration is 

 employed ; only the idea implied in integration is employed to 

 acquire the fact that the integral in question cannot but be a 

 function of three parts of the triangle, of which the centre of 



