Prof. J. J. Sylvester on Spherical Harmonics, 293 



the sphere and the two given points are the apices. The rest 

 of the proof follows as a matter of purely formal or alge- 

 braical necessity from the above fact, conjoined with that of 

 each factor under the sign of integration being subject to 

 Laplace's equation. In this feature of exemption from all use 

 of integration as a process, this proof, I believe, stands alone. 



It is further remarkable that its success depends on the pro- 

 position being stated as a whole ; it would not be applicable, 

 for example, to the simple case, taken per se, treated of at the 

 beginning of this paper. It is by no means uncommon in ma- 

 thematical investigation for this to happen, and (as regards the 

 exigencies of reasoning) for the part to be in a sense greater 

 than the whole — the groundwork of this wonder-striking intel- 

 lectual phenomenon being that, for mathematical purposes, all 

 quantities and relations ought to be considered (so experience 

 teaches) as in a state of flux. In the particular case before us 

 it is not difficult to see a priori why the general proposition 

 should be more easily demonstrable than any special case of it, 

 the reason being that more information as to the form of the 

 function under consideration is made use of in dealing with the 

 general than in dealing with any special case. 



The integral under consideration is 



KB? Oil), 



where 



Call 



ffii 



R2 = x 2 + f.+ s ?-2J l x-2Jci r -2h + h 2 + & + P, 



B' 2 = x 2 + y 2 + z 2 - W% - Wy -2l'z+ V 2 + h /2 + V 2 . 



tf + k 2 + l 2 = r 2 , h /2 + k /2 + l /2 = i 2 , hh' + kk' + U^s. 

 Then ^-p7 ? expanded under the form of a converging series 

 (x,y, z being for a moment regarded as constants), will be of the 



form —multiplied by a rational function of —, — , -4, -~ and of 

 rt L r ? r' 7 r z 7 ir 



1 li' tt V 



-—, 72~, 72-, w when the two points are external, and (more 



t t O 6 



simply) of h, k, I and of h r , k' ', V when they are both internal. 

 I, we know, must turn out to be a complete function of r, s, t. 

 and, when expressed in the form of a series derived from 

 the above expansion, will be the sum of terms of the form 

 r\ s 1 '. t k , where it is obvious that i andj must both be negative 

 when the "pair-point " is exterior, both positive when it is in- 

 terior to the shell, and one positive and one negative in the 

 remaining case. 



