r?0 1> Frof. J, J. Sylvester on Spherical Haniwmcs, 



Thus the desired theorem has been established by virtue of 

 an algebraical necessity of form alone ; and the proof is of 

 course applicable to space in any number of dimensions, sub- 

 stituting for the sphere or spherical surface its analogue in such 

 space, and for the reciprocal of distance the proper power neces- 

 sary for the satisfaction of Laplace's equation, i. e. the (q — 2)th 

 power of the reciprocal, where q is the number of dimensions 

 (supposed to be greater than 2). 



For the case of two dimensions, substituting the logarithm 

 for the reciprocal, so that ex. gr. we are able to affirm that 

 if each element of a circular ring be affected with a density 

 proportional to the product of the logarithms of its dis- 

 tances from two fixed internal points, the mass of such ring 

 will depend only on the product of their distances from the 

 centre of the ring and the angle between these distances 



— for this case, writing E = h -pr +k-rr and W — h f -^r-. + k' -jy-, 

 & elk ah dlv elk 



in the equation (F-F)L=0, F = (E*) 2 and F' = (EV) 2 ; and 



if the two points are interior, every term in r——-, will be of the 



form Cr* . s 1 ' . t Jc , i and k being both positive, and we must have 

 i 2 + 2ij — k 2 — 2kj = 0, and consequently i = k — the other solu- 

 tion, i + k + 2j = 0, being applicable to the case of one point 

 being external and the other internal. If the points are both 

 external there will be four sets of terms. One set will consist of 

 the single term A log r log t ; a second, of terms of the form 

 c log r . 7 J sH k ; a third, of terms of the form c log t . r l &t k ; 

 and the last set, of terms of the form cr i sH k : and it is easy 

 to see that F(logrlog0 = 0, W(logr\ogt) = 0, 

 (F — FQlogr.rW 



= (({ + jf log r — (k +j) 2 log r + 2(i— &))rW, 



and consequently i=k for the second and third set ; as regards 

 the fourth set, i — k for the same reason as in the case of three 

 dimensions. Hence 



1 = A log r log t + log r<j>(rt, s) + log tty(rt 3 s) + co(rt, s) ; 

 and as r and t are interchangeable, we must have (j> = ^ 3 and 

 consequently 



I— A\ogr\ogt = F(rt, s) ; 



so that not now the mass of the ring, but the difference between 

 it and the mass due to the density at the centre is invariable 

 when rt and s are given. 



For greater simplicity, and as bearing more immediately on 

 the theory of spherical harmonics, I have hitherto regarded 

 the points of the pair-point at which the " bipotential " is 



