302 Prof. J. J. Sylvester on SpJierical Harmonics. 



inverses (or electrical images) of each other in regard to the 

 origin, and consequently lif + hf + If less than unity. This is 

 going to the heart of the matter. So I may observe that if 

 we would go to the root of the relation between positive- and 

 negative-degreed solid spherical harmonics, the more logical 

 mode of proceeding is not (as is usually done) to infer this by 

 a lengthy a posteriori process, but immediately from the fact 

 that since 



1 



V (*» + if + z 2 ) -2(hx + ky + Iz) + (K 2 + P + P) 

 is nullified by the operator 



(l)'+(|)X!)' 



so also must the same operator nullify the radical 



1 



Before proceeding further, I ought to observe that H in 

 the above series for the bipotential may easily be shown to be 



4;7T 



-~ — —j multiplied into the coefficient of 0* in the expansion of 

 *; or, in other words, if the distances from the 



s/i-2st+et- 



* s, it will be remembered, is hh'-\-kk'-\rW, and 8 is the product 

 £#J_{_#2_j_j2) (#*4-* , *+2 ,s )« The statement in the text follows as a conse- 

 quence from the fact that (1 — 2st-\~6t 2 )~^ obeys Laplace's law, and, when 

 expanded according to powers of t, is of the form found for 1^, and must 

 consequently be identical with it to a factor pres, that factor being a function 

 of n, whose value is easily found by making h = h' and k, I ; k' , V each zero. 

 In like manner it may be shown that in higher space of n dimensions the 

 corresponding value of PI^ is a function of /z multiplied by the coefficient 



of t in (l - 22M7+ (27i 2 )2(7i r2 ) t 2 f "*" ; and I find that this function, say 

 $(/*, n), (as will be shown in a sequel to this paper) is always a rational 

 function in m, containing in the denominator, when n is odd, one factor of 

 the form 2m-}-j\ all the others being of the form m-\-i — and when n is even, 

 factors all of the form m + i. Whatever the form of these linear factors 

 had been for even numbers, we could see a priori that the Bipotential for 

 space of even dimensions could contain only algebraic and inverse circular 

 or logarithmic functions. But as regards the case of space of odd dimen- 

 sions, the fact of there being no factors except of the form m-\-i, 2?n+j, is 

 prepotent in determining the form of the result. For space of two dimen- 

 sions the Bipotential does not appear readily to yield to summation in 

 finite terms. Thus at one blow the theory of spherical harmonics has been 

 extended to " globoidal " harmonics in general ; and the chief cases of sta- 

 tical distribution of electricity heretofore solved may be regarded as vir- 

 tually solved mutatis mutandis for space of any number of dimensions, 

 of course with the proviso that the law of attraction (in consonance with 



