300 Prof. J. J. Sylvester on Spheriedl Sarmonick. 



ables, provided in place of - we write -^ when n exceeds 2, 



and log* 1 when >? = 2, and consider r/S to be the element of 

 what in n dimensions corresponds to a spherical surface in 

 three-dimensional space. 



The method employed, of first using two Laplacian operators 

 in combination to determine one property of the form under in- 

 vestigation and then a single one of them to act on the form 

 thus partially determined , reminds one very much of the method 

 for obtaining invariants of given orders from their two general 

 partial differential equations. Combined, these two equations 

 express the lav/ of isobarism ; then, assuming the isobarism, a 

 single one of the two serves to determine the special values of 

 the coefficients. The analogy between that process and the 

 one here employed seems to me to be exact, although the sub- 

 ject-matter is so very unlike in the two problems — and is the 

 more interesting on that very account. 



The bipotential in the case where the two points of prise 



are both internal being known under the form F (— ^,cos«), 



where a is the radius of the sphere, its value for the case 

 where these points are both external, and for the case where 

 they are one internal and the other external, may be assigned 

 without any further calculation as follows : — 



1. Suppose r greater than the radius of the sphere, but r f 

 less. We know a priori from the result previously obtained 

 (and stated in a footnote), that the bipotential for this case is of 



1 // \ 



the form - Gr ( — , cos « J . Now in place of r, r f substitute 



a, — ; then the bipotential becomes - GM— > cos«). 



But we may by an easily justifiable application of the prin- 



ar'~ 

 ciple of continuity now regard a (as well as — ) as the distance 



of an internal point from the centre. Hence we have 

 -G(~ 7 cos*)=F^cos a ) ; 



or 



;&(7> C0S ")=^ F (^ C0Sa } 



which is the value of the bipotential of a spherical surface cut 

 by the line of prise, r being the distance of the external point 

 of prise from the centre. 



2. Suppose r and / to be each greater than the radius, and 



