462 Mr. R. Hargreaves on 



by u a belonging to the topography o£ the ellipsoid which 

 fiVures as source. Further 



^.irAS n.s so 



© 



4tt 



J day _ 47r _ 47T 

 w P * ~ A /Ap ~~ A p 



/a; a, i-v 



and the factor l—^p 2 is therefore cancelled in the special 

 case where u v replaces u a . 



§37. If the various processes used in this theory of 

 a3olotropic potential are examined, it will be found that the 

 arguments are stated in a form suited to any number of 

 dimensions. It was not considered advisable to write out the 

 work as for n dimensions, and set down the results for three 

 dimensions as a particular case, because the latter has the 

 wider interest and appeals to physicists as well as to pure 

 mathematicians. Analytically a general scheme would be 

 very little more difficult or laborious. It is proposed to 

 indicate some of the points where a difference in constants 

 appears, and to give attention mainly to the later section, 

 which has analytical applications. 



For the scheme of equations, §21, a more general nota- 

 tion is needed, say 'for the p's p u , p 12 . . . p n % with p r s—psr, 

 and the like for a's and as. There are then n groups of 

 n equations (not all independent) for the a's, and their 

 determinant solution is expressed as before. The potential 

 for a volume distribution is given as before by 



dX 



that for a conductor by 



kp_C a d\ 

 2»J A n/J" 



If we use for base form the special ellipsoidal homaloid which 

 has the status of the sphere, viz., %{¥ r rXr 2 + 2P,.«#,.#,) =eAp, 

 the seolian surfaces are similar surfaces with e + A, for e, and 



lH — ) . Then comparing with § 25, ($6), 



4j A ( € + X)sL (e+X)A; J 



. . (96)» 



*-%. 



(€+X)A, 



n n 

 e 2 A p 2" 1 



kp 4 



4:'n(n— -2)' ~ ~]} 



where to facilitate comparison the results are marked with 



