AGolotropic Potential. 463 



their original numbers, and a subscript n appended. The 

 volume of a sphere radius c in n dimensions is 



c*|r(i)}» lY" + l) ; for an ellipsoid C* is replaced by 1/AJ, 



and here the elements " a " being P/eAp, A B :==l/6 n Ap. 



Introducing the volume 



* 4 „(„-» mi , r i(( „ u ,... ,,„),»-> 



and this represents with dr lor t the law of potential. If 

 this is integrated to give the value at the centre and the 

 result compared with the solution, we have 



i. e., if we write 



0=2{rg)}"r(^-l),wehave 



the integration on the left embracing all values for which 

 MaC^i • • . #») > 1, i. e. 2(a,.,.# r 2 + 2a rs x r .i' s ) > 1. We then intro- 

 duce a?|=^r, . . . x n =lnr, with 2/ 2 =l, and useo) as before for 

 the angular element : the result is 



A,, J -*j ..-.^.j . . (U1) . 



J being symmetrical with regard to a and p. [The angular 

 element can be expressed by an extension of the usual 

 spherical co-ordinate system and probably in other ways.] 

 The connexion between J and the determinant is 



A(X)=A a w - 1 J(X), 



where the members of A(\) are A n +j> n \A a , . . . ; or if we use 



/-i for \£ a , and 



«— 1 t— 3 



d\/^.j=dX.X, * %/&(\)=dft.£ia * L/A{p), 



^ 



