JSolotropic Potential. 429 



It contains the correct number of constants, and these are 

 the values of u.... for \ = 0. 



§ --. The mode of construction of the eqnipotential surfaces 

 corresponding to the ellipsoid u a —l as a conductor, is readily 

 shown. Reciprocate w a =l with regard to the sphere 

 ;<;-'/- + :- = /(•-, and the result is 2 (As 8 + 2 Myzf « £ 4 A a . 

 Then reciprocate 



Z{(A+p\A a )x> + 2(A.'+p>\A a ) ! , S }=V\ 



with regard to the same sphere, and the reciprocal is 



S(A(\)^+2A'(X)^) = A(X)/A a , 



which, since A(X)A a =aA(X), is the surface tf a =l. 



In the isotropic problem the ultimate form of the equi- 

 potential surfaces of a conducting ellipsoid is a sphere, and 

 the interval between the two forms is bridged by surfaces 

 confocal with the ellipsoid. In the a?olotropic problem the 

 ultimate form is an ellipsoid reciprocal to the ellipsoid con- 

 stant = u p , having the coefficients of the aeolotropic differential 

 equation. The interval between the primary form u a =l 

 and the ultimate form k p = \A p is bridged over by the system 

 of quadrics nsed here. They are not confocal : it is proposed 

 to call them aolian quadrics. 



\\ e now give some properties of the coefficients a. . ., which 

 it is convenient to place together, though not all are imme- 

 diately required. 



First, then, we have the determinant properties 



A a A(X) = A*, A a A(A.)=A^(A+^AA a ). . (84) 

 [f we introduce J as defined above, 



JA.=:A a , JA;=A(X), 1 



JA m =A+p\b a , i.e. J(/3y-a'*)=A+pkA a , I . . (85) 

 and J(fi'y' — aa')=A'+/''XA a , J 



the first showing the mode of dependence of the volume of 

 "*=1 on X. 



A\ e next prove the important relations 



J2Q>*+2pV) = ^, and J2(Aa + 2A'a')=A ( /3J-X^) {86) 



Tin* mode of differentiating a determinant gives 



~ =A fl 2{ i >A(\) + 2 1 /A'(\ ^=A(\)2(p*+2pV) 



