﻿38 Mr. R. Hargreaves on Ellipsoidal 



xyz is made through 



u e J(6)^J(6)+J z (\-d)(fjL-e)(v-0), . . . (1) 

 where 



J(<9) = 1 + 6X(pa f 2 P W) +' 6> 2 2(PA + 2P'A0 + (9 3 A P Aa, 

 or l + 0J 1 + 0»J 2 + 03J s , say. 



As (1) is true for all values of we may differentiate it, 



UeJ{0)+Uei(O)=J{0) + J s ^'^] J ^dV=0; 



and then write X for in the result and use u k =l. Thus 



ttxJ(X)==-r-J 8 (X-/i)(X-v) (2) 



If corresponding results for fi and v are used, the seolo- 

 tropic equation of Laplace is 



'V.'0= (A-hV-^) { 4J(X) §V +2j(X) |x} + termsin fandv. (3) 



If the double brackets are proportional to <j> in each case, 

 or to X</>, /*(/>, vcj) in the respective cases, the sum vanishes. 

 Thus we have solutions of V/<£ = in the form of products 

 of similar functions of X, fi, v respectively, the first of which 

 satisfies the equation 



4J(\)|*+2J(X)g = m(m + l)J 3 (X-e)0. . (4) 



We get rid of J 3 by putting 



J = Ji(\-O Q X\-0 O ')(\-0 O "), 



and the mode of dealing with Lame's functions proceeds on 

 normal lines ; but the coefficients of the cubic now depend 

 on the constants of seolotropy as well as on those of the 

 ellipsoid. 



§ 2. It is proposed to go a step further, and show that the 

 parametral equation can for each type of solution be pre- 

 sented in a form dependent on a single constant of the 

 ellipsoid. In Heine's work this valuable property appears 

 to belong exclusively to the solutions K and JN" (in Heine's 

 notation), and his work for L and M would support that 

 view. But K may be an odd polynomial in Heine's X, or 

 may contain \/\ + a 2 in the present notation for the isotropic 

 case, and so correspond to a solution with a linear factor x. 

 Also N may be an odd polynomial, in which case there are 

 three linear factors (x, y, z for isotropy) ; or may be even, 



