﻿40 Mr. R. Hargreayes on Ellipsoidal 



The transformation with the root 6 is to be applied when 



(ii.) </> = Lx \/\—0 o . 

 The order of the harmonic is 2n + 1, the equation in L is 



= (2w + l)(2n + 3)J 8 (\-e)L, 

 which transformed is 



= (2n + l)(2tt + 2)(£-e')L (7 ii.) 



The transformation with O is to be applied also when 



(iii.) </»<* lVO-<V)(A-0 o ") = lVJ/(X-tf ). 

 The order of the harmonic is 2n + 2, the equation in L is 



= (2*i+2)(2» + 3)J 8 (X-6)L, 

 and the transformed equation is 



m?-H P ) J +2(7{»-4f+,) J + (6f-l)L 



= (2n + 2)(2ra+3)(J-eQL (7 iii.) 



The parametral equation in e', or the equation which gives 

 the condition that L may be a polynomial of degree n in A,, 

 is of degree n + 1 in e' and in each case contains the single 

 constant p in linear form. To connect the above with 

 Heine's notation take the isotropic case 



l=p = q = r, 0=p' = q' = r', = a' = b' = c\ 



and suppose the root O to be l + 6 a = 0. Then 



-2pJ 3 ==J(0 ) = 2(J 2 + 3e o J 3 ) = 2(ah + bc + ca)-6bc, 

 and or P = 2a~ l -b- l -c- 1 ; 



p 2 pJ d =zJ(6 o )=J 1 + 20 o J 2 + 36J i J d =%a-2a- l tab + Sa- 2 . abc, 

 or p 2 p = (a~ l — b~ ] )(a~ 1 ^ c _1 ). 



When the principal semi-axes are used, 



p = 2a 2 -b 2 -c 2 and p 2 p = (a 2 -b 2 )(a 2 -c 2 ) ; 



