﻿Harmonics, JEolotropic and Isotropic. 39 



when only two appear, viz. ?/, z. What is true for x as factor 

 in the one case, for yz as factors in the other case, must 

 apply also to other factors ; and a method of establishing the 

 property will now be given in a general form. 



Consider the transformation of variable from A, to f through 

 ^ — 6 =pt;. Then, since J (A) is of the third degree and 

 J(0 O ) =0, we have 



j(x) = (x-* ) j(0 o ) +i(\-e Q Y j(0 o ) + (\-0 e ) 3 j 3 



Now write 

 JW = -2pJ„ J(e )=p 2 pJ 3 , and so 4J 3 j(0 o ) = p{J(0 o ) ^. (5) 



Then 

 JM=i> 3 J 8 e(f-5+ri, J(A)=p 2 J 3 (3f 2 -2f + p), n 

 J(A)=2^J 3 (3?-l),and J(A)/(X-^)=p 2 J 3 (r-f+^); j> . (6) 



whi le # = 1# **_!_** 



dX~pd%> d\*~p*dp' 



Taking the various solutions of (4) according to the classi- 

 fication above, in (i.) </> is itself a polynomial L of degree n 

 in A, and m = 2n is the degree of the harmonic in xyz. The 

 above substitution gives 



= 2n(2n + l)(f-€')L. . (7 i.) 



The equation contains a parameter e' and one constant p 

 depending on the constants of the ellipsoid and those of 

 seolotropy. Each root of J(\)=0 may be made the basis of 

 transformation for this case, and also for 



(iv.) where </>=L V J or <x L </(\-d )(\-d ')(\-6 "). 



If the order of: L is n y that of the harmonic is 2n + 3, 

 Lame's equation becomes 



4J^+6J^+2JL = (2n + 3X2» + 4)J s (X-€)L, 



and the transformed equation is 



•?+p)J J +6(3f- 



= (2n + 3)(2re + 4X?-e')L (7 iv.) 



WP-t+p) J +6(3f-2f + P )g +4(3f-l)L 



