﻿Harmonics, JEolotropic and Isotropic. 37 



solutions of V 2 H = o£ the types 



(i.) H 2rt =Rn, 



(ii.) H 2 „+i = (#, y, or z) R„, 

 (iii.) H 2 „+2=(^, yfi, or zx) x R„, 



and (iv.) H 2n+ z = xyzxB. n , 



the subscript giving the degree of the harmonic in xyz. The 

 polynomials in Xjxv involved in H n are distinct in the four 

 cases, but their coefficients depend on a parameter satisfying 

 an equation of degree n + 1 in each case, so that there are 

 n + 1 values of each R„. This statement gives the sym- 

 metrical classification according to the degree of H as 

 follows : — 



(a) degree ?neven = 2n: there are 4n+l, i.e. 3n + (n + l) 

 solutions; the n + 1 solutions have the form R ;l , and there are 

 n solutions of each of the types (xy, yz, or zx) x R„-i ; 

 (b) degree m odd = 2>i + 1 : there are An + 3, i. e. n + 3(n + 1) 

 solutions; the n solutions have the form xyzxR n -i, and 

 there are n + 1 solutions of each of the types (x, y, or z) x R, t . 



For seolotropy a more general cubic takes the place of 



(a*+\)(5 2 +\)(c 3 +X), 



and the radicals of its factors appear in place of *Jx + a 2 , . . . ; 

 but the linear factors in xyz cease to have the simple form 

 varying as x, y, or z. 



It is the second object of the paper to consider the 

 Cartesian form of harmonics, with special reference to 

 aeolotropy in the equation of Laplace- ; and to give proofs of 

 a few fundamental points in a manner independent of the 

 curvilinear treatment. 



§ 1. The notation of the paper on aeolotropic potential * is 

 followed, but u K or u 9 is used (instead of u a there) for the 

 quadric 2 (ax 2 + 2a' yz) , when it is necessary to distinguish 



the variable X or 6 underlying the coefficients a We 



found that when <f> depends only on \, 



Ve<?> u x \ <M> J(\) d\r 



the whole expression of — \J/(f> in terms of \/jlv requires the 

 addition of corresponding terms in yu and v, and ii K is to be 

 expressed in terms of X/uv. The connexion between Xf.iv and 



* "iEolotropic Potential," Phil. Mag. April 1905. References written 

 as JE. (86). 



