﻿Harmonics, AZolotropic and Isotropic. 41 



these are the quantities which Heine calls p and q f i. e. p is 



here the ratio g : p 2 . 



When n = \ or the polynomial is linear =\ + k y the 

 results are — 



(i.) 6k= 6e'-4, (3 e '_l)2 = i_3 p; 



(ii.) 10£ = 12e'-9, (126 / -5) 2 = 4(4-15 / 9) ; 



(iii.) 14£ = 20e'-9, (20e'-5) 2 = 4(4-7p) ; 



(iv.) 18*=30e'-16, (156 / -5) 2 =9(l-3p). 



The radical in the solutions (i.) and (iv.) is independent o£ 

 the root used ; it is <v/J 2 2 — 3^3 in the general case, and 

 ^/'Ea i — ^b 2 c 2 in the special notation of the isotropic case. 



§ 3. We now give a brief sketch of some points in the 

 Cartesian treatment, dwelling specially on the features of 

 difference due to seolotropy*. With the notation in the 

 preamble f=ve ~1> there is in the value o£ Ve 2 (R* or/x .../«) 

 a term in which V« 2 is applied solely to f\ with the result 



22(p« 1 4 2 i V« 1 ') = 2J(0 1 )/J(0 l )> cf. iE. (86), 



and this multiplies the product f%...fn* There is a term in 

 which V e 2 is applied jointly to f Y and / 8 , the term being 



the bracket by (9) below is 



4(«,-tti) or *CA-/i) ] 



#1 — ^2 #1 ~~ #2 



and the product / 2 .../„ has the multiplier 8/(0i — 2 )« 



The whole result of V e 2 R« is a series of products of n— 1 

 factors / and will vanish if the coefficient of each of such 

 products vanishes, i. e. if 



iW +^-+ -J— -o r«n 



and rc — 1 similar conditions are fulfilled. This is the result 

 of writing in (4) the value \ = xi assuming that <j> is there a 

 polynomial L of degree n in X, and so 



Uej=9=e~ % ..J[=d. and 't(0 1 )fL(0 l ) = i(ff^0 t + ■O^ff} 



* The isotropic case is treated at length bv W. D. Niven in a paper 

 « On Ellipsoidal Harmonics," Phil. Trans, vol.* 182 (1891) A. 



