﻿Harmonics, JEolotropic and Isotropic. 45 



Since 



dx ^ p + mV ' + nq ^ + ' ' ' ~ 2 * l( ^ + my + nz ) =2 ( Lv + m V + nz)l(0 x - Q ) , 

 ^ { l p + mr ' + n q') + ...^2(lx + my + nz)(^^ + ...Y 

 and y e 2 (lx + my + nz)R n = 



1(0) 1 1 1 



and 7i—l other similar conditions are satisfied. This being 

 the modification of (8) required for one linear factor, a 

 second linear factor will involve the addition of 1/2(^ — 0/) 

 to (8 ii.), and for three linear factors the condition is 



4j^J+^7^ + -^^-°- • • • M 



For the external harmonic of type (ii.) we seek a func- 

 tion % of X which makes 



S7 e 2 X {lz + my + m)Ti n = 0. 



The forms in (10 6) have the additional factor Ix + my + nz, 

 and there is a further term 



2 % R »[|^(^ + ^ + ^) + •••} 



which 



= 4% R* (fa + m# +nz)/(\- O ) 



by using (14) and (15). Thus in lieu of (11) we have 



JC+ 2J{\) + \-0 V>-0i"'\-0n) ' 

 and _ p dX 



% "J. (X-6> )(X-6> 1 ) a ...(X-^)VJ(V). 



The condition for the simplification of the integral is got 

 by writing 



f(a) = i/(x - O ) (a - o 2 y ... (x - n y 



in the argument leading to (13), and is precisely the equa- 

 tion (8 ii.). The value of x tnen shows the terms in (13) 

 with the further divisor X - O) i. e. 



1 1 f x 



~ 2 for 



*i-0. JW {L(0i)) 2 ' 



