﻿Harmonics , jEolotropic and Isotropic. 43 



In virtue of the condition (8) this integral may be referred 

 to the form j*(AX 2 + BX + C) ~, or as jjd\/j 3 / 2 is inte- 

 grate to the forms [d\{l or X)/J 3/2 [or, if we please, to 

 \d\/\/J and f<iX/J 3/2 ], i.e. such integrals as occur in the 

 calculation of force. Thus from 



r (\+y')d\ = x+y _ r <& fi-^+Wx 



Ja(X-0)VJ (X-^^/J 1 (\-0)y/J L 2J J 



we see that the condition that X — shall disappear from the 

 denominator of the integral is 2J(0) = (0 + y')J(0), or, if 

 we write + y' = y y the condition is y = 2J(6)/J(0) and the 

 integral is 



C"dX ( 1 7 \_ 7 f (2J-yJM X 



J A v^U-* X=0|V~ (X-^v^J J, 2^-^)J 3;2 ' ' l ' 



The condition as regards 7 gives a special value to the ratio 

 of numerators of \—0 r ? and X— X when 1/X — 0j\ 2 ...\ — QJ 

 is expressed by a series of partial fractions. Thus with 

 F(X) = l/\ — # 2 j 2 ...X— 0„| 2 , the condition is 



frg.) _i _F'(g.)_ r 1 , , 1 i 



which is precisely (8). The value of % or I 2 is then 



J a. ^ v J 



s 1 r l _ 



, 1 ("° {J(x)M)-J(x)J(gi)}<fl n „« 



+ 2J(^) J, " (x-^){J(x)P J- • (io; 



§ 4. In the isotropic solution the linear factors which may 

 occur are m, y, and z ; for seolotropy they are to be found, 

 i.e., for example, <j>=(lj; + my + nz)'R, l being assumed to be a 

 solution, the ratios I : m : n are required. Now 



V« 2 (te + my + nz) E„ = (la + my + nz) V„ 2 R„ + 2 [— * (If + rarf + nq') 



