k 



;< 



458 Mr. R. Har greaves on 



formula, V 2 f + % = ^. We set out from 



d(0 _ 4-7T * 



which is used in the form 



dw 47T 



and differentiate with regard to a 2 , giving 

 P 3/ 2 <ia> _ 4-7T 



J (Sa»P + X)t - (5 + X)7pH)(« ! + X) ^+x) • ( 141) 

 Sinceymdco/(2)a 2 / 2 -l-X)i and other products vanish, we get 

 by multiplying (141) by x?y 2 z 2 , 



J' d{la? + 7ny + nz) 2 dcQ _ 4tt / ocr y z 2 \ 



(2a 2 / 2 f X)t " ^/( )( )( ) I ^+X + 6 2 +\ + c 2 + \/ 



J:7T P d(0 



~V( ")rK~)"J(2^+^" 



In virtue of this equation^ =0, and therefore 



The part of V 2 "^ due to explicit differentiation is 

 _ , P ^o) Airp Q abc 



~ P ° abC )(XaV + \)t y/(<f+\)P+\)(?+\) '' 



the part due to implicit differentiation through \ is 

 3p abc P (Z# + my + n ~) do* (-.dX d\ d\\ 



~Tj &an 2 +\)C~\;fa +m fcj + n fa) 



__ __ r== 27rp abc _ y #_ d\, ^^ 



V / (a 2 + \)(6 2 + X)(c 2 + X) a 2 + \dx y J 



= y L° by a simple property of X. 



* Independently proved thus : - 



= c ^ w _ f 2,r r +i — 



J(ia*P)i = J ^ I [o^cos 2 <^+6 2 sm r ^T^ 2 (c 2 -« 2 cos 2 (f>-b 2 sin 2 0)]f 



a a cos 2 + & 2 sin a (£ |_ f^cos^+a 2 sin 2 <£+rc 2 (c 2 — « 2 cos 2 <£-& 2 sin 2 #)}iJ _ x 



■I 



2dcf> 



2* 



c(a a cos 2 $+6 a sin 2 <f>) — abc' 



