﻿42 Mr. R. Hargreaves on Ellipsoidal 



We now give the proof of the theorem 



\-o ' 41/ a* a* + " +r W §F + §F§^r-J~ ' ' (9) 



the analogue of 



required above. 



To deal with the coefficient of x 2 we derive from iE. (81) 



\A a { a„(p« + rV + q f p\ + 7 „'(/a + qy f + p'/3')x + &'(/« +pV + ^x } 

 = ^a^--|«a(Aa + CV + B'^)x+7e / (C'« + B 7 ' + AWx 

 +/V(B'a + Ay + C/3')J. 



In each double bracket the expression is symmetrical as 

 regards \ and ; if the formula is written with X and 6 

 interchanged and subtracted from the above, the result 

 expresses the fact that the coefficient of a? in (9) vanishes. 

 As (9) is true with any values of the parameters we may 

 make %=1, and the first member is then —fo/(\—0). If 9 

 is made =M, the square bracket in (9) =0, this equation 

 replacing the orthogonal condition in isotropic work. 



The solution R M applies to space within the ellipsoid ; we 

 shall prove that it is a normal solution by finding a function 

 X of X such that )(R n is an external solution. Now, if: <f> is 

 expressed in terms of xyz and A, we have 



-%V/*= -<^<t> (explicit) + 2 1^ J +4p 



+2 m~A p ^ +r w +q ^r- =0i • (10) 



and therefore for <£=^R n 



M«+ tf /JH* { 2r (p£ +' £ +• £).+■■•} =°- ( 10 *> 



Since -= — '= S/ 2 .-./» -r-^ if we apply (9) the double 



bracket is 4R n 2- — ^-; the factor R^ may be removed, and 

 A, — Ui 



X is given by 



whence " ' 



.(\-«»)VJ(x). 



