436 Mr. R. Hargreaves on 



§27. The special forms suited to the motional problem 

 with the ellipsoid referred to principal axes are got by putting 

 a' = V = c' = 0, A'=B'=:O' = ; and writing as in (72) 1-jj 2 

 for p, —qr for p' ... , which involve 



P = l-q 2 -r 2 , F'=gr, A, p =l-Sp 2 . 



The equation to an yeoiian surface is then 



l + \Za(l-2f)-{-\ 2 Zhc(l-q 2 -r 2 )+\hibc(l-%p 2 ) 



= %ax 2 { 1 + X {bl^f + cT^?) + \ 2 bc (l-q 2 - r 2 ) } 



+ 22bcqryz\(l+\a), . . . (100) 



the equation giving the value of J on the left hand, those 

 of J a.,. . . J a! , . . on the right. Or if we write 1/a 2 for a, .,-. so 

 that the new a's are principal semi- axes, 



a W + \ Xb 2 c 2 {l-p 2 ) + X 2 %a 2 {l-q 2 - r 2 ) + \*{l-p 2 - q 2 - r 2 ) 

 = tx 2 lb 2 c 2 + \{c 2 (l-q 2 )+P(l-r*)}+\ 2 {l-q 2 -r 2 )] 



+-2Sqryz\[a 2 + \) . . (101) 



We may use J' for the expression on the left, and then in 

 terms of J / we have for a conductor 



♦-Wt. s= ^(i-v)r^ =g £(i^!)^ (102 , 



b7r Ja vJ' 1<07r J° v J u 



and for a volume distribution 



* = ~ ( ^ 



^ ^e 2 (l-Xp 2 ) r d\ =P e(l-Xp 2 ) 



:i-x P 2 ) r ax 



407T J v/j/- 10 ^ *°' 



>- (103) 



the term multiplying J' -1 in ^ being the right-hand member 

 of (101). 



§ 28. We may now consider the derivation of T and E 

 from S, and the comparison of their values with the statical 

 case. First notice the form of J', viz. 



J=^r + X)(5 2 + X)(c 2 + X)-/X(?; + \)(r+X)-fX(r + X)(a 2 + X) ) 



-rX(a-+X)(i 2 + X) ] 



=<-* +x w + m* + x)[i-&-&-£d >> (104) 



