LIQUID SPHEROID OP FINITE ELLIPTICITY. 
195 
From equations (5) we obtain 
2.(l-K=)V = «|-|t 
2a>(l -k3) W = 
Multiplying by 0^c/0N, 0y/0N, 0^/0N, and adding, we find at the surface 
2.(1-k^)(U^ + v|+w| 
(34). 
= LK 
d-yj/' dx d-yfr dy g d\Jr dx V /0-v/r ^ 
dx dN dy dN ' ^ dz dN) \dy 0]Sr 
dyjr dy 
dx 0]sr 
But U0a;/0N + V’0y/0N + W0z/0N is the normal velocity of the liquid relative to 
the moving axes, and is therefore equal to dh/dt or to ‘ZccoKh. We have, therefore, 
4ct)®(l — k^)kIi 
_ d^/d'\lrdx d‘y\rdy „d-\lrdz\ 0f /d^dx dijrdy\ 
^ dN \dx 0f dy 0f ^ dz d^/ 0N \02/ 0f dx d^J 
= KC% 
^ y d^ I X d'yp' y dy^r 
dN \c^ + 1) 0a:: + I) dy 
= KC%^I- " 
d_^ 
"T 
dz ) to' + 1 dN \ dy ^ dx 
ifo 0l/r 
y dyfr , z' dyjr 
+ 
=0 dN (V - 1) 0a: F- (V - 1) dy k\^ dz') + 1 dN 00 
_ di; \ Kc^ df _ I 0f ] 
Ii/qF 0Z/ ^,^ + ldcf)j . 
Now, taking 0 as given by (21), we have 
moreover, since by (23) 
dyjr 
= ‘®''' 
(36); 
T.'(>.) = (,^-1)«D-P„(.), 
where the symbol D stands for differentiation with respect to v, therefore 
DT,/ (v) = - ly^ {y + sv/{v^ ~ 1) . D^P, (v) ] . 
Also at the surface jx' is equal to /r. 
2 c 2 
(37). 
