ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 501 
{ — daze — 26w*ze*} = — daze“ 
@ 
1 “tO. te Vente * ie 
= ipw*ze* — 2ipw*ze*} = — haxric 
WwW, = ho (x + wy)e*, 
whence 
u = — gaze*, v= — dare '*9, w= dpo(x + ty) &*9, 
and, in the corresponding real solution, - 
u = — 2fwz cos (wt + €), v = 2da2 sin (wt 4 €), 
w = 2ha[x cos (ot + e) — y sin (wt + e)]}. 
These are the component velocities relatively to the moving axes. ‘The velocities 
parallel to the instantaneous positions of the moving axes are 
u— Yo, v+ xo, Ww. 
The velocities parallel to the fixed O€, On, OZ, are therefore 
(u — yo) cos ot — (v+ wo) sinot = — 2fwz cose — wn, 
(v + zo) cos wt + (wu — yo) sin wot = 2do2 sin € + o€, 
w = 2bw {Ecose — yn sine}. 
Thus the fluid motion is a motion of pure rotation, the component angular velocities 
about the axes being 
— 2¢w sin e, — 2a cos «, o. 
The resultant of these angular velocities is an angular velocity » about the line 
whose direction cosines are 
— 2sin e, — 2¢ Cos €, il: 
The similar case for the spheroid with a free surface has been already discussed by 
Bryan (‘ Phil. Trans.,’ 1889). 
(6). Next take 
A= o[1 + (q+) (1 + gf = (1 + E) say 
where 
E=4(q+«) (1 +9). 
Substituting this value of \ in (46) and putting a = y (1 + 4), &c., we obtain 
