on 
S 
ve 
ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 
0, cos wt — 8, sin wt = 2¢ cos [(X — @) t 4 «|, 
6, sin wt + 6, cos wot = — 2¢ sin [(A — @) t + e€]. 
Thus the motion of the principal axis consists approximately of a simple harmonic 
motion in period 27/(\ — w) = 27/Ka, in virtue of which it describes a small cone 
about its mean position in a direction opposite to that in which the system is rotating. 
The position of the instantaneous axis of rotation of the shell is defined by the 
direction cosines 
b,/0, A,/, iL, 
or 
S aA 
— 26 * sin (Ai+e), — 20 > 008 Qt+e«), 1; 
and since X is approximately equal to w, this axis will be very nearly coincident with 
the principal axis. 
From (48) we have also 



and therefore from (34) 
Ste ead Pe ienbisen « 
w=, “39 ida” (az + wz) e ; 
to the same order of approximation we have 
t — eae a ry ee Oc Oe) 
Whence, in the corresponding real solution 

Le — doz cos(At+ €), v= egy wz sin (At + €), 
cee 

Dap dw | — x cos (At + €) + y sin (At + €)]. 
The velocity components relative to the fixed axes are therefore 
. 6 soo 
(wu — yw) cos wt — (v + ow) sin of = Tay poz cos (A — wt + €) — on, 

: 6 == 
(v + za) cos ot + (uv — yo) sin of = Teor poz sin (A — ot + €) + o€, 
6 : meee ! 
i ~ 1 + 24% {Ecos (A — wf + €) — nsin(A — wt + €)}. 
