ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 505 
therefore . 
0, = $/(m + 9a)e*, 
Oy = — Ip r/ (ity + Jen). 
The corresponding real solution is 
0, = 2b4/(m, + 941) cos (M + €), 
0, = 26 \/ (ky + Gey) sin (Aé + €). 
The angular displacements about O€, Oy are therefore 
9; cos wt — 8, sin wt = $ {o/ (mq, + Ge) + (Ka + Geo) cos[(w + d)t + €] 
+b {V(r + 96) — A (Ke, + Je2)} cos [(o — A) t—e], 
§, cos wt + 6, sin wot = o{4/(K, + Ge) + V(Ko + Ge)} sin [(w + A)t + e€] 
+ ob {J (e + 9a) — (ke + Ge)} sin [(w — A)t — €] 
The motion of the principal axis in space consists, therefore, of a combination of 
two simple harmonic motions, the period of each being approximately equal to 
the period of rotation of the system, and the amplitudes being in the ratio 
J (Ky + 96) = VA (Ka + 9) 2 A (KH, + Ge) + A (ka + Gea); in virtue of each of these 
oscillations, the principal axis will describe a cone of revolution in the direction in 
which the system is rotating. In the event of the system being symmetrical about 
the axis of rotation x, = «, and ¢ =, in this case the amplitude of one of the 
oscillations reduces to zero. 
We have likewise 
— ~ 26 * /(m, + ge:) sin (At + €), 
6, r 
*= + 2¢ is / (Kz + qéq) cos (At + €), 
@ 
#1 08 wt 7 3, ol ot = — > * J («, + 96) + S(k2 + Geo)} sin (w + At + e€) 
—¢ ~ iV (er + 941) — V/ (ke + Ge2)} Sin (v/@ — Mt — €), 
Cae 6, : : 
(> Snot + ~-cos wt = > VAC + ge) + / (ky + Geq)} cos f(a + A)t + ef 
+b 6 (1, + Ge) — V/ (He + Ge)} cos (=A) eh. 
The motion of the instantaneous axis of rotation of the shell is therefore in all 
respects similar to that of the principal axis, but the semi-vertical angles of the cones 
described are smaller in the ratio \: @. 
MDCCCXCV.—A. 3 °T 
