ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 473. 
On expanding and arranging according to powers o, X, this determinant 
reduces to 
MTA (B° + y*) + v (8 — 7°) [Ble + 7?) +e (2 — 7°) 
— ot? (a? + y*) (BP + 7’) (B — C — v) (A — C — p) + 4028? (A + v) (B+ B) 
ey (Be Cea Ane 13] 
=o {4028? (B—'C — yp) (A —€ = a) 05 ks Sete ee Se I): 
§ 2. Case of Shell without Inertia. 
If the shell be so thin that we may neglect its inertia compared with that of the 
fluid, we may put A = B = C= 0, and equation (5) then becomes 
(22 — yf) (B= 7) X= wR? fy — 3 (2 +B) YP + dB") + dole’ = 0 (6); 
when the system is symmetrical about the axis of rotation «? = 6, and this equation 
reduces to 
AM (a2 — y2)? — wd? (a2 — y?) (5a? — 2) + 4o!a2B? = 0, 
a lea ala i ee ess eet rt) 
These are the same as the values obtained by Bryan (‘ Phil. Trans.,’ 1889, A, 
p- 208), for the case of a spheroid whose surface is free. As is there indicated, the 
modes of oscillation corresponding to these periods are such that the surface of the 
spheroid maintains its shape, but changes its position. Such oscillations wiil, 
of course, not be affected by supposing the fluid contained in a rigid shell without 
inertia, and we might have expected to obtain the same values for the periods, when 
the figure of the shell agrees with a possible figure of equilibrium of the fluid rotating 
freely. : 
From (7) we see that the roots, if real, are positive; in order that they may be 
real, we require that 9a” — y? and a — y? must have the same sign. 
Hence a necessary condition for ordinary stability is 
the roots of which are 

vy > 9a" Nor “V7 <a 
1.€., y must not lie between « and 3a. 
Returning to the case where « + £?, in order that the roots may be real and 
positive, we must have 
CHG = Gai 82) > 0. 
(2) y* — 3 (a? + B’) y’ + 5a°P? > 0. 
eet) yt 08 Bem Gar By: a?) (ye 8) Os 
MDCOCKCV.—A. 3 P 
