LIQUID SPHEROID OP FINITE ELLIPTICITY. 
209 
is an ellipsoid, differing in form from the original spheroid by small quantities of the 
first order, whose axes make finite, not small, angles with those of the spheroid. 
Suppose the liquid spheroid is rotating steadily about its axis of figure with angular 
velocity co, and that this axis does not quite coincide with our fixed axis of z, but is 
inclined to it at a small angle, while the axes of x, y rotate about the axis of z with 
angular velocity w. The coordinates of the fluid particles will now no longer be 
constant, but will undergo small periodic changes. In the time 27r/(y both the fluid 
particles and the axes will come round to their original positions; thus, the period of 
the a'p'parent relative oscillations is 27r/&j, although the liquid is in reality rotating 
steadily. This accounts for the root k = ^, which occurs in the period-equation, a 
result which may be completely verified by rigorous analytical methods, 
The movements corresponding to the other two roots are somewhat similar to 
precession, the axis of figure of the spheroid turning about the axis of z, to which it is 
inclined at a small angle. 
Stability of the Spheroid. 
20. We have already alluded to Poincare’s investigations of the condition that 
the spheroid, if viscous, may be secularly stable, which requires that the energy 
of the system for the given angular momentum must be a minimum in the spheroidal 
form. The greatest eccentricity corresponds to the least value of I, which causes any 
one of the coefficients K;/(^) to vanish, and this is shown to be that given by 
Va) = o, whence, as in Thomson and Tait {§ 772), 
1/^ = tan a =/= 1-39457. 
If the liquid be perfectly invdscid, the criteria are very different. So long as the 
roots of the period-equations for the various waves and oscillations are all real, the 
spheroid cannot be unstable. It will, however, become unstable if for any harmonic 
the equation in k has a pair of complex or imaginary roots. For, calling these roots 
I i niL, we get the possible surface displacements 
h = 
h = C,/ T7T T/^ {f) 
compounding into the displacement 
h = C/ CT T,/“^ (/x) cos {s(j) + 2(Dlt) 6-“"-'', 
which increases indefinitely with the time. 
Let us imagine that our spheroid is subject to constraints such as to freely allow of 
its surface undergoing harmonic displacements of degree n and. rank s, but which 
MDCCCLXXXIX.- -A. 2 E 
