of Edinburgh, Session 1881-82. 
611 
that the liquid may be in equilibrium in the shape of two, three, 
four, or more separate rings, with its mass distributed among them 
in arbitrary portions, all rotating with one angular velocity, like 
parts of a rigid body. It does not seem probable that the kinetic 
equilibrium in any such case can be stable. 
( d ) The condition of being a figure of equilibrium being still 
imposed, the single-ring figure, when annular equilibrium is 
possible at all, is probably stable. It is certainly stable for very 
large values of the moment of momentum. 
(e) On the other hand, let the condition of being ellipsoidal be 
imposed, but not the condition of being a figure of revolution. It 
is proved in Thomson & Tait’s Natural Philosophy, that whatever 
be the moment of momentum, there is one, and only one, revolu- 
tional figure of equilibrium. 
I now* find that, 
(1) The equilibrium in the revolutional figure is stable, or 
/ / £* 2 \ 
unstable, according as / f = - — - ) is < or > 1*39457. 
(2) When the moment of momentum is less than that which 
makes /= 1*39457 (or eccentricity = *812663) for the revolutional 
figure, this figure is not only stable, but unique. 
(3) When the moment of momentum is greater than that which 
makes /— 1*39457 for the revolutional figure, there is, besides the 
unstable revolutional figure, the Jacobian figure with three unequal 
axes, which is always stable if the condition of being ellipsoidal is 
imposed . But, as will be seen in (/) below, the Jacobian figure, 
without the constraint to ellipsoidal figure, is in some cases 
certainly unstable, though it seems probable that in other cases it 
is stable without any constraint. 
(/) Referring to Thomson & Tait’s Naturcd Philosophy, § 778, 
and choosing the case of a a great multiple of b, we see obviously 
that the excess of b above c must in this case be very small in 
comparison with c. Thus we have a very slender ellipsoid, long in 
the direction of a, and approximately a prolate figure of revolution 
relatively to this long a-axis, which, revolving with proper augular 
velocity round its shortest axis c , is a figure of equilibrium. The 
* Proof of these results, (1), (2), and (3) will he found in the forthcoming 
new edition of Thomson & Tait’s Natural Philosophy, vol. i. part ii. 
VOL. XI. 4 c 
