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Proceedings of the Boyal Society 
motion so constituted, which, without any constraint, is a 
configuration of minimum energy or of minimax energy or of 
maximum energy, for given moment of momentum, is a configura- 
tion of minimum energy for given moment of momentum, subject to 
the condition that the shape is constrainedly an ellipsoid. From 
this proposition, which is easily verified, in the light of § 778 of 
Thomson & Tait’s Natural Philosophy , I infer that, with the 
ellipsoidal constraint, the equilibrium is stable. The revolutional 
ellipsoid of equilibrium, with the same moment of momentum, 
is a very flat oblate spheroid; for it the energy is a minimax, 
because clearly it is the smallest energy that a revolutional 
ellipsoid with the same moment of momentum can have, but it is 
greater than the energy of the Jacobian figure with the same 
moment of momentum. 
(g) If the condition of being ellipsoidal is removed and the 
liquid left perfectly free, it is clear that the slender Jacobian 
ellipsoid of (/) is not stable, because a deviation from ellipsoidal 
figure in the way of thinning it in the middle and thickening it 
towards its ends, or of thickening it in the middle and thinning it 
towards it ends, would with the same moment of momentum give 
less energy. With so great a moment of momentum as to give an 
exceedingly slender Jacobian ellipsoid, it is clear that another 
possible figure of equilibrium is, two detached approximately 
spherical masses, rotating (as if parts of a solid) round an axis 
through their centre of inertia, and that this figure, is stable. It is 
also clear that there may be an infinite number of such stable 
figures, with different proportions of the liquid in the two 
detached masses. With the same moment of momentum there are 
also configurations of equilibrium with the liquid in divers propor- 
tions in more than two detached approximately spherical masses. 
(h) Ho configuration in more than two detached masses, 
has secular stability according to the definition of (k) below, 
and it is doubtful whether any of them, even if undisturbed by 
viscous influences, could have true kinetic stability ; at all 
events, unless approaching to the case of the three material 
points proved stable by Gascheau (see Kouth’s Rigid Dynamics , 
§ 475, p. 381). 
if) The transition from the stable kinetic equilibrium of a 
