OF EQUILIBRIUM OF A ROTATINO MASS OF LIQUID. 
331 
in the energy when the mass of fluid is subject to an ellipsoidal harmonic deformation. 
This section is a paraphrase of M. Poincare’s work, but the notation and manner of 
presentation are somewhat diflerent. The additional terms in the energy are shown 
to involve a certain coefficient, which is called by M. Poincare a coefficient of 
stability. It is clear that when any coefficient vanishes we are at a point of bifurca¬ 
tion. and the particular Jacobian ellipsoid for which it vanishes is also a member of 
another series of figures of equilibrium. 
In § 6 the principal properties of these coefficients, as established by M. Poincare, 
are enumerated. He has shown that the ellipsoid can bifurcate only into figures 
defined by zonal harmonics ; that it must do so for all degrees, and that the first 
bifurcation occurs with the third zonal harmonic. The order of magnitude of the 
coefficients of the several orders and of the same degree is determined. A numerical 
result seems to indicate that as the ellipsoid lengthens, it becomes more stable as 
regards deformations of the third degree and of higher orders, and less stable as 
regards the lower orders of the same degree. 
In § 7 the numerical solution of the vanishing of the coefficient corresponding to 
the third zonal harmonic is found, and it is shoAvn that the critical ellipsoid has its 
three axes proportional to ’GSOGG, *81498, 1‘88583, and that the square of the angular 
0)^ 
velocity is given by —= ‘14200. A short table is also given showing the march 
of the axes of the Jacobian ellipsoids from their beginning on to instability at this 
critical stage. The nature of the formula for the third zonal coefficient of stability 
seems to show that it can only vanish once—a point which it appears that 
M. Poincare found himself unable to prove rigorously. 
A suggestion is made for the approximate determination of the bifurcations into 
the successive zonal deformations, but no numerical results are given. 
In § 8 the nature of the pear-shaped figure is determined numerically, and the 
reader may refer to the figure above, where it is delineated. It will be seen to be 
longer than was shown in M. Poincare’s conjectural sketch. 
If, as M. Poincare suggests, the bifurcation into the pear-shaped body leads 
onward stably and continuously to a planet attended b}" a satellite, the bifurcation into 
the fourth zonal harmonic probably leads unstably to a planet with a satellite on 
each side, that into the fifth to a planet with two satellites on one side and one on the 
other, and so on. 
The pear-shaped bodies are almost certainly stable, but a rigorous and conclusive 
proof is wanting until the angular velocity and moment of momentum corresponding 
to a given pear are determined. To do this further approximation is needed. 
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