FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
307 
position of each molecule, multiplied by the mass of the molecule and summed 
throughout the body. In the system, as rendered statical, it is necessary to add the 
rotation-potential to the gra\dtation potential before effecting the summation. That 
portion, say T, of the whole lost energy which arises from the rotation-potential is 
simply the same thing as the kinetic energy of the mass, when the system is regarded 
as a dynamical one. If we replace W T hj E to denote the whole lost energy ot 
the statical system, the condition that the surface shall be in equilibrium is that the 
variations of E for constant angular velocity shall be stationary. E must then be a 
maximum or a minimum, or a maximum for some variations and a minimum for 
others. 
It might appear at first sight that the condition for the secular stability of the 
figure is that E should be a maximum for all variations, and this is so if certain 
constraints are introduced ; but in the absence of such constraints the figure may be 
stable although E is a minimax. 
It has been shown by M. Poincare that the stability must be determined from the 
variations, subject to constancy of angular momentum, of the total energy of the 
system, both kinetic and potential. The two portions of the total energy, say U, are 
again IF and T; but whereas E involves the lost energy IF of the system under the 
action of the gravitation 23otential, U involves the jiotential energy which is equal to 
— IF. Thus U is equal to — IF -j- T. 
The variation of U with constant angular momentum leads to results for the 
determination of the figure identical with those found from the variation of E with 
constant angular velocity. But there is this imjDortant difference, that to insure 
secular stability JJ must be an absolute minimum. It apjDears, in fact, that, in the 
case of the j^ear-shaped figure, while E is actually a maximum for all the deforma¬ 
tions but one, it is a minimum for that one, which consists of an ellipsoidal strain of 
the critical Jacobian ellijisoid from which the pear-shaped figures bifurcate (§ 19). 
But M. Poincare has adduced another consideration which enaliles us to determine 
the stability of the j)ear by means of the function E, without a dii’ect jiroof that U is 
a minimum for all variations. For he has shown that if for given angular momentum 
slightly less than that of the critical Jacobian ellipsoid, the only possible figure 
is the Jacobian, and if for slightly greater angular momentum there are two figures 
(namely, the Jacobian and the jDear *), then exchange of stability between the two 
series must occur at the bifurcation. If, on the other hand, the smaller momentum 
corresponds with the two figures and the larger with only one, one of the two 
coalescent series must be stable and the other unstable. Now it has been jDroved 
that the less elongated Jacobian elli 2 )Soids are stable, so that if the first alternative 
holds the stability must pass from the Jacobian series to the pear series; and if the 
second alternative holds the pear series must be unstable throughout. The question 
* For the sake of simplicity we may speak of a single pear, instead of two similar pears in azimnths 
180° apart. 
2 R 2 
