FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
33 
27. We must now consider what motion is to be expected in a Jacobian ellipsoid 
which has reached the point of bifurcation at which instability sets in. Poincare 
remarks* that if the pear-shaped figure proved to be unstable, “ la masse fluide 
devrait le dissoudre par un cataclysme subit.” 
After reaching the point 0 in our diagram, the mass cannot move along the 
pear-shaped series, since this would involve a decrea.se of angular momentum. It 
may be thought of as moving along the unstable branch O P' of the series of Jacobian 
ellipsoids for an infinitesimal time until some slight disturbance brings its instability 
into play. 
Now of all the vibrations of this figure, it is known that one only is unstable, 
namely that corresponding to the third zonal harmonic of the ellipsoid. The initial 
motion of the figure must then be one in which the displacement at every point of 
the surface is proportional to the third zonal harmonic. 
Thus the fluid begins by describing exactly the pear-shaped series, but as soon as 
the changes in angular momentum become appreciable, it leaves this series, and passes 
through a series of configurations represented in the region above O in fig. 1. These 
may at first be thought of as lying parallel to the pear-shaped series, but above O. 
If there is a stable branch such as P/S' which ultimately passes above 0, it is 
conceivable that the series of non-equilibrium configurations might ultimately coalesce 
with the series of equilibrium configurations P'S', and the motion would be continued 
along this series. In this case, M. Poincare’s “ cataclysme subit ” would consist in a 
jump from the stable series PO to the stable series P'S'. 
Judging from the results of my parallel investigation on the configurations of 
rotating cylinders, this possibility does not seem at all likely. It is, I think, much 
more probable that the pear-shaped series lies like the series OP in my figure. 
In this case also the liquid would move through a series of configurations which 
would initially be close to the series OP, but would get continually futher removed 
from configurations of equilibrium. The protuberance resulting from the initial third 
harmonic displacement would develop in a manner somewhat similar to that of the 
pear-shaped figure, but as the motion would necessarily be possessed of a considerable 
amount of kinetic energy, the phenomenon would be a dynamical and not a statical 
one. If the configuration represented at 0 in fig. 1 is the highest stable configuration 
possible for a single mass of liquid, this kinetic process can end in only one way, 
namely in the separation of the mass of liquid into two parts. As the third harmonic 
displacement develops, the region of tire pear which moves with greatest velocity is 
known to be the extreme end of the protuberance. It is, therefore, natural to suppose 
that this part of the figure would be shot away first. Moreover as the departure 
from a figure of equilibrium is probably pretty pronounced before the separation takes 
place, it is likely that the mass in question will be shot away with a considerable 
velocity. 
* Letter to Sir G. Darwin in the latter’s ‘Coll. Works,’ III., p. 315. 
VOL. CCXVII.-A. F 
