OF ROTATING COMPRESSIBLE MASSES. 
159 
regarded as the data of the problem, the interior arrangement of the matter is 
uniquely determined. 
3. The first consequence of the foregoing theorem is that when the boundary of a 
mass is fixed, all the internal vibrations are necessarily stable. For the change from 
stability to instabilit)^ can only occur through a vibration of zero-frequency, and this 
would require that there should be two contiguous equilibrium arrangements of the 
interior matter, a possibility which is excluded by the result just obtained. 
The only possible configuration for a mass at rest and under no tidal forces will 
clearly be one in which the boundary is spherical and the surfaces of constant density 
are also spherical and concentric with the boundary. # If the mass is set into slow 
rotation, this system of concentric spheres will give place to a system of concentric 
spheroids. As the rotation increases further, the surfaces of equal density will no 
longer be strictly spheroidal, but it is clear that there must always be a linear series 
of configurations of equilibrium in which the boundary has the shape of a figure of 
revolution. This series of course reduces to the series of Maclaurin spheroids when 
the matter is incompressible. 
Excluding for the present the case in which <r (the density at the boundary), 
vanishes, it can be shownf that there must be an infinite number of points of 
bifurcation on this series of figures of revolution, these corresponding to the different 
sectorial harmonics of the figure. The first point of bifurcation corresponds to the 
second sectorial harmonic : when this is reached the circular cross-section gives place 
to a slightly elliptic cross-section, and this leads to a series of figures having three 
planes of symmetry and three unequal axes, these figures reducing to the Jacobian 
ellipsoids when the matter is homogeneous and incompressible. It is convenient to 
refer to these two series as the series of pseudo-spheroids and pseudo-ellipsoids 
respectively. The first point of bifurcation on the series of pseudo-ellipsoids corre¬ 
sponds to a third harmonic displacement and leads to a series of pear-shaped figures. 
It seems almost certain, although it has not been rigorously proved, that this last 
series of figures ends by fission into two detached masses revolving round one another 
as in the ordinary binary star formation. Assuming this, we may regard the passing of 
the first point of bifurcation on the series of pseudo-ellipsoids as the beginning of the 
process of fission. Until this stage is reached we have seen that the only possible 
figures of equilibrium for a rotating mass are pseudo-spheroids and pseudo-ellipsoids. 
For an incompressible mass the pear-shaped figures are unstable, so that spheroids 
and ellipsoids are the only possible figures of stable equilibrium. If the pear-shaped 
* The special application of this to the figure of the Earth has been discussed in a separate paper 
(‘ Roy. Soc. Proc.,’ A, vol. 98, p. 413). 
t ‘ Monthly Notices of the R.A.S.,’ vol. 77, p. 189. In this paper I was mistaken in thinking that 
o- = 0 presents a true exception to the general theory. The general argument there given failed to prove 
the result in the special case of o- = 0, but the result is true nevertheless, as will appear from the present 
paper. 
Y 2 
ft 
