any internal currents generated in the liquid
by the cooling are annulled so quickly by fluid friction as to be
insignificant; further let the liquid always remain at any instant
incompressible and homogeneous. All that we are concerned with is that,
as time passes, the liquid star shrinks, rotates in one piece as if it
were solid, and remains incompressible and homogeneous. The condition is
of course artificial, but it represents the actual processes of nature
as well as may be, consistently with the postulated incompressibility
and homogeneity. (Mathematicians are accustomed to regard the density
as constant and the rotational momentum as increasing. But the way of
looking at the matter, which I have adopted, is easier of comprehension,
and it comes to the same in the end.)

The shrinkage of a constant mass of matter involves an increase of its
density, and we have therefore to trace the changes which supervene as
the star shrinks, and as the liquid of which it is composed increases
in density. The shrinkage will, in ordinary parlance, bring the weights
nearer to the axis of rotation. Hence in order to keep up the rotational
momentum, which as we have seen must remain constant, the mass must
rotate quicker. The greater speed of rotation augments the importance of
centrifugal force compared with that of gravity, and as the flattening
of the planetary spheroid was due to centrifugal force, that flattening
is increased; in other words the ellipticity of the planetary spheroid
increases.

As the shrinkage and corresponding increase of density proceed, the
planetary spheroid becomes more and more elliptic, and the succession
of forms constitutes a family classified according to the density of
the liquid. The specific mark of this family is the flattening or
ellipticity.

Now consider the stability of the system, we have seen that the spheroid
with a slow rotation, which forms our starting-point, was slightly less
stable than the sphere, and as we proceed through the family of ever
flatter ellipsoids the stability continues to diminish. At length when
it has assumed the shape shown in a figure titled "Planetary spheroid
just becoming unstable" (Fig. 2.) where the equatorial and polar axes
are proportional to the numbers 1000 and 583, the stability has just
disappeared. According to the general principle explained above this
is a form of bifurcation, and corresponds to the form denoted A. The
specific difference a of this