family must be regarded as the excess
of the ellipticity of this figure above that of all the earlier ones,
beginning with the slightly flattened planetary spheroid. Accordingly
the specific difference a of the family has gradually diminished from
the beginning and vanishes at this stage.

According to Poincare's principle the vanishing of the stability serves
us with notice that we have reached a figure of bifurcation, and
it becomes necessary to inquire what is the nature of the specific
difference of the new family of figures which must be coalescent with
the old one at this stage. This difference is found to reside in the
fact that the equator, which in the planetary family has hitherto been
circular in section, tends to become elliptic. Hitherto the rotational
momentum has been kept up to its constant value partly by greater speed
of rotation and partly by a symmetrical bulging of the equator. But now
while the speed of rotation still increases (The mathematician familiar
with Jacobi's ellipsoid will find that this is correct, although in
the usual mode of exposition, alluded to above in a footnote, the speed
diminishes.), the equator tends to bulge outwards at two diametrically
opposite points and to be flattened midway between these protuberances.
The specific difference in the new family, denoted in the general sketch
by b, is this ellipticity of the equator. If we had traced the planetary
figures with circular equators beyond this stage A, we should have found
them to have become unstable, and the stability has been shunted off
along the A + b family of forms with elliptic equators.

This new series of figures, generally named after the great
mathematician Jacobi, is at first only just stable, but as the density
increases the stability increases, reaches a maximum and then declines.
As this goes on the equator of these Jacobian figures becomes more
and more elliptic, so that the shape is considerably elongated in a
direction at right angles to the axis of rotation.

At length when the longest axis of the three has become about three
times as long as the shortest (The three axes of the ellipsoid are
then proportional to 1000, 432, 343.), the stability of this family of
figures vanishes, and we have reached a new form of bifurcation and must
look for a new type of figure along which the stable development will
presumably extend. Two sections of this critical Jacobian figure, which
is a figure of bifurcation, a