The planetary figure becomes unstable 551 
According to Poincaré’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 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 
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Axis oF | Rorarion 
Fig. 2. 
Planetary spheroid just becoming unstable. 
sketch by 8, 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 4+6 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 de- 
clines. 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. 
1 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. 
