The planetary figure becomes unstable 



551 



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 1 , 

 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 



Fig. 2. 

 Planetary spheroid just becoming unstable. 



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 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. 



