552 The Genesis of Double Stars 
At length when the longest axis of the three has become about 
three times as long as the shortest}, 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, are shown by the 
dotted lines in Fig. 3; the upper figure is the equatorial section at 
right angles to the axis of rotation, the lower figure is a section 
through the axis. 
Now Poincaré has proved that the new type of figure is to be 
derived from the figure of bifurcation by causing one of the ends to 
be prolonged into a snout and by bluntening the other end. The 
B 
Fig. 3. 
The ‘ pear-shaped figure” and the Jacobian figure from which it is derived. 
snout forms a sort of stalk, and between the stalk and the axis of 
rotation the surface is somewhat flattened. These are the character- 
istics of a pear, and the figure has therefore been called the “ pear- 
shaped figure of equilibrium.” The firm line in Fig. 3 shows this new 
type of figure, whilst, as already explained, the dotted line shows the 
form of bifurcation from which it is derived. The specific mark of 
this new family is the protrusion of the stalk together with the other 
corresponding smaller differences. If we denote this difference by ¢, 
while A +6 denotes the Jacobian figure of bifurcation from which 
it is derived, the new family may be called 4 +6-+c, and ¢ is zero 
initially. According to my calculations this series of figures is stable’, 
1 The three axes of the ellipsoid are then proportional to 1000, 432, 343. 
2 M. Liapounoff contends that for constant density the new series of figures, which 
M. Poincaré discovered, has less rotational momentum than that of the figure of bifurca- 
tion. If he is correct, the figure of bifurcation is a limit of stable figures, and none can 
