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 1 , 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 Poincar^ 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 



Axis 



o 



A 



: o c 



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 difibrences. If we denote this difference by c, 

 while A+b denotes the Jacobian figure of bifurcation from which 

 it is derived, the new family may be called A + b + c, and c is zero 

 initially. According to my calculations this series of figures is stable 2 , 



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. Poincar6 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 



