550 The Genesis of Double Stars 



A real star radiates heat, and as it cools it shrinks. Let us 

 suppose then that our ideal star also radiates and shrinks, but let 

 the process proceed so slowly that any internal currents generated 

 in the liquid by the cooling are annulled so quickly by fluid friction 

 as to be insignificant ; further let the liquid always remain at 

 any instant incompressible and homogeneous. All that we are con- 

 cerned with is that, as time passes, the liquid star shrinks, rotates 

 in one piece as if it were solid, and remains incompressible and 

 homogeneous. The condition is of course artificial, but it represents 

 the actual processes of nature as well as may be, consistently with the 

 postulated incompressibility and homogeneity 1 . 



The shrinkage of a constant mass of matter involves an increase 

 of its density, and we have therefore to trace the changes which 

 supervene as the star shrinks, and as the liquid of which it is com- 

 posed increases in density. The shrinkage will, in ordinary parlance, 

 bring the weights nearer to the axis of rotation. Hence in order 

 to keep up the rotational momentum, which as we have seen must 

 remain constant, the mass must rotate quicker. The greater speed 

 of rotation augments the importance of centrifugal force compared 

 with that of gravity, and as the flattening of the planetary spheroid 

 was due to centrifugal force, that flattening is increased ; in other 

 words the ellipticity of the planetary spheroid increases. 



As the shrinkage and corresponding increase of density proceed, 

 the planetary spheroid becomes more and more elliptic, and the 

 succession of forms constitutes a family classified according to the 

 density of the liquid. The specific mark of this family is the flatten- 

 ing or ellipticity. 



Now consider the stability of the system. We have seen that 

 the spheroid with a slow rotation, which forms our starting-point, 

 was slightly less stable than the sphere, and as we proceed through 

 the family of ever flatter ellipsoids the stability continues to diminish. 

 At length when it has assumed the shape shown in Fig. 2, where 

 the equatorial and polar axes are proportional to the numbers 1000 

 and 583, the stability has just disappeared. According to the general 

 principle explained above this is a form of bifurcation, and corre- 

 sponds to the form denoted A. The specific difference a of this 

 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. 



1 Mathematicians are accustomed to regard the density as constant and the rotational 

 momentum as increasing. But the way of looking at the matter, which I have adopted, 

 is easier of comprehension, and it comes to the same in the end. 



