253-255] Internal Condition of Rotating Stars 287 
On taking the integration round a circle of constant angular velocity <f >, 
this takes the form 
so that ®- 2 ^» remains constant as the star shrinks. If the shrinkage is uniform, 
<j) changes uniformly, and is always proportional to w -2 or to p . Thus the 
average rotation &> of the core will remain proportional to p^ throughout the 
shrinkage of the star, and the value of a>‘ i J27ryp will be proportional to pK 
Although it has been convenient to regard our different configurations as 
belonging to masses of a given density p, we have seen that each separate shape 
of configuration corresponds to a given value of co^^rryp, so that there are an 
infinite set of configurations of any specified shape corresponding to the 
different values of cu and p which are consistent with a^j^-nyp having an 
assigned value. The angular momenta of the various configurations of any 
given shape are proportional to Mr 2 a>, where r is any linear dimension of the 
mass, or, since M is proportional to pr 3 and tffôiryp is assigned, the angular 
momenta are proportional to M 3 /A Thus if M is kept constant and p is 
allowed to increase, while the angular momentum remains constant, the mass 
will pass through the same sequence of configurations as though M and p were 
both kept constant while the angular momentum increased. This is precisely 
the series of configurations we have had under review. • 
• 255. We now see that the shrinkage which accompanies the passage of an 
actual star from one region of stable configurations to another—or if we so 
prefer to put it, the shrinkage which accompanies the ionisation of one ring of 
electrons—may result in fission of the star into two distinct masses. 
As we have seen, the final stages of the process of fission are probably 
cataclysmic, with the result that it is difficult to follow them out dynamically. 
Had the pear-shaped figure proved to be stable, the final stages of the fissional 
process would have consisted merely in a gradual deepening of the furrow 
round the pear until we were left with two detached masses in contact rotating 
as one single rigid body, the periods of rotation of the two masses and their 
period of revolution about one another all three coinciding. There is no longer 
the same justification for this supposition when it is recognised that fission 
occurs only after cataclysmic motion. 
We may, however, notice that only one vibration is unstable at the point 
of bifurcation at which cataclysmic motion begins, this vibration being one in 
which neither half of the mass gains upon the other, either in rotation or 
revolution. When the elongation of the pear-shaped figure first takes place, 
the pointed end of the pear must lag somewhat behind the rotation of the 
blunter end, as a consequence of conservation of angular momentum, but any 
such difference of rotation produces a distortion which corresponds to a stable