200 
MR. J. H. JEANS ON THE CONFIGURATIONS 
can be retained continually decreases. With an allotted amount of hydrogen, the 
ellipsoidal point of bifurcation may or may not be reached before matter has begun 
to be ejected from the equator of the figure. 
The simplest case for accurate discussion occurs when the core is treated as 
incompressible. In fig. 3 the thick curve represents the cross-section through the 
axis of rotation of an incompressible mass at the Maclaurin-Jacobian point of 
bifurcation, while the thin curve represents the equipotential Q = C^, which has a 
series of double points round its equator. If the volume of light material 
surrounding the core is just equal to the volume between these surfaces, matter will 
begin to be thrown off at the equator precisely at the moment at which the 
ellipsoidal point of bifurcation is reached. If the volume of light matter is at first 
greater than this critical volume, matter will be thrown off equatorially before the 
ellipsoidal point of bifurcation is reached, the amount ejected being such that the 
volume is just reduced to the critical volume when this point is reached. Conversely, 
of course, if the volume of light matter is initially less than this, the point of 
bifurcation will be reached before any matter is thrown off equatorially. 
A rough estimate shows that the critical volume, when the matter is incompressible, 
is about one-third of the volume of the core. For a compressible core for which 
y > 2'2, it is of course less, becoming equal to zero when y = 2'2 (approximately). 
This completes our collection of theoretical results. They may now be recapitulated 
and discussed with reference to actual astronomical conditions. 
Summary and Discussion of Results. 
45. Our discussion began with a general survey of the types of configurations 
which can be assumed by compressible astronomical masses in rotation. The result 
announced in a previous paper* that the incompressible mass provides a good model 
from which to study the behaviour of compressible masses has on the whole been fully 
confirmed. A mass of incompressible matter, shrinking while rotating, will assume 
first the shape of a spheroid, then that of an ellipsoid, afterwards becoming unstable 
* ‘Phil. Trans.,’ A, vol. 213, p. 457. 
