4'2 Ellipsoidal Configurations of Equilibrium [CH. n 



the mass must be supposed to rotate freely in space so that its angular mo 

 mentum remains constant. As it shrinks, its density will continually increase 

 and this may or may not result in an increase of angular velocity. To studj 

 the problem by the most direct method, we should have to look for series o 

 configurations of constant angular momentum and varying density. It is how 

 ever a convenience to suppose that the density remains constant while th< 

 angular momentum increases, and it is easily seen that this leads to exactb 

 the same mathematical problem. We accordingly proceed to study the sta 

 bility of the Maclaurin and Jacobian series, supposing p to remain constan 

 while the angular momentum is made continually to increase. 



In this problem the angular momentum is given in the last columns o 

 the tables on pp. 39 and 40, and in a diagram in which the angular mo 



Fig. 6. 



mentum is taken for ordinate, the series will be found to be as in fig. 6. Clearb 

 the Maclaurin spheroids will be stable up to the point at which they meet thi 

 Jacobian ellipsoids. At this point of bifurcation they lose their stability, am 

 since the series of Jacobian ellipsoids turns upward at this point, it followi 

 that stability passes to them. 



If the mass is constrained to remain ellipsoidal there is no further pom 

 of bifurcation on the Jacobian series, and, as the angular momentum con 

 tinually increases along this series, it follows that all configurations on it ar< 

 stable. But it will be found later (Ch. V) that when the constraint to remaii 

 ellipsoidal is removed, the Jacobian series loses its stability at a certain stag< 

 by meeting a series of non-ellipsoidal (pear-shaped) configurations. This ha; 

 been anticipated in our diagram. 



