213 
195, 196] Ellipsoidal Configurations 
through this point being the Jacobian ellipsoids. As this latter series turns 
downwards from the point of bifurcation, the Jacobian ellipsoids also are 
unstable. Thus there are no stable configurations of equilibrium for a rota 
tion greater than that given by &> 2 /27ryp = 0*18712. When the rotation 
exceeds this amount, the problem ceases to be a statical one and becomes 
a dynamical one; here we shall not attempt to follow it. 
Stability when the angular momentum is increased. 
196. The problem just considered is of interest as illustrating the theory 
of points of bifurcation, but it entirely fails to represent the conditions which 
prevail in astronomical events. To represent natural conditions the mass 
must be supposed to rotate freely in space so that its angular momentum 
i 
» 
I 
remains constant. If the rotating body shrinks, its density will increase, and 
this may or may not result in an increase of angular velocity. To study the 
problem by the most direct method, we should have to look for series of 
configurations of constant angular momentum and varying density. It is, 
however, a convenience to suppose that the density remains constant while 
the angular momentum increases, and it is easily seen that this leads to 
exactly the same mathematical problem. We accordingly proceed to study 
the stability of the Maclaurin and Jacobian series, supposing p to remain 
constant while the angular momentum is made continually to increase. 
In this problem the angular momentum is given in the last columns of 
Tables XVI and XVII (pp. 210, 211), and in a diagram in which the angular 
momentum is taken for ordinate, the series will be found to be as in fig. 25. 
Clearly the Maclaurin spheroids will be stable up to the point at which they