132 



Cataclysmic Motion 



[CH. vi 



II. THE ROTATIONAL PROBLEM 



132. The diagram of the equilibrium configurations of a rotating mass 

 of incompressible liquid has been seen to be of the type shewn in fig. 25. At 

 first the mass moves along the series of Maclaurin spheroids SM until it 

 comes to the point of bifurcation M. At this point the Maclaurin spheroids 

 lose their stability, and the motion proceeds along the series of Jacobian 

 ellipsoids MJ' until the point of bifurcation J is reached. At this point the 

 Jacobian ellipsoids lose their stability. The second series through J is, as 

 we have seen, a series of pear-shaped figures such as JP in the diagram. The 



s 



Fig. 25. 



angular momentum of these figures decreases as we proceed along the series 

 from J, so that the series is unstable and the curve JP turns downwards in 

 the diagram after leaving J. Thus there is no stable configuration beyond J, 

 and dynamical motion of some kind must occur as soon as shrinkage has pro- 

 ceeded so far that the angular momentum is greater than that represented 

 by the point J. 



In the tidal problem we saw that the dynamical motion, when it occurred, 

 was along the unstable series through the point at which the dynamical 

 motion commenced. In the present problem such a solution is impossible, 

 since the angular momentum must remain constant through the dynamical 

 motion and equal to that at J. 



Judging from the analogy of the two-dimensional problem, we may be 

 fairly confident that the series of pear-shaped figures JP ends in a configura- 

 tion P at which the mass divides into two parts, and so may be regarded as 



