39-44] The Rotational Problem 41 



42. As a problem suggested by Plateau's experiments, let us examine 

 what would be the sequence of configurations if a mass of gravitating matter 

 had its angular velocity continually increased by some mechanical means 

 such as the spinning at an ever increasing rate of a pole through jts^centre. 



The configurations of equilibrium are those already discussed ; so long as 

 the mass is constrained to remain .ellipsoidal, they consist of Maclaurin 

 spheroids and Jacobian ellipsoids. To examine the stability of these figures 

 we draw a diagram in which the angular 

 velocity is the vertical coordinate (see 

 fig. 5)." 



We find at once that the Maclaurin 

 spheroids remain stable until the ro- 

 tation is given by co 2 /27rp = '18712. At . ^ n 

 this stage a point of bifurcation occurs, o^ ** 

 the branch series being the Jacobian ^/ 

 ellipsoids. The Maclaurin spheroids / 

 accordingly lose their stability, and I 

 since the Jacobian ellipsoids turn down- / 

 wards from the point of bifurcation, ' 



these also are unstable. Thus there 



Fig. 5. 



are no stable configurations of equili- 

 brium for a rotation greater than that given by o> 2 /27r/o = '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. 



43. Suppose, as an alternative problem, that the mass had been con- 

 strained to remain a figure of revolution. The Jacobian series of figures 

 would then have no existence, and the point defined by co*/27rp = '18712 on 

 the Maclaurin series would have no physical significance except as being the 

 point at which the newly imposed constraint first came into play. The Mac- 

 laurin spheroids now remain stable up to the point defined by &> 2 /27rp = '225. 

 This is the maximum value which &> can have for a spheroidal configuration, 

 and when w exceeds this value there are no possible configurations of equi- 

 librium at all subject to the constraints which we have supposed to be 

 imposed. Again the problem becomes a dynamical one, and again we shall 

 not attempt to trace out this part of the motion. 



Stability when the angular momentum is increased 



44. The problems just considered are of interest as illustrating the theory 

 of points of bifurcation, but fail entirely to represent the conditions postulated 

 in the rotational theory of planetary evolution. To represent these conditions 



