596 EQUILIBRIUM OF ROTATING MASSES OF LIQUID. [CHAP. XII 



component momenta corresponding to the ignored coordinates 

 unaltered. The two criteria become equivalent when the disturb- 

 ance considered does not alter the moment of inertia of the 

 system with respect to the axis of rotation. 



The second form of the problem is from the present point of 

 view the more important. It includes such cases as Maclaurin's 

 and Jacobi's ellipsoids, provided we suppose the nucleus to be 

 infinitely small. As a simple application of the criterion we may 

 examine the secular stability of Maclaurin's ellipsoid for the 

 types of ellipsoidal disturbance described in Art. 319*. 



Let n be the angular velocity in the state of equilibrium, and h the 

 angular momentum. If 7 denote the moment of inertia of the disturbed 

 system, the angular velocity, if this were to rotate, as rigid, would be h//. 

 Hence 



and the condition of secular stability is that this expression should be a 

 minimum. We will suppose for definiteness that the zero of reckoning of V 

 corresponds to the state of infinite diffusion. Then in any other configuration 

 V will be negative. 



In our previous notation we have 



c being the axis of rotation. Since 6c=a 3 , we may write 



where /(a, 6) is a symmetric function of the two independent variables a, b. 

 If we consider the surface whose ordinate is / (a, b), where a, b are regarded 

 as rectangular coordinates of a point in a horizontal plane, the configurations 

 of relative equilibrium will correspond to points whose altitude is a maximum, 

 or a minimum, or a * minimax,' whilst for secular stability the altitude must 

 be a minimum. 



For a=oo, or 6=00, we have f (a, 6) = 0. For a = 0, we have F=0, and 

 /(a, b) oc 1/6 2 , and similarly for 6=0. For a =0, 6=0, simultaneously, we have 

 / (a, 6) = oo . It is known that, whatever the value of h, there is always one 

 and only one possible form of Maclaurin's ellipsoid. Hence as we follow the 

 section of the above-mentioned surface by the plane of symmetry (a = 6), the 

 ordinate varies from oo to 0, having one and only one stationary value in the 



* Poincare, I. c. For a more analytical investigation see Basset, "On the 

 Stability of Maclaurin's Liquid Spheroid," Proc. Camb. Phil. Soc., t. viii., p. 23 

 (1892). 



