1892.] Maclcmrin's Liquid Spheroid. 29 



the original one. The quantity which remains unchanged is the 

 angular momentum. If the angular velocity were supposed to 

 remain unchanged, it would be necessary for the disturbance to 

 consist in part of a couple about the axis of rotation, and a 

 disturbance of this kind could not be applied to a figure of revolu- 

 tion by means of an external pressure, although it might of 

 course be applied to a Jacobi's ellipsoid. The constancy of the 

 angular momentum leads to an additional term in the conditions 

 of stability, and it is the existence of this term which renders 

 Maclaurin's spheroid stable for a spheroidal displacement. We shall 

 presently see, that if the disturbance is such that the figure 

 assumes the form of any other surface of revolution which is 

 symmetrical with respect to the equatorial plane, the spheroid 

 will be unstable when its excentricity is sufficiently great. 



8. I shall now give Poincar^'s investigation. 



Let V be the total potential due to gravitation and centri- 

 fugal force ; then 



=/ 



dm 1 „ „ 



where dm' is an element of mass, and R its distance from any point 

 X, y, z. 



Since the angular momentum h is connected with to by the 

 equation h = /qW, 



where Iq is the moment of inertia of the figure about its axis, the 

 equation for V becomes 



The potential energy of the system due to gravitation is 

 Trr 1 ([dm' dm 



2 j j R ' 



where Wj is the work which must be done in removing a spherical 

 mass of liquid of equal volume to infinity against the attraction of 

 its parts. 



In evaluating this integral, Poincare divides the elements into 

 two parts dm, dm', and dfjL, d/jf, where the first two refer to the 

 original figure, and the last two to the stratum of liquid whose 

 superposition may be conceived to constitute the disturbed figure. 

 The integral may, therefore, be written 



1 [[dm' dm 1 [ [dfidfi 1 Udjxdm' 1 ff dfi'dm , 



2 R ^2} R "2jJ R 2jj R 



