[ 254 ] 



XXVIIl. On the Oscillations of a Rotating Liquid Spheroid 

 and the Genesis of the Moon. By A. E. H. Love, B.A* 



1. TD lEMANN'S investigations f of the motion of a liquid 

 a\j ellipsoid contain tlie condition of stability of the form 

 of steady motion usually referred to as Maclaurin's spheroid, 

 when the liquid is perfectly inviscid. The equation for the 

 critical value of the excentricity of the spheroid is 



e(3 + 4^2)^/(1-62)= (3 + 2e2-4e^)sin-»e; 



and, if we take the notation of Thomson and Tait in which 

 [l-\-f^) {1 — e^) =1, the critical value of / is very nearly 3'14. 

 If/ be greater than this value the motion is unstable. If the 

 liquid be viscous, it is stated by Thomson and Tait that the 

 motion is secularly unstable, however slight the viscosity may 

 be, if/ exceeds 1*39457. This statement has been recently 

 proved by Poincare %, who has also shown that the motion is 

 thoroughly stable for all displacements when / is less than 

 this value. 



In this paper is given an investigation by the method of 

 Greenhill and Basset of the equations, first obtained by 

 Riemann, determining the lengths of the axis of a liquid 

 ellipsoid which rotates about one of its principal axes and 

 moves in such a way as to remain ellipsoidal ; and these 

 equations are then applied to find the small oscillations about 

 that state of steady motion in which the free surface is an 

 oblate spheroid, and the liquid rotates as if rigid about the 

 polar axis ; the displacement contemplated being of such a 

 kind that the axis of rotation remains fixed in space, and the 

 surface is always ellipsoidal and has this axis for one of its 

 principal axes. It appears that there are two periods of 

 oscillation, 27r/wi and ^irfn^, where 



V = 167r7p/(3 +/2) - 20)2(3 + 8/2 +/^)/(3/2 +/^) 

 and 



n.^^Uirr^pil +f)/i3 +/2)2- 0)2(1 +/2) (27 + 18/2-/^)/(3 +/2)2. 



In these p is the density of the liquid, w the angular velocity, 

 and 7 the constant of gravitation, and w and /are connected 



* Communicated by the Author, having been read before the British 

 Association, September 6, 1888. 

 t Abh. km. Ges. Wiss. Qott. 1860. 

 \ Acta Mathematica, vii. (1885). 



