44 Waves on a Liquid Spheroid of finite Ellipticity. [Nov. 22, 



Taking next the tesseral harmonics of degree n and rank s, we find 

 that they determine n s 4- 2 periodic small motions. These are 

 essentially tidal waves rotating with various angular velocities about the 

 axis of the spheroid, the angular velocities of those rotating in opposite 

 directions being in general different. All but two of the values of K are 

 numerically less than unity, the periods of the corresponding tides at 

 a point fixed relatively to the liquid being greater than the time of a 

 semi-revolution of the mass. 



The mean angular velocity of these n s + 2 waves is less than 

 that of rotation of the mass by 2/{s(n s + 2)} of the latter. 



In the two waves determined by any sectorial harmonic, the relative 

 motion of the liquid particles is irrotational. The harmonics of 

 degree 2 and rank 1 give rise to a kind of precession, of which there 

 are two. 



I have calculated the relative frequencies of several of the principal 

 waves on a spheroid whose eccentricity is l/\/2. 



The question of stability is next dealt with, it being shown that in 

 the present problem, in which the liquid forming the spheroid is sup- 

 posed perfect, the criteria are entirely different from the conditions 

 of secular stability obtained by Poincare for the case when the liquid 

 possesses any amount of viscosity, which latter depend on the energy 

 being a minimum. In fact for a disturbance initially determined by 

 any harmonic (provided that it is symmetrical with respect to the 

 equatorial plane, since for unsymmetrical displacements the spheroid 

 cannot be unstable), the limits of eccentricity consistent with stability 

 are wider for a perfect liquid spheroid than for one possessing any 

 viscosity. If we assume that the disturbed surface initially becomes 

 ellipsoidal, the conditions of stability found by the methods of this 

 paper agree with those of Riemann. 



The case when the ellipticity, and therefore the angular velocity 

 are very small is next discussed, it being shown that all but two of 

 the waves, or all but one of the oscillations for any particular har- 

 monic become unimportant, their periods increasing indefinitely. 



In the case of those whose periods remain finite for a non-rotating 

 spherical mass, the effect of a small angular velocity u of the liquid 

 is to cause them to turn round the axis with a velocity less than that 

 of the liquid by io/n. 



Finally the methods of treating forced tides are further discussed. 



The general cases of a " semi-diurnal " forced tide or of permanent 

 deformations due to constant disturbing forces are mentioned in con- 

 nexion with some peculiarities they present, and the paper concludes 

 with examples of the determination of the forced tides due to the 

 presence of an attracting mass, first when the latter moves in any 

 orbit about the spheroid, secondly when it rotates uniformly about the 

 spheroid in its equatorial plane. 



