188 
MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
liquid about its axis, the values of k are the roots of a rational algebraic equation, 
and depend only on the eccentricity of the spheroid, as well as the degree and rank 
of the harmonic, while the number of diflPerent free waves depends on the degree of 
the ecpiation in k. At any instant the height of the disturbance at any point of the 
surface is proportional to the corresponding surface harmonic on the spheroid, inulti- 
23lied by the central jjeiqDendicular on the tangent j^lane, and is of the same form for 
all waves determined by harmonics of any given degree and rank, whatever be their 
frequency; but the motions of the fluid particles in the interior will differ in nature in 
every case. 
Taking first the case of zonal harmonics of the degree, we find that, according 
as n is even or odd, there will be ^ or \{n 1) different periodic motions of the 
liquid. These are essentially oscillatory in character and symmetrical about the axis 
of the sjDheroid. In all but one of these the value of k is essentially less than unity, 
that is, the period is greater than the time of a semi-revolution of the liquid. 
Taking next the tesseral harmonics of degree n and rank s, we find that they 
determine n — .? -j- 2 j^eriodic 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 ojojDOsite 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 j^oint 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/{5 — 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 \/2. 
The question of stability is next dealt with, it being shown that in the j)resent 
jDroblem, in which the liquid forming the spheroid is supposed perfect, the criteria are 
entirely different from the conditions of secular stability obtained by Poincare for 
the case wdien the liquid possesses any amount of viscosity, and which latter dejiend 
on the energy being a minimum. In fact, for a disturbance initially determined by 
any harmonic (jirovided that it is symmetrical with respect to the equatorial plane, 
since for unsymmetrical disjilacements 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 ellijisoidal, the conditions of stability found by the methods of this paper 
agree with those of Piemann. 
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 liarmonic, become unimportant, their periods increasing 
