LIQUID SPHEROID OF FINITE ELLIPTICITY. 
189 
indefinitely. In the case of those whose periods remain finite for a non-rotating 
spherical mass, the effect of a small angular velocity co of the liquid is to cause them 
to turn round the axis with a velocity less than that of the liquid by co/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 connection with some peculiarities they present; 
and these are followed by 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. The effects of such a body in destroying the equilibrium of the spheroid where 
the forced tide coincides with one of the free tides form the conclusion of this paper. 
Poincare’s Differential Equations for Waves or Oscillations of Rotating Liquid. 
2. Suppose a mass of gravitating liquid is in relative equilibrium when rotating as 
if rigid about a fixed axis with angular velocity w, and that it is required to determine 
the waves or small oscillations due to a slight disturbance of the mass. 
Let the motion be referred to a set of orthogonal moving axes, of which the axis of 
z IS the fixed axis of rotation, while the axes of a?, y rotate about it with angular 
velocity w. In the steady or undisturbed motion the positions of the fluid particles 
relative to these axes will remain fixed. In the oscillations, let U, V, W Pe the 
small component velocities of the fluid at the point (x, y, z) relative to the axes. The 
actual component velocities referred to axes fixed in space and coinciding with our 
axes of X, y, z, at the time considered, will be U — V + oix, W, and the equations 
of hydrodynamics may be written ''' 
0U 
dt 
CO (V + a;a:) + U g- 
dt 
+ co(U 
dw 
dt 
+< +''f 
CO 
+ w 
+ w 
+ w 
0U 
0 ^~ 
dz 
0W 
^ ] being the potential due to the attraction of the liquid and any forces which may 
act on it, ^3 the pressure, and p the density. 
For small disturbances we may neglect squares and products of the relative 
velocities U, V, W (as is usual in wave problems), and, therefore, the above equations 
reduce to 
* Basset, ‘ Hydrodynamics,’vol. I, §23; or Greenhill, ‘Encyclopsedia Britannica,’ article “ Hydro¬ 
mechanics.” 
