LIQUID SPHEROID OE FINITE ELLIPTICITY. 
215 
original spheroid, as we should most naturally expect. But in the last case, since xjj is 
finite, tliere will be a finite relative motion of the liquid with respect to the moving 
axes. That this must be the case may be seen as follows ;—The spheroid is supposed 
to be deformed from its original form by the action of the given conservative forces 
and surface pressures. The displacement does not vanish at the ec^uator ; hence, if we 
consider the fluid particles at the surface, forming a circle round the ecjuator, tlie 
displacement must necessarily increase or diminish the size of this circle. By 
Thomson’s circulation theorem, the circulation in this circuit must remain the same as 
before, since the liquid is supposed perfect; hence, the angula,r velocities of the fluid 
particles in this circle must be altered, and they can no longer continue to rotate 
about the axis of the spheroid with the original angular velocity co. Therefore the 
disturbance must produce permanent relative motions of the liquid, unless there be 
any viscosity present, in which case the mass will ultimately rotate as if rigid in the 
deformed figure, and the “ equilibrium theory ” will again become applicable. 
Tides due to Action of a Satellite. 
25. We shall conclude by showing how to determine the forced tides due to the 
presence of a small satellite of mass m revolving in any orbit about the spheroid. 
If we take (fd to be the longitude of any point on the spheroid, measured from a 
plane fixed in space, with which the moving plane of {y, z) coincides at time t=0, 
then, (f) being the longitude measured from the latter plane, we have 
(f) (J) . 
Let (p,^, <f)\) be the spheroidal coordinates of the mass m at time t, <j)\ being 
measured from the fixed initial plane. Then, at any point (p, ^') whose distance 
from the mass is B, we have ' 
Vg = my/R.(99). 
Since 1 /R can be expanded in spheroidal harmonics by the formula 
1/R= i/cs;y (2» + 1)[P.(,.)J,.(0 
+ 2 s;:” {(«-*)!/(» + s)!} - <l>\)] (^OO). 
Since the motion of the satellite is supposed known, (p^, (j)\) are known 
functions of t. In order to complete the solution we must suppose the quantities 
cjn (Cl), (pi) < (^i) cos and TJ*> (p^) (Ci) sin S(j)\ expanded by 
Fourier’s theorem in simple harmonic functions of the time. If the period of the 
satellite in its orbit be 27r/L, the expansion will only involve circular functions of 
