WITH THE TIDES OF A VISCOUS SPHEROID. 
585 
where 
x—r sin 6 cos ((f) — (of)" 
y—v sin 6 sin ((f) — cot) h. 
z—r cos 0 
Now consider the case when the viscosity is infinitely small: here e is small, and 
• 38 vo) 
sin 2e=tan 2e= _- 
5 Qwa* 
Hence —— sin 2e=r—which is independent of the viscosity. 
38v 5g« 2 r J 
By substituting this value in (81), we see that however small the viscosity, the 
nature of the motion, by which each particle assumes its successive positions, always 
preserves the same character; and the motion always involves molecular rotation. 
But it has been already proved that, however slow the tidal motion of a fluid 
spheroid may be, yet the fluid motion is always irrotational. 
Hence in the two methods of attacking the same problem, different first approxi¬ 
mations have been used, whence follows the discrepancy of 79 instead of 75. 
The fact is that in using the equations of flow of a viscous fluid, and neglecting 
inertia to obtain a first approximation, we postulate that w ^> w w< ^ are ^ ess i- m “ 
portant than vV 2 a, vV~/3, v V : y ; and this is no longer the case if v be very small. 
It does not follow therefore that, in approaching the problem of fluidity from the 
side of viscosity, we must necessarily obtain even an approximate result. 
But the comparison which has just been made, shows that as regards the form of 
the tidal spheroid the two methods lead to closely similar results. 
It follows therefore that, in questions regarding merely the form of the spheroid, 
and not the mode of internal motion, we only incur a very small error by using the 
limiting case when v= 0 to give the solution for pure fluidity. 
In the paper on “Precession” (Section 7), some doubt was expressed as to the 
applicability of the analysis, which gave the effects of tides on the precession of a 
rotating spheroid, to the limiting case of fluidity; but the present results seem to 
justify the conclusions there drawn. 
The next point to be considered is the effects of inertia in— 
The forced oscillations of an elastic sphere. 
Sir William Thomson has found the form into which a homogeneous elastic sphere 
becomes distorted under the influence of a potential expressible as a solid harmonic 
of the points within the sphere. He afterwards supposed the sphere to possess the 
power of gravitation, and considered the effects by a synthetical method. The result 
is the equilibrium theory of the tides of an elastic sphere. When, however, the 
disturbing potential is periodic in time this theory is no longer accurate. 
4 F 2 
