470 MR. 8S. S. HOUGH ON THE OSCILLATIONS OF A 
vol. 7). This analysis reduces the determination of the motion of the fluid to the 
problem of finding a single function ¥, subject to certain boundary conditions, which 
in our case take a very simple form. In the case where the surface of the fluid is 
ellipsoidal, it is found that, when the system is oscillating in one of its normal modes, 
w will be expressible as the sum of a series of Lamé products of a single order 7 only. 
When n is different from 2, the types of oscillation are such that no disturbance of 
the shell is involved, and a period equation for the oscillations of the fluid may be 
deduced in a manner similar to that given by Porncarn. 
The types of oscillation corresponding to n = 2 demand exceptional treatment, in 
consequence of the motion communicated to the shell when they exist. The fluid 
motion, however, is found to be such that the molecular rotation is everywhere the 
same. Mr. Bryan has suggested to me that this circumstance may be made use of 
in order to treat the oscillations which involve motions of the shell by a simple 
analysis previously employed by GREENHILL (‘ Proc. Camb. Phil. Soc.,’ vol. 4, p. 4) 
which does not involve Lamé functions. To facilitate the reading of the paper, the 
results are first deduced by this method, and the Lamé analysis by which they were 
originally obtained is reserved for an appendix. 
The oscillations under consideration are found to be of two types. One of these 
corresponds to an oscillation previously discussed by Hopkins in his “‘ Researches in 
Physical Geology” (‘ Phil. Trans. 1839). This exists only in consequence of the 
contained fluid, and in it the oscillations of the sheil are similar in character to the 
“forced” nutations of the earth produced by the action of the sun and moon. In 
the other type the motion of the shell is closely analogous to the motion of a rigid 
body when slightly disturbed from a motion of pure rotation about a principal axis, 
and, in fact, identifies itself with such an oscillation in the event of the inertia of the 
fluid becoming negligible. 
On applying the problem to the case of the earth, the latter mode is that on which 
the variations of latitude depend. The period, however, is found to be shorter than 
it would be if the fluid were solidified, and thus, in this particular case, M. Forte's 
results are contradicted. It appears to me to be highly probable that any such 
freedom in the interior of the earth as that supposed by M. Foie, provided the 
surface does not undergo deformation, would have the effect of reducing, instead of 
extending, the period, and the true explanation of the phenomenon is probably that 
given by Newcomps (‘ Monthly Notices of the Royal Astronomical Society,’ March, 
1892), who shows that the elasticity of the earth, as a whole, would have the effect 
of prolonging the period. 
§1. The Period Equation. 
Let us refer to rectangular axes coincident with the principal axes of the ellipsoidal 
cavity. 
