ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. A477 
or 
= ow. (kK; + Ge) (Ky + Je) 
to the same order, and this is the value obtained above (10). 
§ 4. Application to the Case of the Earth. 
The nature of the two types of oscillation will be found fully discussed in the 
Appendix. It is there shown that the oscillation corresponding to the root w (1 + E) 
is that previously examined by Horxtns in his ‘ Researches in Physical Geology,’ 
whereas the second type is analogous to the motion of a rigid body when disturbed 
from a motion of rotation about a principal axis. 
If the Earth could be regarded as a system such as we have been considering, we 
see that in addition to the ordinary Solar and Lunar Nutations, which would be of 
the same nature as when the Earth is supposed solid throughout, there might exist 
certain free nutations the amplitude of which could only be determined by observa- 
tion. If the amplitudes were sufficiently large, the oscillations corresponding to the 
root \ = w (1 + E) would render themselves visible in the same way as the Solar and 
Lunar Nutations, namely, by small periodic displacements common to all stars. The 
period of these displacements would be 1/E sidereal days, and a knowledge of it would 
enable us to determine EH, a quantity which depends on the form of the internal 
surface and the thiokness of the crust. 
The oscillations which correspond to the root \ = w/{(K, + qe) (kK, + Ge)} 
would manifest themselves in a different manner. They are, in fact, similar to the 
“Eulerian” nutation (vide TissERAND, ‘ Mécanique Céleste,’ vol. 2, p. 494), and will 
involve a small periodic change in the latitude of places on the Earth’s surface, as 
found by meridian observations of a circumpolar star, this change taking place in a 
period of {(k, + qe) (Ky + qe) }~* sidereal days. 
Now it appears probable that in oscillations of long period, such as Precession, the 
effects of fluid friction would be to make the internal fluid move with the crust 
as if rigidly connected to it (TisszRAND, ‘ Mécanique Céleste,’ vol. 2, p. 480, or 
Lord Ketytn, ‘Popular Lectures and Addresses,’ vol. 2, p. 244). Hence, if A, 
A, €& be the principal moments of inertia for the Earth as a whole, supposed 
symmetrical about its axis of rotation, the Theory of Precession will still enable us 
to determine the value of es aS go5- 
cy 
But, if we put x, = k,, €, = «,, and denote by M the mass of the fluid, 
C€=C+2M2=C {14+ q(1 + 24)}, 
A=A+G5M(?+y)=A+4qC(1 +). 
