ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 479 
Now, although in the light of Professor Newcoms’s work (‘ Astron. Soc. Monthly 
Notices, March, 1892), it appears probable that this effect would be modified by the 
elasticity of the Crust, it could scarcely be reversed if the fluid nucleus were of any 
considerable extent. We must, therefore, conclude that the observations on Jatitude- 
variation, so far from establishing the existence of a fluid interior, as supposed by 
M. Forts, rather tend to confirm the views hitherto maintained by physicists on 
other grounds, that there can be no internal fluid mass of any considerable extent. 
APPENDIX. 
TREATMENT OF THE PROBLEM BY LAMé& ANALYSIS. 
§ 1. Equations of Motion of Flud. 
Let us refer to rectangular axes rotating with angular velocity w about the 
axis of z. The fluid is supposed to have no motion relatively to these axes other 
than that due to the small oscillations with which we are dealing. 
Let u, v, w, be the velocity-components at any point 2, y, 2 relatively to these 
axes : we shall, as is usual in small-oscillation problems, neglect squares and products 
of the small quantities u, v, w. 
The actual velocity-components parallel to the instantaneous positions of the moving 
axes will be 
U—wy, V+oxr, w, 
and the differential equations of motion of the fluid are therefore (Basset, ‘ Hydro- 
dynamics,’ p. 22) 
ee ca al -+), 
ie MO Vv, if 
ot w(u — ay) + on = a - +) 
: Be Hie zi) 
at By ea 
where V, is the gravitation-potential of the forces to which the fluid is subject, p the 
fluid-pressure, and p, the density. 
Putting 
eV Anineee(sbo) oe as ae 
the above equations reduce to 
du/ot — 2ov = oys/0x 
Bless Wo — MEN oo Se eo eo 8 0 (2) 
ow/ot = 0y/ 02 
