ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 495 
where, for brevity, we have put 
v=smp)-p (p'—b*)* (p> — c°) (c? — B'), w= smi. p(p°—L*) (p?— cc? . (38). 
The terms in ” arising from ()) are due to the centrifugal force of the fluid, and 
occur in consequence of the axis of rotation of the shell not accurately coinciding with 
that of the fluid, while the terms (c) are due to the effective inertia of the fluid. 
§ 8. Dynamical Equations of Motion of the Shell. 
Let A, B,C be the principal moments of inertia of the shell; p, g, 7 the angular 
velocities about the principal axes. 
The position of the shell at time ¢ + 6¢ may be found from its a, at time ¢ by 
dé, “ 
le O° ay OF 

giving it a small rotation w d¢ about Oz, 
about the positions of the axes Ow, Oy, Oz at time ¢ + dé. 
The direction-cosines of these latter axes referred to their position at time ¢ are 

(i, W800). ae U(Groty 10)s 5 #(OrNO". 1) 
Hence, resolving the rotations in the directions of the axes Ox, Oy, Oz at time ¢, 
the component angular displacements are 
Oe Gh (ae O85, 
and the angular velocities about the axes Ox, Oy, Oz are 
b,, 6,, o + 5. 
Resolving these about the axes Ox,, Oy,, Oz, we see from the scheme (35) that 
p=6,—ah,. g=6,+ a6, r=a+6,. .,. .. (39): 
EULER’s equations of motion are 
Ap —(B—C)qr=L 
Bg — (C — A)rp = M, 
Cr —(A — B)pg=N 
where, if the system be subject to no external disturbing force, L, M, N have the 
