ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 471 
Let a, 8, y be the principal semi-axes of this cavity; A, B, C the principal 
moments of inertia of the shell. 
Suppose the motion of the fluid at any instant consists of a rigid body rotation 
with angular velocities & y, € about the axes of the ellipsoid, compounded with the 
irrotational motion consequent on giving the shell additional angular velocities 
GeO, Ox. 
The velocity-potential of the irrotational motion will be 
Be 
Bare oe 
cea ca 
e+ Be” 

Y yz 5 200, + - 
The velocity-components will therefore be 






EN Vises GH mace end aril 
US ee a raging Ostrard Saty Zn | 
Bg P= 
cama s+ ar ee eRe (1). 
pee poeta =, — ay + yé | 
B+ ese ie 
Hence, if hy, hy, h, be the components of angular momentum, and M denote the 
mass of the fluid, p, its density, 

’ 1 
=A(O) + 4) + [[[orde dy dz (wy — v2) =A (0, + £) $y | 
+ 3M (6° +) & ! 
hy = B(Q,+ 7) + [|r de dy de (uz — wx) = B(O, + 9) + # = Og (2) 
+ 3M (7° + 2’) », | 
bs : ae sé e (2 — By 
hs = C(O; + f) + [ [fox de dy dz(vx — wy) = C (9, + €)+ 5M e+ Os | 
! 
+ IM (2 + B)L j 
where 
p= (eowr), vse og) 6 6 ies ss (OO) 
If the system be disturbed from a motion of pure rotation, with angular velocity a, 
about the axis of z; & 7, 0), Q,, Og, will all be small quantities, while ¢ will be 
approximately equal to w, and hence, on omitting small quantities of the second order 
and putting ¢= w» in small terms, the equations of angular momentum, viz. :— 
hy — her +-heq = 0 | [ p =O+¢ 
hy — hep + hyr = 0 iga) qg=O+7 
hs — Ing + hor = 0 r=O,+¢ 
