ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 483 
that is to say, that it is an ellipsoid whose principal axes coincide with the axes of 
reference. 
We will take as the standard case, that in which 
: po > & 1b S10. 
Put 
poesia) 
The equation to the auxiliary surface f(a, y, 72’) = 0 becomes 
x AE ae 
aN a a eta hae (EO) 

a? 
S55 
p 
Let us now introduce two sets of elliptic coordinates (p, , v), (p’; »’, v’), connected 
with (x, y, 2) (x, y, 2’) respectively by the equations 






2B yy _V@=P)F=*) ee ESE =e) 
oi Meson UCD Ves = ~Veam 9 ve 
& pv’ y es VJ (uw == b) (8 ize y’) Pa a Vv (6? za pb?) (c? 7S y’) 
p’ => be’? /(p2 — 0?) ne by/ (2 a: b) ’ J (p” os c’) ES VEL (12), 
p’ will be equal to p for points which lie on the surfaces (A) (A’), but not otherwise. 
Let us also put 
X=alp, Yayo —2), Z=2/Mp?— 2) =2/V/(e? — ©), « (13) 
for points on these surfaces; so that X, Y, Z are subject to the relation 
pee 1 Dee Bee eels. (15), 
X, Y, Z may therefore be regarded as the coordinates of a point lying on a sphere 
of unit radius. 
Denote by R, M, N three conjugate Lamé functions of the elliptic coordinates 
p, », v, and by R’, M’, N’ three similar functions of the coordinates p’, p’, v’. 
A form of solution of equation (9) convenient for satisfying boundary conditions at 
the surface (A’) is 
Ui SARIN frie tt tue.-lcceeh. toed. Giemsa (4) 
The effect of the fluid on the motion of the shell will depend only on the fluid 
pressure over the surface, and this by (1) will involve the value of y, at the surface. 
3Q 2 
