ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 481 
§ 2. The Boundary Conditions. 
The position of the shell at any instant may be defined by means of three coordi- 
nates, 6,, 8, 3, which denote the small angular displacements, about the axes of 
reference, of the shell from the position it would occupy in the steady motion. 
The displacements parallel to the coordinate axes of the point of the shell, whose 
coordinates are (a, y, 2), are 
— yO, + 265, — 26, + «6s, — «0, + y0). 
If cosa, cos, cosy be the direction cosines of the normal to the undisturbed 
surface, the normal distance between this surface and the displaced surface will be 
(— 8; + 20.) cos a + (— 26, + «6;) cos B + (— x8, + y,) cosy 
= 6, (y cosy — zcos B) + 0, (z cosa — xcos y) + 4; (wcos B — y cos a). 
The condition to be satisfied at the boundary is that the rate of increase of this 
length must be equal to the component velocity of the fluid, relative to the moving 
axes, in the direction of the normal to the undisturbed surface. Now as these rela- 
tive velocities are all small quantities whose squares we are neglecting, it is 
unnecessary to distinguish between the velocities at the disturbed and undisturbed 
surfaces ; thus, at the latter surface we require 
ucosa+vcosB+weosy = 4, (y cos y — z cos B) 
+ A, (2 cos a — x cosy) + 6, (« cos B — y cos 2). 
Putting 0, = @4e™, &c., and omitting the exponential factor, we obtain 
u, cosa + v, cos B + w, cosy = 1A [ 0, (y cos y — 2 cos B) 
+ 6, (zcosa — «cos y) + &, («cos B — ycos «)]; 
or, putting in the values of u,, v,, w,, W, from (6), 



r Wy ow Ory 4a*\ 
aC ee cosa + By CE a0 ee “08 y ( | 
rn 
2a 0 
eee cos a — cos 
= [9 (y cosy — zcos B) + 0, (z cos a — x cosy) 

10/5 (a COS) B'—YICOs &)I| “MEI Se Bt AES BITES iE AE Vig: 
Let us now put 
Ae : 
eS aie, 
MDCCCXCV.— A. 3 Q 
