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Proceedings of the Boyal Society 
of the internal fluid. The component linear velocities of a point 
(x, y , z) of the shell are 
pZ - cry , (TX — zvZ, -&y — px . •. . (1), 
and the component linear velocities of (x, y , z) are 
d<f> dcf) dcf> / q \ 
dte’ dy ’ cfc/’ ' * ' 
If (x, y , z) be any point of the inner surface of the shell, the 
normal component of velocity (1) must be equal to the normal 
component of velocity (2); or in symbols 
(p2 - <ry)& + (crx - »z) | y + (yy - px ^ 
pz 
P x f7i = 
where 
x 2 y 2 z 2 j 
^ + w + w =l 
dcf> px d<h py dcf) pz 
dx a 2 + dy b 2 cfc c 2 
the axes of coordinates being taken as Coincident with the axes of 
the ellipsoid at the instant considered; (a, b , c) being the three 
semi-axes of the ellipsoid ; and p being the perpendictilar from the 
centre to the plane touching the ellipsoid at (x, y , z,). To satisfy 
this, assume 
c f) = A yz + Bzx + C xy .... (4); 
and determine A, B, C, to fulfil the first of equations (3). We find 
that (3) is now satisfied by 
cf> = 
b 2 
a 2 - b 2 
c* , c* - a* , kjp - u * 
To— o2/« + P-rr-2 zx+<r -2 ~r fy 
b 2 + c 2 c 2 + a 2 a 1 + b 2 
(5). 
It is important to remark that this expression for cf> satisfies the first 
of equations (3) independently of the second, from which we infer 
that with the same angular velocity of rotation, the motion of any 
portion of the contained liquid is independent of the magnitude of 
the ellipsoidal body, and is determinate from the ratios alone of the 
three semi-axes. From (4) we find for the velocity components : — 
u = p 
c 2 + a 2 
2 a 2 —b 2 
2 + <r — — —y 
b * 
V = cr 
-b 2 
x + 
b 2 
a 2 + b 2 b 2 + c 2 
b 2 - c 2 c 2 - a 2 
w = « r 7TT“2 y + P~ 2 ~r- 2 X 
b 2 + c 2 c 2 + a 2 
This solution is given in Lamb’s Fluid Motion , sec. 102. 
( 6 ), 
