revolving freely within a Hollow Spheroid. 201 
both the bounding surfaces of the surrounding spheroid are 
similar. 
From what has before been stated, itisevident that pbecomes=0. 
We can take for the axis of y a line which lies in the plane 
passing through the centre of the fluid mass and the centres of 
the bounding surfaces. In this case n=O and 6=0, and also 
M=M! and N=N!. We shall now examine if an oblate ellip- 
soid of rotation can satisfy the conditions of equilibrium. Sup- 
2 
posing zal +A?, and the density of the fluid = p, then the 
last of equations (4) will become 
“es 
~ $n wpt (~— arc tan dQ) 
2 
Say ae 
— 3 pf 13 (are tan rA— sa + wy? 
2 
If Soap b° taken =H, the equation of condition becomes 
arc tan X 
B= BOG? 43)—5 wee ramets) 
But this equation is just the same as that we obtained in de- 
termining the conditions of equilibrium of a freely revolving fluid 
mass whose particles attract each other. Thus we find that pre- 
cisely the same conditions of equilibrium are involved when the 
fluid is revolving in a hollow ellipsoid with similar but eccentric 
bounding surfaces, and when it is perfectly free. Toa given 
value for > there is therefore always a corresponding angular 
velocity; and to a given angular velocity there corresponds 
either no ellipsoid, or one ellipsoid, or two different ellipsoids, 
according as E is = 0:2246. 
Between the rotation of a fluid mass confined in a hollow sphe- 
roid and a mass which revolves freely, there is, however, this 
important difference, that in the former case the rotation does 
not take place about the axis of symmetry of the fluid unless 
both the bounding surfaces of the spheroid are concentric, but 
about a parallel axis which is the axis of symmetry of the outer 
bounding surface of the spheroid,—the distance between the 
two axes being 
CC ve ° . ° ° ° e ° (7) 
whence 
