revolving freely within a Hollow Spheroid. 199 
ferent if we suppose the fluid mass surrounded by a hollow sphe- 
roid, the two bounding surfaces of which are not concentric. 
The “fluid mass can then, under certain circumstances, actually 
assume a position of equilibrium, even when its axis of rotation 
is entirely external to the mass itself, and when it thus plays the 
part of an inner satellite to the hollow spheroid, with which it 
has a common motion of rotation. The existence of such sin- 
gular states of equilibrium we will now demonstrate. 
Suppose the hollow spheroid, together with the internal fluid 
mass, to revolve about the axis of the outer bounding surface of 
the spheroid, the axis of the inner bounding surface being parallel 
to the axis ofrotation. The fluid mass can then assume a position 
of equilibrium in which its figure is that of an ellipsoid of rota- 
tion, whose axis is parallel to the above-mentioned axes. 
Let the centre of the outer bounding surface be the origin of 
a system of rectangular coordinates, the axis of z comeiding with 
the axis of rotation. Let m, n, and p be the coordinates of the 
centre of the inner bounding surface, and a, 8, y those of the 
centre of the fluid mass. The equation of the surface of the fluid 
will then be | 
poe ae Sa A\2 z— 2 
al al mY 
If the component parts of the attraction parallel to the axes be 
denoted by X, Y, Z, we have 
= —Mz-+ M'(e¢—m)—M"(z—2), 
Y=—My+M'(y—n) —M"y—8), 
Z=—Nz+N!(z—p)—N"(¢—4). 
If therefore the angular velocity be denoted by w, we get for the 
differential equation of the surfaces de niveau, 
(—Mz+M'(2—m)— —M"(¢@—«) ) dae + (— My +M/(y—n) 
—M"(y—f) )dy+ (—Nz+N'(z a on )dze 
peoecda yey 0, . wes si apc sc atuarerge ea Recs) 
where M, M’, M”, N, N’, and N” are pee Ue Me or Bae 
Consequently by integration we get 
(—M+ M'—M"+w?)(a? + y?) +(—N+ N’—N")z? 
+ 2(—M'm + M"a)a + 2(—M'n+ M"B)y 
eI yee O.. ies ye oy te ep ee) 
As equations (1) and (3) are to be identical, we have the fol- 
lowing equations of condition :— 
