FIGURE AND STABILITY OF A LIQUID SATELLITE. 
167 
As this case may be adequately treated by spherical harmonic analysis, it need not 
detain us, and we see that the most interesting cases of Roche’s problem are those 
where X lies between 0 and 1. 
§ 2. Figures of Equilibrium of a Rotating Mass of Liquid and tlieir 
Stability. 
A mass of liquid, consisting of either one or more portions, is rotating, without 
relative motion of its parts, about an axis through its centre of inertia with angular 
velocity co. We choose as an arbitrary standard figure one which does not differ 
very widely from a figure of equilibrium, and we suppose that any departure from the 
standard figure may be defined by two parameters e and /, which may be called 
ellipticities. It is unnecessary to introduce more than two ellipticities, because the 
result for any number becomes obvious from the case of two. We also assume a 
definite angular velocity for the standard configuration. 
Let V (e,f) denote the gravitational energy lost in the concentration of the system 
from a condition of infinite dispersion into the configuration denoted by e, f 
Let I(e,f), co (e,f) denote the moment of inertia and angular velocity about the 
axis of rotation in the same configuration. 
The initial values of these quantities are those for which e =f= 0, and are 
V (0, 0), I (0, 0), co (0, 0). These all refer to the arbitrary standard configuration ; 
they are therefore constants, and I shall write them F, /, co for brevity. 
Let ellipticities e, f be imparted to the system, and let the angular velocity be so 
changed that the angular momentum remains constant. 
Then 
I (e, f) co{e,f) = I (0, 0) w (0, 0) = Ico. 
The kinetic energy of the system is half the square of the angular momentum 
divided by the moment of inertia ; and since the angular momentum is constant it 
is equal to ^ (Ico) 2 /1 (e, f). 
Thus the whole energy of the system, both kinetic and potential, is equal to 
_ vu /■) + i(M 
(,n /(«,/)• 
If V (e, f) = V+ S V, I(e,f) = I+SI, the expression for the energy as far as 
squares of small quantities is 
— (F+SF) + t(M = - V+tI(o 2 -8V-WSI+W^. 
1 + 01 1 
The first two terms may be omitted as being constant and of no interest, and the 
energy with the sign changed, so that it is the lost energy of the system, becomes 
SV + ±oj 2 
(Mi 2 
I ' 
