FIGURE AND STABILITY OF A LIQUID SATELLITE. 
223 
Then 
where 
1= M = cosec P (1 + a cot 2 P + b cot 2 /3), 
a = 7(1+Wo), 
b = ^(l + 5p 0 +fi&Po 2 )- 
Finally 
cosec P (1 + a cot 2 P + b cot 4 P) 
_cosec x (1 + a cot 2 x + & cot 4 x) 
a, = «[ 
which is to be evaluated by quadratures as was proposed 
2 
sec x dx/j, 
for the fourth harmonic. 
18. Moment of Momentum and Limiting Stability. 
The moment of momentum of the system is Ico, but when we are determining the 
configuration of minimum moment of momentum, which is a figure of bifurcation and 
gives us the configuration of limiting stability, the conditions are different according 
as whether we are treating the problem of the figures of equilibrium where both 
masses are liquid, or Roche’s problem in which the ellipsoid denoted by capital letters 
is rigid. 
Accordingly I write 
I = '3 7 rpa' ! 
X ( b 2 +c 2 ) + 
1+X 
(1+\)U 
+ sTrpa, 3 . } 
1 + X 
(. B 2 +C 2 ), 
and for determining the angular momentum of the figures of equilibrium I take the 
whole expression for I, but for Roche’s problem omit the last term. 
a 3 
Since (o = fir/>-g(l + £), I compute for Roche’s problem 
»‘.=i( i +£r 
3/2 
{b 2 +c 2 ) XA 
1+X 
a 
(1+X) 2 a 2 . 
and for the figures of equilibrium 
a 3/2 
p 2 = p-i+ -p(i + £) 1/2 
B 2 +C 2 
5 (1+ X)' a 2 
The moment of momentum is given by 
Ioi = (|7rp) 3 2 a e ’(/Xj or p, 2 ). 
It will be observed that /x x and /x 2 are expressible by numbers for any given solution 
of the problem. 
Suppose now that we have a succession of solutions for equidistant values of y 
differing but little from one another. Then if the solutions lie close to the region of 
limiting stability, we shall find that one of them corresponds to minimum moment of 
