EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
409 
But 
5 CO“ 
167T 
and the moment of inertia of the mean sphere is ra 5 ; hence, if we write 
f=°£ + i(f)th(h + h) + 3™*} 
F = + I (ff [i (H, + 4) + 3JfJ, 
the moments of inertia, i and /, are given by 
i = (1 +/), 
i= iW(i + n 
477 
We already have in (71) 
o 2 
0( j ) 
(!+*)• 
Hence the sum of the rotational momenta of the two masses is 
/ 47t\" 
(i + i> = f^) [a«(l +/) + H*(l + ^)] 
1 + 
A\ 3 
(l+Kf- 
The whole system revolves orbitally about the centre of inertia with an angular 
velocity co : hence the orbital momentum is 
■§77 [a°w 6? -j- A 3 oj I) ] . 
But 
7 M 3 c 
a = —-jt 
a 3 + A 3 
a 3 c 
D = “7 
a 8 + M 3 
Hence the orbital momentum is 
and this is eqtial to 
4 77 A 3 a 3 
A 3 + A 
coc , 
It will be convenient to refer the mass to the radius of a sphere of the same mass 
as the sum of the two. 
Let b be the radius of such a sphere ; then 
I/' = A 3 -f a 3 . 
Thus the whole moment of momentum is 
(fry v 
AY° 
(1 +/)+ T (1+^) 
7\ + (f 
0 +K)K 
We shall therefore compute the coefficient of (fryb 5 , 
MDCCCLXXXVII.—A. 3 G 
