ATTENDED BY SEVERAL SATELLITES. 
503 
h — k 2 y 1 is greater or Jess than i~ \ Xp^p, 
and tlie second has, or has not, a pair of real roots, according as 
li — k x x' is greater or less than Xp/cp. 
Now if we substitute for the X’s and k s their values, we find that 
\i K i- 
A l K l — n\( 
\p/c 2 *=the same with m. 2 in place of m x . 
Now let y 1 and y 2 be two lengths determined by the equations 
Mm x „ Mm % , ~ 
--~y 2 = C. 
itf + m x x M + m 2 ; 
Or in words—let y l be such a distance that the moment of inertia of the planet 
(concentrated at its centre) and the first satellite about tlieir common centre of inertia 
may be equal to the planet’s moment of inertia about its axis of rotation; and let y 2 
be a similar distance involving the second satellite instead of the first. 
And let aq, Wo be two angular velocities determined by the equations 
fcipyi' 3 &jpy 2 3 = /x(df-|-m 2 ). 
Or in words—let oq be the angular velocity of the first satellite when revolving in a 
circular orbit at distance y x , and <w 2 a similar angular velocity for the second satellite 
when revolving at distance y 2 . 
Now 
So that 
C 
= 0) iYi an( I 
Mm l *_1 Mm x 
0 M+m x 
7i 
1 Mm x „ 
h' K i=n m i ™ "lXi 
C M+m x 
and similarly Xp«:p=the same with the suffix 2 in place of 1. Hence the first of the 
two equations (26) has, or has not, a pair of real roots, according as 
C(h — K 2 y } ) is greater or less than "lYm 
and the second has, or has not, a pair of real roots, according as 
C(/i — k,x v ) is greater or less than % ■ 2 fe).,y.d. 
v 1/0 3* M + m 2 " 
It is obvious that Mm^y^/^M-^m^) is the orbital momentum of the first 
satellite when revolving at distance y x , and similarly Mmyoyyy/(M + m 2 ) is the 
orbital momentum of the second satellite when revolving at distance y 2 . 
