ATTENDED BY SEVERAL SATELLITES. 
505 
Then treating these biquadratics in the same way as before, we find that they have, 
or have not, two real roots, according as h is greater or less than 
Now 
iMI 
C 
TO, 
TO 0 
(M + 3 (M + Too) 3 _ 
Therefore there is, or there is not, a pair of real solutions of the equations 
n—D, x ={ly, according as the total momentum of the system is, or is not, greater than 
Ti l x 
C>JP 
TO, 
TO, 
And this is also the criterion whether or not there is a maximum, or minimum, or 
maximum-minimum point on the energy surface. 
In the case where the masses of the satellites are smali compared with the mass 
of the planet, we may express the critical value of the momentum of the system in 
the form 
4 ^ + TO g )fi 
3* ^‘(if+TOi + TO# 
A comparison of this critical value with the two previous ones shows that if the two 
satellites be fused together, and if y be such that 
M(m 1 + m 2 ) 3 _ 
M+m 1 + to ^ ’ 
and if w be the orbital angular velocity of the compound satellite when moving at 
distance y, then the above critical value of the momentum of the whole system is 
4 M(m l + Too) 3 
3 ' M+ m l + in"! 
and this is 4/3* times the orbital momentum of the compound satellite when revolving 
at distance y. 
Hence if the masses of the satellites are small compared with that of the planet, 
there are, or are not, maximum or minimum or maximum-minimum points on the 
surface of energy, according as the total momentum of the system is greater or less 
than 4/3* times the orbital momentum of the compound satellite when moving at 
distance y. 
In the case where the masses of the satellites are not small compared with that of 
the planet, I leave the criterion in its analytical form. 
There are thus three critical values of the momentum of the whole system, and the 
actual value of the momentum determines the character of the surface of energy 
according to its position with reference to these critical values. 
