268 Mr. D. Vaughan on the Stability of Satellites in small Orbits, 
and (16); and as this may be found by simple equations for all 
except the semiaxes A, B, and C, it is to their values alone that 
we must look for imaginary expressions. The formule referred 
to give 
ee ee 
in (P—N)+ oar V «?(P—N)?—8SMa, } 
x 1 ) 
Re , 1 
~ 2M 2M 
Now it is evident that none of the above radicals can become 
imaginary except the first ; and the stability of the body ceases 
to be possible when a?(P—N)?—8MSz, in passing from a posi- 
tive to a negative value, becomes equal tonothing. In this case 
x 
A= AM (P—N). e ° ° ® ry (22) 
But by comparing the expressions given in my last article for 
centrifugal force and the disturbance of the primary at the extre- 
mity of the major axis of the satellite, it appears that the latter 
C= V/ AMSz + R22?. 
is double the former, or that N equals aa We may deduce 
the same result by considering that the orbital velocity of the 
satellite’s centre is equal to “Mz; and from this the rotatory 
velocity of the extremity of the greater axis 1s equal to 7 Ma, 
or Ay/ M Calling this », 
x 
v MA 
N= RX? or N= Sie ; : 2 Y : (28) 
This value being substituted for N in the last equation, gives 
A= ea li whence Pela » 2 (24) 
z w 
The diminished force of gravity which, at the extremity of A, 
is represented by P—N— a thus becomes 
a . ap thease 
the satellite were a homogeneous fluid, the stability must become 
impossible when more than three-fifths of the attraction along 
the major axis is neutralized by centrifugal force and the disturb- 
ing influence of the central sphere. 
The cause of the unstable equilibrium in such cases will be 
rendered more intelligible by a further examination of equa- 
