266 Mr. D. Vaughan on the Stability of Satellites in small Orbits, 
the direction of A, Y the sum of those in the direction of B, 
and Z the sum of those in the direction of C, it appears from 
(1), (2), (6), and (7) that 
aP aN 2aM 
le 
bQ bN 6M 
15 = ay B = + epi? a eh 6. cs ee (9) 
Tres oh eg 
Now it is well known that, to satisfy the conditions of equili- 
brium, or to make gravity perpendicular to the surface in all 
parts of the satellite, it is necessary that 
Xda+Vdbt7de=0.. ws oe te pe 
Substituting their values for X, Y, and Z, there results 
P—N 2M Ree M R.M 
(Sa - S )aaat (AR M54 (B 4% o \cde re 
Q 
But the equation of an ellipsoid is a3 -- ap + = 1, and its 
differential, multiplied by the constant quantity 8S, becomes 
Sada _ Shdb , Sede 
=r tape + ce =0. 0" 
A comparison of the corresponding terms of equations (12) and 
(13) will enable us to fix the necessary relations of the constants 
for satisfying the former. It thus appears that 
See at ME oe 
rye + oy — BY ay tat Ieee (15) 
R ti bade, 
These relations being independent of the values of the coordi- 
nates a, b, and c, they will be the same for every part of the sur- 
face of the body; and it follows that an equilibrium established 
at any one locality must extend to every part of the entire mass. 
Accordingly the relative magnitudes which the axes A, B, and C » 
must possess, to make gravity perpendicular to the sur face at any 
intermediate pot, must give gravity a like vertical direction in 
all places, and secure the same stability to every portion of the 
satellite which has an ellipsoidal form. This, however, would 
not appear to be rigorously correct if, in the expressions for the 
