270 Mr. D. Vaughan on the Stability of Satellites in small Orbits, 
section and density of the fluid as unity) will be 
Se ie 
The integral of this expression, taken within the limits of a=0 
and a=A, gives for the central pressure of the fluid in the 
longer tube 
4(p-n—=4). 4 Ly ii 
A similar process applied to the fluid in the tube coinciding with 
the minor axis, will give for the differential of pressure, 
R M 
G+) ede rae meee 
and a similar integration will give for its pressure at the centre, 
C MC 
B (BSS). obey <i walecrneaanl 
For stability it is necessary that the contents of both tubes should 
press to the centre with the same amount of force, or that ~ 
4(p—n—“4)_E(r+*2)=0. . (80) 
Now from the peculiar position which the third tube is sup- 
posed to occupy on the surface, the general equation for the 
equilibrium of its contents will become Xda+Zdc=0, or by 
substitution, 
(<> - oe )ada+ (e+ =) ele=0. sucked 
Integrating within the limits of a=A, c=0, and a=0, c=C, 
this becomes 
(p-n—2*)_S(R+%2)=0. . (82) 
The identity of equations (80) and (82), and the relation 
between (26), (28), and (31), show that the equilibrium of the 
internal and external parts of the mass depend on precisely the 
‘same conditions, and that the fluid should rush to the most pro- 
‘minent parts of the satellite from the surface, as well as from its 
internal regions, whenever gravity along the major axis was dimi- 
nished more than 60 per cent. by the disturbing forces. Brevity 
compels me to omit the more lengthy investigation which would 
be required to show that such consequences are not peculiar to 
special localities, but are the same on all parts of the surface of 
the body. 
