and the Theory of Saturn’s Rings. 269 
tion (14), which, on multiplying its members by A, becomes 
2MA § 
P—N as Tee — A’ e e ° ° . 
The terms of the first member constitute the expression for the 
force of gravity at the extremity of A; and the impossible root 
merely shows that gravity at this point, after having lost over 
three-fifths of its intensity by the disturbances, cannot amount 
(25) 
S brits : 
to —, and consequently can no longer maintain the inverse ratio 
A 
to the length of the axis. This peculiar relation between the 
length of each axis and the gravity at its extremity has already 
been deduced from formula (20), and is indispensable to the 
equilibrium of similar columns of fluid extending from these 
points, either to the centre of the body, or through shells of 
matter equally dense, and bounded by the surfaces of concentric 
ellipsoids similar in position and dimensions. ‘This leads to the 
conclusion maintained on different grounds in my last commu- 
nication, in which I regarded the rupture of the satellite as n- 
evitable, when an increase of elongation would fail to give any 
preponderance to the pressure along the greater axis, or when 
the ellipticity required to be increased to an infinite extent to 
counteract a very slight augmentation of the disturbing forces. 
My former estimates, indeed, do not agree very closely with the 
present investigation in determining the amount of disturbance 
necessary to bring stabiiity to an end; but in these estimates 
the eccentricity of the elliptical section containing the mean and. 
minor axes of the satellite was neglected ; and from more exact 
calculations, which are not yet in a condition to be published, it 
appears that some reduction must be made in the value I first 
assigned to the smallest orbit in which a homogeneous satellite 
could be preserved. 
To furnish another proof that the central and the superficial 
conditions of equilibrium necessarily lead to the same results in 
every respect, let us suppose a portion of the fluid to be enclosed 
in three tubes; two of which are connected at the centre and 
extend to the nearest and most distant part of the surface, while 
the third stretches along the surface to mect their extremities, 
while it coincides with the plane in which they are situated. 
Now the force of gravity being (PN oe at the extre- 
mity of the major axis, it must be reduced to — _ oe a 
along this line at a distance from the centre denoted by a, and 
the element of the pressure in the tube (taking the transverse 
