264 Mr. 1). Vaughan on the Stability of Satellites in small Orbits, 
less difficult road to the solution of the curious problem, and to 
seek a clue to the stability of the annular appendage of Saturn 
by investigating the form which matter must necessarily assume 
in very great proximity to a central body. 
In my communication published in the Philosophical Maga- 
zine for last December (1860), I treated on the equilibrium of 
satellites revolving extremely near to their primaries; and I 
endeavoured to give an estimate of the smallest orbits which they 
could describe in safety. In the cases I considered, the satellite 
was supposed to have its movements adjusted for keeping the 
same point of its surface always directed towards the primary, 
not merely because the hypothesis facilitated the investigation, 
but because observation lends it every support, and the principles 
of natural philosophy furnish most cogent reasons for its adop- 
tion. In describing a very small orbit without such an adjust- 
ment, a satellite must experience, not only excessive tides in its 
seas, but even incessant commotions in its solid matter; and the 
destruction of power by friction necessarily involves a continual 
change in the rotatory movement of the subordinate world, after 
a manner analogous to that which I described in a paper pre- 
sented to the British Association for the Advancement of Science 
in 1857. This must have the ultimate effect of establishing a 
synchronism of the orbital and diurnal movements, together with 
a coincidence of the planes in which they are performed ; so that 
the disturbimg force may give the secondary planet a permanent 
elongation, without rendering it a prey to the effects of violent 
dynamic action. From late researches, however, | am convinced 
that a want of these peculiar conditions would not seriously 
affect the fate of a large satellite when brought into dangerous 
proximity to its primary; and would not change, to any great 
extent, the magnitude of the orbit in which its dismemberment 
must be inevitable. ; 
A homogeneous fluid satellite, having its motions adapted for 
keeping one part of its surface in perpetual conjunction with the 
primary, must find repose in a form differing little from an ellip- 
soid. This proposition, which in my last article was assumed as 
true, may be proved by showing that the relation between the 
forces exerted on every part of the fluid mass is almost precisely 
such as is necessary for equilibrium when the figure is an ellip- 
soid, the dimensions being small compared with the diameter of 
its orbit. For this purpose put A, B, and C for the major, mean, 
and minor semiaxes of the ellipsoid, while P, Q, and R express 
the forces of attraction at their extremities in the absence of all 
disturbances. Now at any point in the surface, the coordinates 
of which referred to the centre are represented by a, ), and ¢, 
the components of the attraction in the direction of each axis 
