to its primary. From these and other considerations it may be legitimately concluded that the rings of Saturn are the remains of two former satellites; and their origin and their present condition must be regarded as the ultimate consequence of a rare medium disseminated through space.
Though the dismemberment which I have shown to be inevitable in small orbits differs, in some features, from that supposed to take place in the nebular hypothesis, yet in tracing the effects of both it is necessary to be guided by certain mathematically demonstrated principles in regard to stability. Two homogeneous fluid planets varying widely in size, but having the same density and occupying the same time in their diurnal movement, would be similar spheroids, or have the same relative deviation from a true sphere. Stability would cease to be possible in both if they were as dense as the earth and turned once in two hours and twenty-five minutes. If they were as rare as hydrogen gas at the level of our seas, they could not endure a rotation which took place in a less time than twenty-five days. In the investigations which I have given in the "Philosophical Magazine" in regard to secondary planets close to their primaries, the results have a like generality. Twelve hours would be very nearly the shortest time of revolution for an homogeneous fluid satellite as dense as water, whether its diameter were a hundred or a thousand miles, or whether it revolved around the earth or around Jupiter. Such a body, however, would require to be about thirteen times as dense as its primary in order to circulate in safety a little beyond the surface of the latter orb. A small satellite composed of fluid quicksilver would be capable of maintaining a planetary form if revolving just outside the atmosphere of Saturn or of Neptune; but it would be doomed to dismemberment if moving in a similar proximity to the surface of the earth or even of Jupiter.
The results are not very different even in cases of the greatest possible deviation from the homogeneous character I have ascribed to the bodies. If both planet and satellite were composed of rare gas enveloping a central nucleus, the smaller body would require to have an average density nearly eight times that of the greater, in order to preserve its integrity in such a dangerous proximity. A modern advocate of the nebular hypothesis supposes that each planet, when formed from the rarefied matter of a previous solar ring, was fifteen times less dense than the sun would be if it were an homogeneous sphere inclosed by the planetary orbit. In his own words, "After all their contraction during their condition as rings, and during their aggregation into globes, we may assume at a moderate estimate that when their rotation began they were fifteen times less dense than the average density of the sun expanded to their orbits." To change from a ring to a planet, however, the nebulous matter should have about one hundred times the density which the writer ascribes to it. Before it could become dense enough for the transformation, the nebulous expanse would, like
