independent of the earth's position in her orbit. There is no difficulty, however, in explaining these appearances, even on the supposition of the solidity of the system. Such explanation will serve to introduce and render intelligible the corresponding explanation on the hypothesis we are actually examining. Consider the great division in the rings: it is perfectly clear that if the rings were indefinitely thin, this division would appear to be bounded by two exactly similar and concentric ellipses, and it would therefore appear broadest at the ends of its longer axis and narrowest at the ends of its shorter axis. But now suppose the rings to be of appreciable and uniform thickness-then it is clear that this circumstance will operate to make the division appear narrower at the ends of the shorter axis, while it will not affect the apparent breadth of the division at the ends of the longer axis. For at the ends of the shorter axis the apparent breadth will be the angle between two lines of sight, one passing over the upper edge of the nearer boundary of the division, the other passing under the lower edge of the farther boundary; and it is clear that as the angle diminishes at which the ring is viewed, the apparent breadth of this part of the division would rapidly diminish, until at length the line passing over the upper edge of the nearer boundary would fall upon the opposite face of the division, so that the division would no longer be visible at this point. After this, as the angle at which the ring is viewed continued to diminish, the arc along which the division is invisible would gradually extend more and more towards the extremities of the longer axis of the apparent outline of the division; but until the angle became very small it is clear that the apparent breadth of the division would be very little affected at the ends of the longer axis, for here the lines of sight to the edges of the division would fall (approximately) along, and not across, the bounding faces of the division. Similar remarks apply to the division in the outer ring; but this division being so much narrower than the great division, would disappear much sooner at the ends of the shorter axis, as the ring closed, and the arc along which it is invisible would extend much more rapidly towards the extremities of the longer axis. Now, imagine the

formation of the rings to be that exhibited in fig. 5, Plate IX.; that is, that each ring is formed of a number of concentric hoops of uniform thickness, but the breadths of which diminish, while the intervals between them grow gradually wider towards the inner boundary of each ring. Then it is clear, either from the considerations detailed above or from an examination of the figure, which represents the appearance of such a system of rings, that dark spaces must be visible at the ends of the longer axis of the inner boundary of each bright ring.* These shaded spaces would vary in form according to the manner in which the rings and the divisions between them varied in width, and might either be bounded by definite outlines or toned off by insensible gradations. It is clear, however, that if the width of the rings diminished, and the width of the spaces between them increased, by any uniform law, the shadings would present oval forms similar to those presented by the Saturnian system.

The explanation of these appearances on the supposition that the rings consist of flights of disconnected satellites, is similar to the above though not so convenient for illustration-whether we suppose the satellites to travel in narrow rings, or, which is more probable, to be in general less regularly disposed. We have only to imagine that the satellites are strewn more densely near the outer edges of the bright rings, and especially of the inner bright ring, and that this density of distribution gradually diminishes inwards. For instance, we may conclude that along the inner edge of the inner bright ring the satellites are so sparsely strewn that, at the extremities of the apparent longer axis of that edge, the dark background of the sky becomes visible through the gaps between the satellites. If these gaps were separately visible we should find, as the eye travelled across the breadth of the bright ring at this part, that they became smaller and less numerous as the satellites became more and more densely crowded; but as the eye travelled round the ring we should find the gaps becoming smaller and less numerous from another cause. For a satellite would

*We must suppose these narrow rings to be so numerous, and, therefore, the divisions between them so narrow, that neither rings nor divisions would be separately visible even in the most powerful telescopes.

appear of the same size at whatever part of the ring it appeared, and thus, if separately visible, would occupy a much smaller part of the breadth of the rings when seen near the longer axis, where this breadth is greatest, than when seen near the shorter axis, where this breadth is least. Hence a flight of satellites which, in a telescope of sufficient power, might be resolvable into its component satellites when in the former position, might, from such foreshortening, become irresolvable in the latter, though the separate satellites maintained their relative positions unchanged. If such a flight of satellites could be traced in its motion from the longer to the shorter axis of the system, the discs of the component satellites would be seen gradually to approach, then to overlap each other, until, finally, all the dark spaces between them would disappear. If the satellites were not separately visible, such a flight would appear dusky in the former position, and would become gradually smaller and brighter, until in the latter position it would be as bright as the outer parts of the bright ring. Now the ring may be considered as made up of flights of satellites; and though the members of such flights in no case maintain their relative positions unchanged, even for a few seconds, yet the general average of density along any band of the ring remains tolerably uniform. Hence we can readily understand that there should be a gradual increase in the brightness of the rings, whether the eye travels across their width from within outwards, or along any circle concentric with the outlines of the rings from the longer to the shorter (apparent) axis of the system. Further, as it appears impossible to offer any other explanation of these shaded spaces, we may conclude that in the inner bright ring, and probably in each member of the outer double bright ring, the distribution indicated actually prevails—that is, that the component satellites are crowded along the outer boundaries of the bright rings, and more sparsely distributed along the inner boundaries; and that, although there may be local irregularities-such as strips, along which for an interval satellites are more or less crowded than in the neighbouring spaces-yet, on the whole, the density with which the satellites are strewn increases gradually outwards in each bright ring.

The appearances observed by Mr. Wray and M. Otto Struve,

which seem altogether inexplicable on either of the hypotheses before considered, may be readily explained on the supposition we are examining at present. For it is conceivable that the disturbing attractions of Saturn's outer satellite may draw the satellites composing the ring from the plane of Saturn's equator (or the mean plane of the ring), so that when the edge of the ring is turned to the observer the satellites thus disturbed present the nebulous appearance described. Furtner, the more densely the satellites. composing any part of the ring are crowded, the more efficient will be their common action to check such disturbances; so that the gradual increase in the width of these nebulous appendages, as they (apparently) approach the disc of the planet, is, perhaps, a further indication of the diminution of density inwards mentioned above. But this phenomenon may be satisfactorily explained in another manner :-The number of satellites at a given distance from the central plane of the ring must rapidly diminish as that distance increases; thus, when this distance is very small, the disturbed satellites may be strewn with sufficient density to become visible near the extremities of the ansæ, where the line of sight passes through a small range of satellites; but that the sparsely strewn satellites at a greater distance from the central plane of the rings should become visible, it may be necessary that the line of sight should pass through a much greater range, that is, should fall much nearer the disc of the planet. Thus, clearly, the apparent breadth of these appendages would be greater near the planet's disc, even though there were not an increase inwards in the numbers of satellites disturbed from the mean plane of the ring. It is very probable, however, that there is such an increase, and that the effects resulting from both causes combine to render the peculiar apparent shape of these appendages more distinct than it would be if either cause operated alone.

The investigation of the motions of a crowd of satellites traveling in rings about a central attracting globe, is a problem of too great complexity to be exactly resolved. If the motion of our moon is of so complex a nature that even yet all its inequalities have not been exactly determined, it will readily be conceived that a problem which deals with the motions of hundreds of moons,

disturbed by and disturbing each other, must lie far beyond the range of our most powerful modes of mathematical analysis. Even if we knew the exact size, shape, and position of each satellite, and the rate and direction of its motion at any instant, the exact investigation of the subsequent motions of the system would still lie utterly beyond the grasp of the acutest human intellect. But of all those elements we are ignorant. All that we know certainly is that the bodies constituting the system are very numerous; we may also conclude from the analogy of other parts of the solar system that they are not uniform either in size or density.

Notwithstanding the difficulty of the problem, and the uncertainty of all its conditions, highly interesting general results may be deduced from its consideration.

And first, while we cannot assert that such a system is actually permanent, it is undoubtedly safe from sudden destruction. We speak of the orbits of our earth and of the planets as permanent, because, though they undergo various changes, these are oscillatory, and produce no lasting effect. But rings of satellites, subject like all the members of the solar system to numerous disturbing attractions, and mutually disturbing each other, undergo changes of form that proceed continuously. Whether such development results in the destruction of the rings (as rings) is not certain. It appears probable, however, that under certain conditions the destruction of the rings might be indefinitely postponed.

We may consider separately two forms of disturbance, chiefly due to the varying attractions of Saturn's eight satellites, but partly to the attractions of the other members of the solar system: each form of disturbance also generates the other, or modifies disturbances already existing.

In the first place, the members of these rings will be subject to perturbations out of the general plane of the system. If it were possible to trace the motion of a single satellite, it would be found that its orbit has its ascending and descending nodes on the ring's plane, and (at each instant) a definite inclination to that plane. These elements of the satellite's orbit would be found to be continually changing; the nodes at one time advancing, at another regreding--the inclination now diminishing, now increasing.