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.

*

Considering the whole system, the result of these extra-planar motions and their variations would be a series of waves, wrinkling (so to speak) both surfaces of the ring. These waves would vary in extent, and would move with various velocities-travelling neither directly across nor in circles concentric with the rings. They would not of themselves produce any marked or permanent effects upon the extent of the rings,-that is, on their diameters internal and external. Their effects on the development of the system would arise chiefly from their influence in generating the form of disturbance next to be considered. But their effects on the appearance of the rings when the edge is turned towards the earth are, as we have seen, very observable; for it is undoubtedly to such waves as these that the changes observed by M. Maradi and others at the disappearance of the ring, and the nebulous appearances already considered, are to be attributed.

Secondly, the members of these rings will be subject to variations in their distances from the centre of their gigantic primary. If a single satellite were tracked as before, it would be found that its orbit has its peri-saturnium, and its apo-saturnium, and (at each instant) a definite eccentricity. These elements, like those just considered, would be found to vary continually; the line of apsides advancing at one time and regreding at another, the eccentricity now diminishing and now increasing. Considering the whole system the result of these variations would be a series of waves of concentration and dispersion.† These would travel

*It must be remembered that it is not the motions of the satellites themselves that are here spoken of, but the motions of the waves of disturbance resulting from irregularities in the motions of those bodies. The two kinds of motion are as distinct as the motion of a wave on the ocean from the motion of the particles of the ocean; the wave itself may travel hundreds of miles, while the particles whose successive mo.. tions form the wave may not be displaced more than a few yards.

It is not to be understood that waves of this kind, and waves of the kind before considered, exist separately, and separately travel across or round the ring; they are only considered separately to avoid confusion, but are in reality commingled, and their motions are varied and interchanged in inextricable combinations. If it were possible to view the rings from their common centre, waves of the kind first considered would be visible, apparently travelling round the ring; to an eye placed anywhere in the plane of the rings the same kind of waves would be seen, and their motions round and across the ring would both be visible. If the rings were viewed from a point in the axis of Saturn produced (so that they appeared as in the figures of Plate XII.)

neither directly across nor in circles concentric with the rings; but it appears probable, from the formation of the system of rings, that there would be a continual tendency in waves of the kind we are considering to assume the form of circles concentric with the rings and travelling across their breadth inwards and outwards. Their effects on the appearance of the rings, viewed from the earth, would depend partly on the intensity attained by the wave, and partly on the density with which the satellites are strewn in the particular zone of the ring across which the wave is travelling. If the intensity of the wave is great and the satellites not very densely crowded, the transparent phase of the wave may be traceable in a temporary division or dusky stripe.* Analysis shows that waves of this kind would produce a gradual but continuous increase in the breadth of the system of rings-the inner edge travelling inwards, the outer edge travelling, but much more slowly, outwards. These changes do not, of course, operate only at the edges, but throughout the breadth of the rings; † probably their effects are smaller at the edges than elsewhere: however, it is clear that the only marked change visible to us must be the increase in the breadth of the system.

waves of the second kind would be visible as waves of transparency and opaqueness, travelling, in general, concentrically across the ring, inwards and outwards. A tolerably exact notion of the disturbances to which the rings are subject may be obtained as follows:-Let a semi-transparent fluid be poured into a large circular plate of uniform colour until the bottom of the plate is just hidden; if now this fluid be disturbed in any manner waves will be seen travelling across the surface, crossing and interlacing as they are reflected from the edges of the plate; if the fluid be viewed, however, from above, these disturbances will appear as waves of colour (the colour of the plate and the colour of the fluid); if a motion of steady rotation be communicated to the plate by suspending it from a twisted string, the rotation of the rings, considered as a system, will be illustrated; and it will be found that disturbances can be as readily communicated to the rotating fluid as to the fluid at rest.

*It might be interesting to examine whether the temporary marks that appear on the rings have any motion across the breadth of the system in the intervals during which they remain visible.

It may be suggested as possible that in the great division of the rings we have the indication of a zone along which, at an early stage in the development of the system, the parts of the ring spreading outwards were separated from those spreading inwards. This division may possibly be still increasing in width. The division in the outer ring seems certainly to be increasing in width, since it becomes more distinctly visible as the rings successively attain their greatest opening.

Let us examine the effects of such increase at the inner and outer edges of the system, respectively. It is clear that both changes operate to increase the extent of the rings, and consequently, as the changes proceed, the satellites have more and more space for their movements; but it also appears obvious that among satellites near the inner edge seeking smaller orbits collisions must be much more frequent than among satellites near the outer edge seeking larger orbits. Further disturbance would thus be continually generated among satellites near the inner edge. The satellites no doubt move in the same general direction about Saturn, so that it is only the difference of the velocities of two impinging satellites that comes into play at a collision ;* but the eccentricities of the orbits of the satellites may be very importantly affected in this manner, and it is clear that a satellite which once begins to move in an orbit of considerable eccentricity must continually cause fresh disturbances, until either its orbit is altered to a form of less eccentricity or it falls upon the planet. The general effect of such collisions would be that (after the lapse possibly of many ages) numbers of satellites originally travelling in orbits nearly circular would pursue eccentric orbits. There would still remain a tolerably well defined inner edge; but these orbits would lie partly within and partly without its circle. It appears probable that after a time this process would be checked by the formation of a new ring within the original inner boundary of the system, and that the orbits of the satellites composing this new ring would gradually become less and less eccentric. After a further lapse of time, however, the inner edge of this ring would begin to undergo a series of like changes, ending in the formation of a new ring within it, and so on continually, or until the process were checked or assumed new forms through the approach of the rings to Saturn's equator. The inner edges of outer rings would probably be liable to similar changes, proceeding, however, much more slowly.

* The satellites composing the system being bodies of imperfect elasticity, there is at every collision a loss of a part, however small, of the 'vis viva' of the system, and a corresponding generation of heat. The angular moment' of the system about Saturn is not, however, affected by collisions.