the sun illuminating only the edge of the ring. After this, the sun being on the northern, the earth on the southern side of the ring's plane, the ring continued for a time invisible. Finally, on August 13th, 1862, at 4 A.M., the earth was at E'", Saturn at p′′, and the line of nodes in the position q'8 E"""; thus the plane of the ring passed through the earth, and the earth passing to the northern side of this plane, the ring again became visible. When the earth had reached E (that is, in November, 1862), the line of nodes had passed to the position q'e'R"; thus the northern side of the ring has since continued visible. It appears, then, that while the line of nodes of the ring's plane on the ecliptic was crossing the earth's orbit in the years 1861-62, the plane of the ring passed three times through the earth, that the ring disappeared twice and reappeared twice, and continued invisible during two intervals, the first of nearly ten weeks, the second of twelve weeks. If the earth had been at e' when the line of nodes had reached the position q'e R', it is clear that the plane of the ring would have passed three times through the earth in this case also. The first passage would have taken place before Saturn reached the point P, until which time the rings would have been invisible. They would have become visible when Saturn had passed the point P, and continued so until Saturn had very nearly reached the point p", when -the earth being near e'-the plane of the rings would have passed twice through the earth, disappearing at the first passage and reappearing at the second. Further, if the earth is anywhere on the arc e E e', and for a few degrees beyond the points e and e', when Saturn reaches the point P', the rings' plane will pass three times through the earth, and there will be two disappearances of the rings, and two reappear ances. Similar remarks apply to the passage of Saturn through the arc M'M M' of his orbit. Thus, if the earth is anywhere on the arc e E'e', or a few degrees beyond the points e and e', when Saturn reaches the point м', the rings' plane will pass three times through the earth, and there will be two disappearances and two reappearances of the rings. But if the earth is anywhere on the remaining arc of its orbit when Saturn arrives at M", the rings' plane will pass only once through the earth, and there will be one disappearance and one reappearance. In general, when the plane of the rings has passed through either the earth or the sun, the rings disappear or reappear. For, it is clear, that, if before the passage of the rings' plane through the earth their illuminated side was turned to the observer, then after such passage the earth must be on the darkened side of the rings, and vice versa. And again, if before the passage of the rings' plane through the sun, the earth and the sun are on the same side of the rings' plane-then, after such passage, the earth and the sun must be on opposite sides of the rings' plane, and vice versû. It may happen, however, that the plane of the ring passes through the sun and the earth at the same or nearly the same instant of time; and it is perfectly clear that, in this case, if the ring is ininvisible before such passage it will be invisible after it, and vice versâ. Thus, if the ring's plane passes only once through the earth at this passage of the nodal line across the earth's orbit, there will be no interval during which the ring is invisible (except the brief interval of the passage of the ring's plane through the earth and the sun); and only one such interval if the ring's plane passes three times through the earth. It will readily be seen that in the former case Saturn and the sun would be in conjunction at the double passage, and Saturn therefore invisible;* in the latter case Saturn would be in opposition to the sun, and therefore favourably situated for observation. Again, two passages of the earth through the plane of the ring may coincide. This will occur if, when Saturn is at P′, P", M', or M", the earth happens to be on one of the points separating the two arcs of its orbit mentioned above. Thus, if the earth is a few degrees beyond the point e when Saturn arrives at p', the line of nodes of the ring will overtake the earth (which is here moving in a path inclined at a very small angle to that line); but before the ring's plane has passed beyond the earth, the rapid motion of the latter * To avoid confusion no notice has been taken of the invisibility of Saturn at and near conjunction, in the description of the passage of the nodal line of the ring's plane across the elliptic. carries it (as the angle increases at which the direction of such motion is inclined to the line of nodes) again in front of the line of nodes, and the rings only disappear for the comparatively brief interval during which their plane passes through the earth. A similar double passage will happen if the earth is a few degrees from e' when Saturn reaches p'; for, in this case, when Saturn is approaching p", the earth will have moved round in its orbit (having passed once through the plane of the rings), and have reached the line of nodes; but before the earth has passed through and beyond that line, the direction of the earth's motion will have become inclined to the line of nodes at so small an angle, that that line will again pass in front of the earth. The earth will similarly hang for a short time in the plane of the rings, without passing through and beyond it, if, when Saturn reaches M' or м", the earth is a few degrees from e or beyond e'. In these cases it will be seen that the rings are visible both before and after the double passage. It is plain, also, that at such a double passage the earth would be for a longer time in the plane of the rings than when merely passing through that plane, and thus the phenomena attending the disappearance of the ring would be very favourably seen. The coincidence described is, of course, very uncommon; if, however, the earth is near one of the points mentioned, she hangs longer in and near the plane of the rings than at an ordinary passage through that plane.* If the earth passes through the plane of the ring when Saturn is at or near opposition, an observer on earth will be between the planes bounding opposite faces of the ring for about 8 seconds- the thickness of the rings being as ་ It is stated in Sir John Herschel's 'Outlines of Astronomy' that the plane of Saturn's ring in crossing the earth's orbit generally passes twice through the earth; and in Hind's 'Introduction to Astronomy' that there are usually two, if not three, disappearances about the time of the planet's arrival at the nodes.' The above investigation shows that the earth can never pass twice exactly through the plane of the rings during the passage of this plane across the earth's orbit, but may pass either once or three times. And again, it is clear that there may be two disappearances of the ring, but never three; or, if reappearances and disappearances are both included under the term 'disappearances,' then there may be two or four, but can never be three exactly. The case supposed by Herschel would leave the earth on the same side of the ring's plane before as after the passage of that plane across the earth's orbit; the case supposed by Hind would leave the rings invisible after such passage. sumed to be 100 miles-for he is carried from one plane to the other with a velocity equal to the difference of the velocities of the earth and Saturn, or at the rate of 44,000 miles an hour. If the earth passes through the plane of the rings when Saturn is in conjunction, and Saturn could be seen at such a time, the observer would be carried from plane to plane of the opposite faces of the ring in 4 seconds. On the other hand, when two passages of the earth through the ring's plane coincide, an observer might be nearly nine hours between the planes bounding opposite faces of the rings (supposing the thickness of the rings to be 100 miles),—not being carried through both planes, but twice through one plane. If two passages were very nearly coincident, an observer might be four or five hours between the two planes at one passage, and an hour or two at the other, passing each time through both planes. When the earth passed through the plane of the ring in November, 1861, the time of passage of each point on the earth from plane to plane of the bounding faces of the rings, was very short: the corresponding passage in February, 1862, occupied a longer interval. Another consequence of the earth's motion in her orbit is that the shadow of the rings on the planet, and the shadow of the planet on the rings, become visible. It is clear that to an observer placed at the sun's centre these shadows would at all times be invisible,—since those parts only of the rings and planet are in shadow from which the sun is invisible (through the interposition of parts of the planet and rings respectively), and the sun being invisible from them, they would be invisible to an observer placed in the sun. And the more nearly the line of sight from the earth to Saturn approaches to coincidence with the line from the sun to Saturn, the less conspicuous will the shadows be. Thus, when Saturn is in quadrature to the sun, at which time the angle between these lines is greatest (having an average value of about 6° at that time*), both shadows are, in general, more conspicuous than at any other time; but there is a difference between the two shadows in this respect. When the rings are open to nearly their greatest extent, the shadow of the rings on the planet is not conspicuous, even * When Saturn is in perihelion this angle is 6° 22′ 25′′-6; when he is in aphelion it is 5° 41' 50"-2. when Saturn is in quadrature, for the motion of the earth in her orbit causes very little alteration in the apparent opening of the rings at this time; thus the earth being elevated at very nearly the same angle as the sun above the plane of the rings, those parts of the planet's disc which would be invisible from the sun (that is, in shadow) are also invisible-or only visible along very narrow strips of their surface-to the observer on earth, in whatever part of its orbit the earth may be. On the other hand, the shadow of the planet on the rings is very favourably seen when Saturn is in this part of his orbit, and in or near quadrature; for the portion of the rings concealed from the observer on earth, and the portion hidden from the sun (that is, in shadow), are shifted from coincidence with each other, through an angle of 53° or 61° (according as Saturn is near perihelion or aphelion)* about the centre of the rings, and a large part of the shadow of the globe thus becomes visible on one side or the other of Saturn's disc. Further, since the rings are open to their greatest extent, the shadow, which extends nearly across the width of the rings, is less foreshortened at this time than when the rings are less open. Thus the shadow appears as in fig. 3, Plate I., in the form of a broad curved black space, bounded by two elliptical outlines. This figure represents Saturn as he appeared when near the point XIII of his orbit (fig. 1, Plate VIII.), and in quadrature following opposition (that is, when the earth was between E' and E". The shadow lies to the right of the planet's disc: in reality the shadow lay at this time to the left of Saturn's disc, but in an inverting telescope the shadow appeared as represented. If there were no division in the rings the shadow would have been visible beyond the uppermost point of Saturn's disc, and for a short distance to the left of this point; for, as we have seen, the line of sight from the earth to Saturn was inclined at a slightly greater angle to the plane of the rings than the line from the sun. Since, however, the division in the rings becomes visible above the disc of Saturn, there is no visible shadow at this point or in its neighbourhood.† When Saturn The rings are open to their greatest extent when Saturn is near one or other of the apses of his orbit. In pictures of Saturn by Bond and other observers, the division is thus shown. Pos |