semi-diameters of the moon. The dark ring covers nearly the whole space within the inner edge of the inner bright ring; for the inner edge of the dark ring crosses the horizon 13° 55′ on each side of the meridian, and rises on the meridian to an altitude of only 46', so that only a narrow strip of sky about 28° from point to point and about three semi-diameters of the moon in width on the meridian, is left uncovered by the rings on the southern horizon. Again, it appears from the table that the meridian-altitude of a declination-parallel through the point (called A in the table) in which the outer edge of the system meets the horizon, is 34° 31', or 4° 31' greater than the altitude of this edge where it crosses the meridian (at B); and that a declination-parallel through B crosses the horizon at distances 63° 29′ on either side of the south point, or 6° 7′ nearer than A to the south point: and similarly a declination-parallel through A' crosses the meridian 7° 38′ above B', and a declinationparallel through B' crosses the horizon 12° 7' nearer than A' to the south point. Hence a star rising at a culminates 4° 31′ above B; a star culminating at B is altogether hidden by the bright rings, except for a very brief interval when it crosses the division between the rings; a star rising at A' is altogether hidden by the inner bright ring, being 5° 22′ from the division between the rings even at culmination; and lastly, a star culminating at B is visible (through the dark ring) throughout its path above the horizon. Hence many stars must remain altogether invisible until the slow precessional motions of Saturn's equinoctial points so far alter the declinations of such stars as to remove them from the invisible zone of the Saturnian heavens.

In a similar manner the appearance of the rings for any latitude may be determined from Table XI. At Saturn's equator, the edge of the ring being turned towards the planet, it is probable, from the appearance of the rings when their edges are turned to the earth, that an irregular zone of variable appearance is turned towards the Saturnians. Assuming the dark ring to be only indistinctly visible, and the inner edge of the inner bright ring to be 100 miles in thickness, its appearance would be that indicated in note (3) Table XI. The width of the zone thus presented would at the zenith be nearly two-thirds, at the horizon about one-fourth, of

the apparent diameter of the moon. The absolute extent of surface of the ring-system visible above the horizon is greatest for latitudes near the equator,* but the apparent surface of the celestial sphere covered by the rings attains its maximum extent in higher latitudes. It will be seen from Table XI. that the arcs of the meridian covered, respectively, by the outer ring, the division between the rings, the inner ring, and the dark ring, attain their maximum values in about latitudes 45°, 40°, 32° 30′, and 21°; while the arcs of the meridian covered, respectively, by the system of bright rings, and by the complete system of rings, attain their maximum values in about latitudes 35° and 29°.

It is clear that the bright rings are plainly visible from parts of that hemisphere, only, of Saturn which lies above their illuminated face. From the other hemisphere the rings are traceable in their effects in occulting the stars or other celestial bodies whose arcs above the horizon pass wholly or in part behind the rings. These rings may also reflect a faint light received from Saturn's moons. The dark ring may possibly be visible in both hemispheres, since the satellites composing it are probably separately visible from Saturn's surface. By day, the rings are either altogether invisible, or only appear as clouds of faint light below the sun's diurnal path. It might at first sight be supposed that the circumstance that these rings are composed of disconnected satellites, must have a marked effect, whether such satellites are separately visible or not; that the satellites in different parts of their revolution about the planet must exhibit such phases as our own moon, and that parts of the ring in which all the satellites are 'full' or nearly full, must present a much larger amount of illuminated surface to the planet, than parts in which all the satellites are 'new' or nearly new. A little consideration will show that this is not actually the case. appearance of the system shows that the satellites composing it must be very numerous and closely packed: thus the effects of mutual eclipses and occultations among the satellites counter

The

It is easily calculated that the surface of either face of the ring-system above the horizon at the equator is equal to about ths of the whole surface of either face, considered as extending from the inner edge of the dark ring to the outer edge of the outer bright ring, without regard to divisions.

balance the effects due to their phases, and the question of illumination may be considered precisely as it would on the assumption that the rings are solid bodies. Now the brilliancy of an illuminated surface (beyond the earth's atmosphere) does not vary with the distance of the observer, nor with the angle at which he views the surface; these circumstances affect the apparent magnitude of the object, and, in the same proportion, the total amount of light received by the observer, but the intrinsic brilliancy of the object remains unaltered. The apparent brilliancy of an illuminated surface varies, however, with the angle at which the illuminating body is elevated above that surface.* Hence the apparent brilliancy of the rings at any instant is the same throughout their visible extent, and (cæteris paribus) from whatever part of the hemisphere above their illuminated side they may be viewed; but such brilliancy varies with the sun's changes of declination, increasing gradually from the vernal equinox to the summer solstice, and thence decreasing to the autumnal equinox.

Between the vernal equinox and the summer solstice of either hemisphere, the shadow of the planet on the rings assumes suc

*It appears from these considerations that Professor Challis is in error when he states that a 'distant spherical body shining by reflected light would appear equally bright at all points of the disc' (Article on the Indications by Phenomena of Atmospheres to the Sun, Moon, and Planets,--' Reports of the Astronomical Society,' June 1863). A self-luminous spherical body whose surface is uniformly brilliant would so appear, and therefore we may accept the diminution of brightness near the sun's periphery as an indication that the sun has an atmosphere; but in the case of a sphere shining by light received from a distant luminous body, the illuminated hemisphere is not uniformly brilliant, and therefore the disc presented by it exhibits corresponding variations of brilliancy. An atmosphere surrounding a planet, by tending to equalise the illumination of the planet's surface, would diminish rather than increase this variation; in fact, there is no resemblance between the cases of a selfluminous sphere and of a sphere shining by reflected light, as regards the effects to be attributed to the presence of an atmosphere.

The variation of brilliancy is sufficiently conspicuous in the discs of the planets Saturn and Jupiter; and though these planets never present a gibbous appearance, yet there is a perceptible difference in the illuminations of opposite sides of the disc when the planets are in or near quadrature; see the figures of Plate I. Similar variations of brilliancy are exhibited by Venus and the moon, when horned or gibbous. When the moon is full the variation is also traceable, but less clearly, owing to the irregularities of her surface. An examination of the general brilliancy of different parts of the lunar disc confirms the views of the moon's form (as respects her visible hemisphere) presented in Note C, Appendix I.

N

cessively all the forms indicated in Plate XII.* At the vernal equinox the edges of the shadow are straight, as in fig. 1; at and near the summer solstice the outline of the shadow is part of an ellipse, an extremity of whose longer axis lies within the outer edge of the rings; in all intermediate cases the outlines of the shadow are parts of an ellipse of considerable eccentricity. The interval of time in which the shadow changes from the form indicated in one figure to that indicated in the next is about 384 days. It is easy to determine the manner in which the vast shadow of the planet sweeps over the illuminated face of the rings. At sunset, at and near either equinox, the rings are illuminated throughout their visible extent in all latitudes. Near the equator the shadow of the planet rises in the east, as soon as the sun has set,† eclipsing at once the whole breadth of the rings near the horizon; in higher latitudes the shadow rises later, eclipsing first the outer edge of the rings. Later in the Saturnian year the curvature of the shadow shows its effect; the parallel of latitude within which the eclipse commences along the inner edge of the rings passing higher and higher, until it includes all latitudes within which the rings are visible. Near the summer solstice the outer edge of the outer ring is not eclipsed at all. The shadow also rises later and later to midsummer; but as the nights grow shorter and shorter, and as in high latitudes this change takes place at a greater rate than the change in the hour at which the shadow rises, it will happen that, in high latitudes, great parts of the ring are already in shadow when the sun has set. In all latitudes and at all seasons the central line of the shadow crosses the meridian at midnight. At this hour a very small part of the ring is visible, even from points near the equator, near the time of either equinox; but, for about three years, near the time of the summer solstice, the outer edge of the ring is not in shadow at midnight. At this time the system must present a magnificent appearance, as a vast double arch of light, indented by a broad elliptical shadow. Owing

* The point of view in these figures is supposed to lie in the axis produced of the planet; the lines A A', L L', M M', T T and E E', correspond to the lines similarly lettered in fig. 3, Plate XI.; and the circles n n'n", a a'a", I l'l", m m'm", t t't" and e e'e" to the lines similarly lettered in fig. 2, Plate XI.

Owing to refraction the shadow doubtless rises before sunset, just as the eclipsed moon is sometimes visible while the sun is yet apparently above the horizon.