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were found; for then both Saturn and the Earth would be in this plane; and as the inclination of the ring to the ecliptic is already known, the positions of these two bodies at that instant would be sufficient for determining its nodes. But, as the passage of the plane through the Sun occasions the ring to disappear, as well as its passage through the centre of the Earth, these passages must be distinguished by an examination of the periods in which they take place. The passages through the Sun depend only upon the mo tion of the Sun and that of Saturn; but those through the Earth depend also upon the motion of the Earth about the Sun.

These disappearances and reappearances of the ring succeed each other very exactly, and in the same order, during each sidereal revolution of Saturn: it is there fore concluded, that they result from the passage of its plane through the Sun. The following epochs, in which these phenomena have been observed, are extracted from the work of Dionis Sejour, on the ring of Saturn: they must, however, be regarded only as approximations to the truth; for the very slow manner in which the ring both disappears and reappears, renders it impossible to determine the instant of its passage through either the Earth or the Sun with accuracy.

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If, to the epoch of 1715, there be added the sidereal revolution of Saturn, or 29 Julian years and 166.827213 days, we shall have exactly the epoch of 1744; and those of 1774 and 1803 may be deduced in the same manner. The epoch of 1730 gives, in the same way, those of 1760 and 1789.

This agreement is too exact to be attributed to

chance, and evidently indicates the law which these phenomena follow; and, their return being independent of the Earth's motion about the Sun, they must be referred to the passage of the plane of the ring through the centre of that body. Then, since these phenomena form two distinct series, which proceed through the same periods, but of which the epochs are different, they evidently refer to the two opposite situations in which the plane of the ring ought to meet the Sun in each revolution.

But as, in each series, the passages always return after a complete revolution of Saturn, it follows that the two situations of the ring answer to the same positions of that planet; that is, that the plane of the ring remains constantly parallel to itself upon the orbit of Saturn, and consequently its path upon the plane of the ecliptic ought always to make a constant angle with the path of the orbit.

By examining the preceding epochs of the Sun's passage through the nodes of Saturn's ring, it is evident that the periods between their occurrence are not of equal duration. Between the epoch of 1715 and that of 1730 there is 15 years and 9 months; but between those of 1730 and 1744, there is only 13 years and 8 months. The same is also observed for the other terms. This difference arises from the eccentricity of Saturn's orbit; for if a plane were drawn through the centre of the Sun, and perpendicular to the plane of the ring, it would determine these nodes, and would also divide the orbit of Saturn into two unequal parts, the one containing the perihelion and the other the aphelion of the planet. This body, therefore, passed through its aphelion between 1715 and 1730; and through its perihelion between 1730 and 1744.

It would be easy to predict all the appearances of Saturn's ring, if the epochs of its plane passing through the centre of the Earth were only known, as those are for the centre of the Sun.

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nual motion of the Earth imparts a greater degree of difficulty to this determination. In order to accomplish this, it is necessary to find a number of sidereal revolutions of the planet which corresponds either exactly, or very nearly, with an exact number of sidereal years. By taking the ratio of these periods, or dividing the one by the other, we shall have

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and by reducing this fraction to lower terms, we find the first approximative values are 59 and 3; the other nearer values give very long periods.

The first of these fractions indicates that, in 59 sidereal years, there would be nearly two complete revolutions of Saturn; for if the duration of these two periods be calculated exactly, it will be found that the first exceeds the second by 32.185796 days: hence, after a period of 59 sidereal years, Saturn is this time in advance of the Sun. From the known velocities of the Sun and Saturn in their orbits, it is easy to ascertain the number of days which must be added to the 59 years, to bring the two bodies into the same relative position.

For, let R denote the sidereal revolution of Saturn, and that of the Sun, d the difference, equal to 32.185796 days, and a the number of days to be added to the 59 years, to bring the Sun and Saturn into the same relative position; then, in the time d, Saturn will evidently describe an arc equal to

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in the time a he will describe an arc equal to

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and in the same interval the Sun will describe an arc

equal to

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day, the time which must be added to the 59 sidereal years, to bring the Sun and Saturn to the same relative position as at the commencement of that period.

The period given by the fraction 32 is much more exact; it shows that, in 324 sidereal years, Saturn will have very nearly completed 11 revolutions: the difference is only 5-606304 days, which Saturn is found behind the Sun. By following the same method of calculation as above, we shall find

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which is therefore the time that must be subtracted from 324 years, to bring the two bodies into the position in which they were at the commencement of that period. These calculated periods also agree very accurately with the results of observation.

Some spots have been observed on the surface of the ring, by which its rotary motion has been ascertained. This rotation is completed in the same time as that of the planet, and about the same axis, which is perpendicular to the plane of the ring. This rotation presents one remarkable circumstance. If we conceive a satellite to revolve about Saturn at the distance of the middle of the ring, and calculate the time of its sidereal revolution by the third law of Kepler, it will

be found to be precisely the same as that of the ring. For, if we take the first satellite of Saturn, and compare its revolution with that of the ring, we have its mean distance from the planet equal to 3·08, and the mean distance of the ring equal to 1 + + 3 = 2, the radius of Saturn being taken for unity. Then, since the squares of their times of revolution are as the cubes of their mean distances, and 9427 of a day is the time of one sidereal revolution of the satellite, we shall have for that of the ring

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which differs very little from that given by observations. This circumstance completely explains the manner in which the ring is supported about the planet without being in contact with it; or at least brings it to the general cause which sustains all the satellites, as well as the planets, in their orbits. This will also hold equally if the ring consists of two or more parts, as is generally supposed.—See Biot's Astronomie Physique, tome troisième.

The following dimensions of Saturn's rings (supposing them to be two), in English miles, are given by Dr. Herschel, as the result of his observations: viz.

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