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descending node the loops undergo similar changes in a reverse order; the point of intersection passes to the station following opposition; thence along the retrograde path to the station preceding opposition (so that the opening between the loop and path, which before was towards the east, now lies towards the west); and finally, near the descending node, the path, as at the ascending node, is twisted without a loop. In passing from his descending to his ascending node, Saturn's path is similarly varied, the loop being now south of the ecliptic, or still on the side of Saturn's mean path farthest from the ecliptic.

The causes of these phenomena will be made sufficiently apparent if we consider Saturn's motion during a synodical revolution in each of two extreme cases-viz., first, when he is at a node, and secondly, when he is at his greatest distance from the ecliptic.

Suppose, then, first, that during a synodical revolution Saturn passes from Q to q' (fig. 3, Plate V.), and is in opposition when at his ascending node N. During this time the earth moves from a point slightly to the west of n', through rather more than one complete revolution, to a point slightly to the east of n'. As the earth passes the point n' Saturn passes from the northern to the southern side of his heliocentric path. He remains to the south of that path as the earth moves from n' through E' to n. When the earth is at n Saturn (in opposition at his ascending node) again crosses his heliocentric path and also the ecliptic, passing to the north of both these great circles of the celestial sphere. While the earth moves from n through E to n' Saturn remains to the north of his heliocentric path, passing to the south as the earth passes the point n'.

If, then, we draw the line EN E' (fig. 4, Plate V.) to represent part of the ecliptic, and the dotted line s N s', inclined at an angle of 21° to EE', to represent part of Saturn's heliocentric path, and combine the results of the preceding paragraph with the knowledge already obtained of Saturn's progressions and retrogressions, it is easily seen that Saturn's apparent path on the celestial sphere, during the synodical revolution considered, is of

the form on N n'q'; n, N and n' being the points at which he appears to cross his heliocentric path s s'*.

Next let us consider the nature of Saturn's apparent path when he is at his greatest distance from the ecliptic. Suppose that during a synodical revolution he passes from P to p" (fig. 3, Plate V.), and is in opposition when at his greatest distance from the ecliptic at P'. During this time the earth moves from a point slightly to the west of E' through rather more than a complete revolution to a point slightly to the east of E'. While the earth is moving to n she is on the northern side of the plane of Saturn's orbit, and Saturn is on the southern side of his heliocentric path. He passes to the northern side as the earth passes the point n; remains on the northern side of his heliocentric path as the earth moves from n through E to n' (attaining his greatest departure from that path when in opposition at P'); crosses to the southern side as the earth passes the point n'; and remains on that side throughout the remainder of the synodical revolution we are considering.

If, then, we draw EE', fig. 5, Plate V., to represent part of the ecliptic, and the dotted line s n'n s′ (parallel to E E′ and at a distance from that line corresponding to an arc of 21 degrees on the celestial sphere) to represent Saturn's heliocentric path, it is plain that Saturn's path during the synodical period is of the form PnP'n'r'; n and n' being the points at which he appears to cross his heliocentric path.†

There is no difficulty in applying similar methods to determine the form of Saturn's apparent path when he is in any other part of his orbit. It will be found to vary in the manner

* While traversing parts of this path near Q and a', Saturn is not visible from the earth, being near conjunction. If he were visible in these parts of his orbit, it would be found that at n and n' his departure from the ecliptic is greater than at any other moment during the synodical revolution considered. These points are therefore marked q and q' to indicate their correspondence with the points p and p" in the synodical revolution next considered.

† While traversing parts of the path near P and P", Saturn is not visible from the earth, being near conjunction; if he were visible in these parts of his orbit, it would be found that at p and p", the positions he occupies when the earth is at E', he attains his greatest southern departure from his heliocentric path-or approaches nearest to the ecliptic-in the synodical revolution considered.

described above. The following consideration may assist the student:

Since Saturn is seen on his heliocentric path whenever the earth is at n or n', his geocentric path crosses his heliocentric path once in every six months; now, Saturn completes a synodical revolution in a period exceeding twelve months by twelve days and three quarters; thus the points of intersection of his geocentric and heliocentric paths fall successively farther and farther back, in each successive synodical loop, by the space Saturn traverses in 63 days; they therefore occupy, successively, every part of Saturn's synodical loops.

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Orbits of the planets Mercury, Venus the Earth Mars, Jupiter and Saturn,zone of asteroids, and
orbits of the asteroids Harmonia, Nemausa, Polyhymnia and Nysa.

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