shall have such a figure as fig. 46; where E C represents the plane of the earth's orbit, J E the orbit of Jupiter, and S T that of Saturn. The orbit of Jupiter, by regressing on Saturn's orbit, assumes the position of the dotted line je; but it is plain that the intersection of

FIG. 46.

E

this orbit with the earth's orbit has gone in the opposite direction, or has progressed. If the motion of the node on Saturn's orbit from J to j is regression, the motion of the node on the earth's orbit from E to e must be progression.

(212.) There is a remarkable relation between the inclination of all the orbits of the planetary system to a fixed plane, existing through all their secular variations, similar to that between their excentricities. The sum of the products of each mass, by the square root of the major axis of its orbit, and by the square of the inclination to a fixed plane, is invariable.

(213.) The disturbance of Jupiter's satellites in latitude presents circumstances not less worthy of remark than the disturbance in longitude. The masses are so small, and their orbits so little inclined to each other, that the small inequalities produced in a revolution may be neglected. Even that depending on the slow revolution of the line of conjunctions of the first three satellites, so small is the mutual inclination of

their orbits, does not amount to a sensible quantity. We shall, therefore, consider only those alterations in the position of the planes of the orbits which do not vary sensibly in a small number of revolutions. For this purpose, we must introduce a term which has not been introduced before.

(214.) If the moon revolved round the earth in the same plane in which the earth revolves round the sun, the sun's attraction would never tend to draw the moon out of that plane. But (taking the circumstances as they really exist), the moon revolves round the earth in a plane inclined to the plane in which the earth revolves round the sun; and the consequence, as we have seen, is that the line of nodes upon the latter plané regresses, and the inclination of the orbit to the latter plane remains, on the whole, unaltered. The plane of the earth's orbit, then, may be considered a fundamental plane to the moon's motion; by which term we mean to express, that if the moon moved in that plane, the disturbing force would never draw her out of it; and that if she moved in an orbit inclined to it, the orbit would always be inclined at nearly the same angle to that plane, though its line of nodes had sensibly altered. The latter condition will, in general, be a consequence of the former.

(215.) In order to discover what will be the fundamental plane for one of Jupiter's satellites, we must consider that, besides the sun's attraction, there is another and more powerful disturbing force acting on these bodies, namely, the irregularity of attraction

produced by Jupiter's flatness. The effect of this (as we shall show) is always to pull the satellites towards the plane of Jupiter's equator. If Jupiter were spherical, the only disturbing force would be the sun's attraction, tending on the whole to draw the satellite towards the plane of Jupiter's orbit, and that plane would be the fundamental plane of the satellite. If Jupiter were flattened, and if the sun did not disturb the satellite, the irregularity in Jupiter's shape would always tend to draw the satellite towards the plane of his equator, and the plane of his equator would be the fundamental plane of the satellite. As both causes exist, the position of the actual fundamental plane must be found by the following consideration. We must discover the position of a plane from which the sun's disturbing force tends, on the whole, to draw the satellite downwards, and the disturbing force depending upon Jupiter's shape, tends to draw it upwards (or vice versa) by equal quantities; and that plane will be the fundamental plane. This plane must lie between the planes of Jupiter's orbit and Jupiter's equator, because thus only can the disturbing forces act in opposite ways, and therefore balance each other; and it must pass through their intersection, as otherwise it would at that part be above both or below both, and the forces depending on both causes would act the same way.

(216.) The disturbing force of the sun, as we have seen (82.), &c., is greater as the satellite is more distant; the disturbing force depending on Jupiter's shape is then less, as we shall mention hereafter. Consequently,

as the satellite is more distant, the effect of the sun's disturbing force is much greater in proportion to that depending on Jupiter's shape. Thus, if there were a single satellite at the distance of Jupiter's first satellite, its fundamental plane would nearly coincide with the plane of Jupiter's equator; if at the distance of Jupiter's second satellite, its fundamental plane would depart a little farther from coincidence with the plane of the equator; and so on for other distances; and if the distance were very great, it would nearly coincide with the plane of Jupiter's orbit. If, then, Jupiter's four satellites did not disturb each other, each of them would have a separate fundamental plane, and the positions of these planes would depend only upon each satellite's distance from Jupiter.

(217.) In fact, the satellites do disturb each other. In speaking of the planets (210.), we have observed that the effect of the attraction of one planet upon another, in the long run, is to exert a disturbing force tending to draw that other planet (at any part of its orbit) towards the plane of the first planet's orbit. The same thing is true of Jupiter's satellites. Now, though each of them moves generally in an orbit inclined to its fundamental plane, yet in the long run (when the nodes of the orbit have regressed many times round), we may consider the motion of each satellite as taking place in its fundamental plane. The question, therefore, must now be stated thus. The four satellites are revolving in four different fundamental planes; and the position of each of these planes is to be determined by the consideration

that the satellite in that plane is drawn towards the plane of Jupiter's orbit by the sun's disturbing force, towards the plane of Jupiter's equator by the force depending on Jupiter's shape, and towards the plane of each of the other three satellites, by the disturbing force produced by each satellite: and these forces must balance in the long run.

(218.) The determination of these planes is not very difficult, when general algebraical expressions have been investigated for the magnitude of each of the forces. The general nature of the results will be easily seen; the several fundamental planes will be drawn nearer together (that of the first satellite, that of the second, and that of the third, will be drawn nearer to Jupiter's orbit, while that of the fourth will be drawn nearer to Jupiter's equator). The four planes will still pass through the intersection of the plane of Jupiter's equator with that of Jupiter's orbit. Thus, if we conceive the eye to be placed at a great distance, in the intersection of the planes of Jupiter's orbit and Jupiter's equator, and if the dotted lines in fig. 47 represent the appearance of

the fundamental planes which would exist if the satellites did not disturb each other, then the dark lines will represent the positions of these planes as affected by