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same variations of node and inclination will be caused. In this situation of M, then, the nodes recede during every part of the revolution of P, but the inclination diminishes throughout the quadrant S A, increases again by the same identical degrees in the quadrant A N, decreases throughout the quadrant Nb, and is finally restored to its pristine value at S. On the average of a revolution of P, supposing M unmoved, the nodes will have retrograded with their utmost speed, but the inclination will remain unaltered.

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(511.) Case 2.

Suppose the disturbing body now to be fixed in the line of nodes, or the nodes to be in syzygy, as in the annexed figure. In this situation the direction of the disturbing force, which is always parallel to S M, lies constantly in the plane of P's orbit, and therefore produces neither variation of inclination nor motion of nodes.

(512.) Case 3. Let us take now an intermediate situation of M, and indicating by the arrows the directions of the disturbing forces (which are pulling ones throughout all the semi-orbit which lies towards M, and pushing in the opposite,) it will readily appear that the reasoning of art. 510. will hold good in all that part of the orbit which lies between T and N, and between V and H, but that the effect will be reversed by the reversal of the direction of the motion with respect to the plane of M's orbit, in the intervals HT and N V. In these portions, however, the disturbing force is

feebler than in the others, being evanescent in the line of quadratures T V, and increasing to its maximum

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in the syzygies a b. The nodes then will recede rapidly in the former intervals, and advance feebly in the latter; but since, as H approaches to a, the disturbing force, by acting obliquely to the plane of P's orbit, is again diminished in efficacy, still, on the average of a whole revolution, the nodes recede. On the other hand, the inclination will now diminish during the motion of P from T to c, a point 90° distant from the node, while it increases not only during its whole motion over the quadrant c N, but also in the rest of its half revolution N V, and so for the other half. There will, therefore, be an uncompensated increase of inclination in this position of M, on the average of a whole revolution.

(513.) But this increase is converted into diminution when the line of nodes stands on the other side of S M, or in the quadrants V b, T a ; and still regarding M as fixed, and supposing that the change of circumstances arises not from the motion of M but from that of the node, it is evident that so soon as the line of nodes in its retrograde motion has got past a, the circumstances will be all exactly reversed, and the inclination will again be augmented in each revolution by the very same steps taken in reverse order by which it before diminished. On the average, therefore, of a WHOLE REVOLUTION OF THE NODE, the inclination will be restored to its original state. In fact, so far as the mean or average effect on the inclination is concerned,

instead of supposing M fixed in one position, we might conceive it at every instant divided into four equal parts, and placed at equal angles on either side of the line of nodes, in which case it is evident that the effect of two of the parts would be to precisely annihilate that of the others in each revolution of P.

(514.) In what is said, we have supposed M at rest; but the same conclusion, as to the mean and final results, holds good if it be supposed in motion: for in the course of a revolution of the nodes, which, owing to the extreme smallness of their motion, in the case of the planets, is of immense length, amounting, in most cases, to several hundred centuries, and in that of the moon is not less than 237 lunations, the disturbing body M is presented by its own motion, over and over again, in every variety of situation to the line of nodes. Before the node can have materially changed its position, M has performed a complete revolution, and is restored to its place; so that, in fact (that small difference excepted which arises from the recess of the node in one synodical revolution of M), we may regard it as occupying at every instant every point of its orbit, or rather as having its mass distributed uniformly like a solid ring over its whole circumference. Thus the compensation which we have shown would take place in a whole revolution of the node, does, in fact, take place in every synodic period of M, that minute difference only excepted which is due to the cause just mentioned. This difference, then, and not the whole disturbing effect of M, is what produces the effective variation of the inclinations, whether of the lunar or planetary orbits; and this difference, which remains uncompensated by the motion of M, is in its turn compensated by the motion of the node during its whole revolution.

(515.) It is clear, therefore, that the total variation of the planetary inclinations must be comprised within very narrow limits indeed. Geometers have accordingly demonstrated, by an accurate analysis of all the circumstances, and an exact estimation of the acting forces,

that such is the case; and this is what is meant by asserting the stability of the planetary system as to the mutual inclinations of its orbits. By the researches of Lagrange (of whose analytical conduct it is impossible here to give any idea), the following elegant theorem has been demonstrated:

"If the mass of every planet be multiplied by the square root of the major axis of its orbit, and the product by the square of the tangent of its inclination to a fixed plane, the sum of all these products will be constantly the same under the influence of their mutual attraction." If the present situation of the plane of the ecliptic be taken for that fixed plane (the ecliptic itself being variable like the other orbits), it is found that this sum is actually very small: it must, therefore, always remain So. This remarkable theorem alone, then, would guarantee the stability of the orbits of the greater planets; but from what has above been shown, of the tendency of each planet to work out a compensation on every other, it is evident that the minor ones are not excluded from this beneficial arrangement.

(516.) Meanwhile, there is no doubt that the plane of the ecliptic does actually vary by the actions of the planets. The amount of this variation is about 48′′ per century, and has long been recognized by astronomers, by an increase of the latitudes of all the stars in certain situations, and their diminution in the opposite regions. Its effect is to bring the ecliptic by so much per annum nearer to coincidence with the equator; but from what we have above seen, this diminution of the obliquity of the ecliptic will not go on beyond certain very moderate limits, after which (although in an immense period of ages, being a compound cycle resulting from the joint action of all the planets,) it will again increase, and thus oscillate backward and forward about a mean position, the extent of its deviation to one side and the other being less than 1°21'.

(517.) One effect of this variation of the plane of the ecliptic, that which causes its nodes on a fixed plane

to change,—is mixed up with the precession of the equinoxes (art. 261.), and undistinguishable from it, except in theory. This last-mentioned phænomenon is, however, due to another cause, analogous, it is true, in a general point of view to those above considered, but singularly modified by the circumstances under which it is produced. We shall endeavour to render these modifications intelligible, as far as they can be made so, without the intervention of analytical formulæ.

(518.) The precession of the equinoxes, as we have shown in art. 266., consists in a continual retrogradation of the node of the earth's equator on the ecliptic, and is, therefore, obviously an effect so far analogous to the general phænomenon of the retrogradation of the nodes of the orbits on each other. The immense distance of the planets, however, compared with the size of the earth, and the smallness of their masses compared to that of the sun, puts their action out of the question in the enquiry of its cause, and we must, therefore, look to the massive though distant sun, and to our near though minute neighbour, the moon, for its explanation. This will, accordingly, be found in their disturbing action on the redundant matter accumulated on the equator of the earth, by which its figure is rendered spheroidal, combined with the earth's rotation on its axis. It is to the sagacity of Newton that we owe the discovery of this singular mode of action.

(519.) Suppose in our figures (arts. 509, 510, 511.) that instead of one body, P, revolving round S, there were a succession of particles not coherent, but forming a kind of fluid ring, free to change its form by any force applied. Then, while this ring revolved round S in its own plane, under the disturbing influence of the distant body M, (which now represents the moon or the sun, as P does one of the particles of the earth's equator,) two things would happen:-1st, Its figure would be bent out of a plane into an undulated form, those parts of it within the arcs V c and Td (fig. art. 511.) being rendered more inclined to the plane of M's orbit, and

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