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out of the plane in which its orbit would be performed if undisturbed, and, by so doing, causes it to describe a curve, of which no two adjacent portions lie in one plane, or, as it is called in geometry, a curve of double curvature. Suppose, then, APN to be the orbit which P would describe about S, if undisturbed, and suppose it to arrive at P, at any instant of time, and to be about to describe in the next instant the undisturbed arc Pp, which, prolonged in the direction of its tangent P p R, will intersect the plane of the orbit M L of the disturbing body, somewhere in the line of nodes SL, suppose in R. This would be the case if M exerted no disturbing power. But suppose it to do so, then, since it draws both S and P towards it, in directions not coincident with the plane of P's orbit, it will cause them both, in the next instant of time, to quit that plane, but unequally: — first, because it does not draw them both in parallel lines; secondly, because they, being unequally distant from M, are unequally attracted by it, by reason of the general law of gravitation. Now, it is by the difference of the motions thus generated that the relative orbit of P about S is changed; so that, if we continue to refer its motion to S as a fixed center, the disturbing part of the impulse which it receives from M will impel it to deviate from the plane PSN, and describe in the next instant of time, not the arc P p, but an arc P 7, lying either above or below Pp, according to the preponderance of the forces exerted by M on P and S.

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(498.) The disturbing force acts in the plane of the triangle S P M, and may be considered as resolved into two; one of which urges P to or from S, or along the line S P, and, therefore, increases or diminishes, in so far as it is effective, the di

CHAP. XI. ESTIMATION OF DISTURBING FORCES.

319

rect attraction of S or P; the other along a line PK, parallel to S M, and which may be regarded as either pulling P in the direction P K, or pushing it in a contrary direction; these terms being well understood to have only a relative meaning as referring to a supposed fixity of S, and transfer of the whole effective power to P. The former of these forces, acting always in the plane of P's motion, cannot tend to urge it out of that plane: the latter only is so effective, and that not wholly; another resolution of forces being needed to estimate its effective part. But with this we shall not concern ourselves, the object here proposed being only to explain the manner in which the motion of the nodes arises, and not to estimate its amount.

(499.) In the situation, or configuration, as it is termed, represented in the figure, the force, in the direction P K, is a pulling force; and as P K, being parallel to S M, lies below the plane of P's orbit (taking that of M's orbit for a ground plane), it is clear that the disturbed arc P q, described in the next moment by P, must lie below P p. When prolonged, therefore, to intersect the plane of M's orbit, it will meet it in a point r, behind R, and the line S r, which will be the line of intersection of the plane S P q, (now, for an instant, that of P's disturbed motion,) or its new line of nodes, will fall behind S R, the undisturbed line of nodes; that is to say, the line of nodes will have retrograded by the angle RS r, the motions of P and M being regarded as direct.

(500.) Suppose, now, M to lie to the left instead of the right of the line of nodes, P retaining its situation, then will the disturbing force, in the direction P K, tend to raise P out of its orbit, to throw P q above P p, and

in advance of R. In this configuration, then, the node will advance; but so soon as P passes the node, and comes to the lower side of M's orbit, although the same disposition of the forces will subsist, and P q will, in consequence, continue to lie above Pp, yet, in

this case, the little arc P q will have to be prolonged backwards to meet our ground plane, and, when so prolonged, will lie below the similar prolongation of Pp, so that, in this case again, the node will retrograde.

(501.) Thus we see that the effect of the disturbing force, in the different states of configuration which the bodies P and M may assume with respect to the node, is to keep the line of nodes in a continual state of fluctuation to and fro; and it will depend on the excess of cases favourable to its advance over those which favour its recess, in an average of all the possible configurations, whether, on the whole, an advance or recess of the node shall take place.

(502.) If the orbit of M be very large compared with that of P, so large that M P may, without material error, be regarded as parallel to M S, which is the case with the moon's orbit disturbed by the sun, it will be very readily seen, on an examination of all the possible varieties of configuration, and having due regard to the direction of the disturbing force, that during every single complete revolution of P, the cases favourable to a retrograde motion of the node preponderate over those of a contrary tendency, the retrogradation taking place over a larger extent of the whole orbit, and being at the same time more rapid, owing to a more intense and favourable action of the force than the recess. Hence it follows that, on the whole, during every revolution of the moon about the earth, the nodes of her orbit recede on the ecliptic, conformable to experience, with a velocity varying from lunation to lunation. The amount of this retrogradation, when calculated, as it may be, by an exact estimation of all the acting forces, is found to coincide with perfect precision with that immediately derived from observation, so that not a doubt can subsist as to this being the real process by which so remarkable an effect is produced.

(503.) Theoretically speaking, we cannot estimate correctly the recess of the intersection of the moon's orbit with the ecliptic, from a mere consideration of

CHAP. XI.

MOTION OF THE NODES.

321

the disturbance of one of these planes. It is a compound phænomenon; both planes are in motion with respect to an imaginary fixed ecliptic, and, to obtain the compound effect, we must also regard the earth as disturbed in its relative orbit about the sun by the moon. But, on account of the excessive distance of the sun, the intensity of the moon's attraction on it is quite evanescent, compared with its attraction on the earth; so that the perturbative effect in this case, which is the difference of the moon's attraction on the sun and earth, is equal to the whole attraction of the moon on the earth. The effect of this is to produce a monthly displacement of the center on either side of the ecliptic, whose amount is easily calculated by regarding their common center of gravity as lying strictly in the ecliptic. From this it appears, that the displacement in question cannot exceed a small fraction of the earth's radius in its whole amount; and, therefore, that its momentary variation, on which the motion of the node of the ecliptic on the moon's orbit depends, must be utterly insensible.

(504.) It is otherwise with the mutual action of the planets. In this case, both the orbits of the disturbed and disturbing planet must be regarded as in motion. Precisely on the above stated principles it may be shown, that the effect of each planet's attraction on the orbit of every other, is to cause a retrogradation of the node of the one orbit on the other in certain configur ations, and a recess in others, terminating, like that of the moon, on the average of many revolutions in a regular retrogradation of the node of each orbit on every other. But since this is the case with every pair into which the planets can be combined, the motion ultimately arising from their joint action on any one orbit, taking into the account the different situations of all their planes, becomes a singular and complicated phanomenon, whose law cannot be very easily expressed in words, though reducible to strict numerical statement, and being in fact a mere geometrical result of what is above stated.

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(505.) The nodes of all the planetary orbits on the true ecliptic then are retrograde, although (which is a mostmaterial circumstance) they are not all so on a fixed plane, such as we may conceive to exist in the planetary system, and to be a plane of reference unaffected by their mutual disturbances. It is, however, to the ecliptic, that we are under the necessity of referring their movements from our station in the system; and if we would transfer our ideas to a fixed plane, it becomes necessary to take account of the variation of the ecliptic itself, produced by the joint action of all the planets.

(506.) Owing to the smallness of the masses of the planets, and their great distances from each other, the revolutions of their nodes are excessively slow, being in every case less than a single degree per century, and in most cases not amounting to half that quantity. So far as the physical condition of each planet is concerned, it is evident that the position of their nodes can be of little importance. It is otherwise with the mutual inclinations of their orbits, with respect to each other, and to the equator of each. A variation in the position of the ecliptic, for instance, by which its pole should shift its distance from the pole of the equator, would disturb our seasons. Should the plane of the earth's orbit, for instance, ever be so changed as to bring the ecliptic to coincide with the equator, we should have perpetual spring over all the world; and, on the other hand, should it coincide with a meridian, the extremes of summer and winter would become intolerable. The enquiry, then, of the variations of inclination of the planetary orbits inter se, is one of much higher practical interest than those of their nodes.

(507.) Referring to the figure of art. 498., it is evident that the plane S P q, in which the disturbed body moves during an instant of time from its quitting P, is differently inclined to the orbit of M, or to a fixed plane, from the original or undisturbed plane PSp. The difference of absolute position of these two planes in space is the angle made between the planes PSR and PS r,

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