but in the line of nodes, the sum of the arcs D A and EC exceeds a semicircle, and that the more, the nearer M is to a position at right angles to the line of nodes. Secondly, the arcs favourable to the recess of the node comprehend those situations in which the orthogonal disturbing force is most powerful, and vice versa. This is evident, because as P approaches D or E, this component decreases, and vanishes at those points (612.). The movement of the node itself also vanishes when P comes to the node, for although in this position the disturbing orthogonal force neither vanishes nor changes its direction, yet, since at the instant of P's passing the node (A) the recess of the node is changed into an advance, it must necessarily at that point be stationary.* Owing to both these causes, therefore, (that the node recedes during a longer time than it advances, and that a more energetic force acting in its recess causes it to recede more rapidly,) the retrograde motion will preponderate on the whole in each complete synodic revolution of P. And it is evident that the reasoning of this and the foregoing articles, is no way vitiated by a moderate amount of excentricity in either orbit.

(628.) It is therefore a general proposition, that on the average of each complete synodic revolution, the node of every disturbed planet recedes upon the orbit of the disturbing one, or in other words, that in every pair of orbits, the

It would seem, at first sight, as if a change per saltum took place here, but the continuity of the node's motion will be apparent from an inspection of the annexed figure, where bad is a portion of P's disturbed path near the node A,

concave towards the plane G A. The momentary place of the moving node is determined by the intersection of the tangent be with A G, which as b passes through a to d, recedes from A to a, rests there for an instant, and then advances again.

node of each recedes upon the other, and of course upon any intermediate plane which we may regard as fixed. On a plane not intermediate between them, however, the node of one orbit will advance, and that of the other will recede. Suppose for instance, CA C to be a plane intermediate between P P

and M M the two orbits. If pp and m m be the new positions of the orbits, the node of P on M will have receded from A to 5, that of M on P from A to 4, that of P and M on C C respectively from A to 1 and from A to 2. But if FAF be a plane not intermediate, the node of M on that plane has receded from A to 6, but that of P will have advanced from A to 7. If the fixed plane have not a common intersection with those of both orbits, it is equally easy to see that the node of the disturbed orbit may either recede on both that plane and the disturbing orbit or advance on the one and recede on the other, according to the relative situation of the planes.

(629.) This is the case with the planetary orbits. They do not all intersect each other in a common node. Although perfectly true, therefore, that the node of any one planet would recede on the orbit of any and each other by the individual action of that other, yet, when all act together, recess on one plane may be equivalent to advance on another, so that the motion of the node of any one orbit on a given plane, arising from their joint action, taking into account the different situations of all the planes, becomes a curiously complicated phænomenon whose law cannot be very easily expressed in

words, though reducible to strict numerical statement, being, in fact, a mere geometrical result of what is above shown.

(630.) The nodes of all the planetary orbits on the true ecliptic, as a matter of fact, are retrograde, though 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.

(631.) 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. It is otherwise with the moon, and that owing to two distinct reasons. First, that the disturbing force itself arising from the sun's action, (as appears from the table given in art. 612.) bears a much larger proportion to the earth's central attraction on the moon than in the case of any planet disturbed by any other. And secondly, because the synodic revolution of the moon, within which the average is struck, (and always on the side of recess) is only 291 days, a period much shorter than that of any of the planets, and vastly so than that of several among them. All this is agreeable to what has already been stated (art. 407, 408.) respecting the motion of the moon's nodes, and it is hardly necessary to mention that, when calculated, as it has been, à priori from an exact estimation of all the acting forces, the result 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.

(632.) 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.

(633.) Referring to the figures of art. 610. et seq., it is evident that the plane SP 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 P Sp. The difference of absolute position of these two planes in space is the angle made between the planes PSR and P Sr, and is therefore calculable by spherical trigonometry, when the angle R Sr or the momentary recess of the node is known, and also the inclination of the planes of the orbits to each other. We perceive, then, that between the momentary change of inclination, and the momentary recess of the node there exists an intimate relation, and that the research of the one is in fact bound up in that of the other. This may be, perhaps, made clearer, by considering the orbit of P to be not merely an imaginary line, but an actual circle or elliptic hoop of some rigid material, without inertia, on which, as on a wire, the body P may slide as a bead. It is evident that the position of this hoop will be determined at any instant, by its inclination to the ground plane to which it is referred, and by the place of its intersection therewith, or node. It will also be determined by the momentary direction of P's motion, which (having no inertia) it must obey; and any change by which P should, in the next instant, alter its orbit, would be equivalent to a shifting, bodily, of the whole hoop, changing at once its inclination and nodes.

(634.) One immediate conclusion from what has been

pointed out above, is that where the orbits, as in the case of the planetary system and the moon, are slightly inclined to one another, the momentary variations of the inclination are of an order much inferior in magnitude to those in the place of the node. This is evident on a mere inspection of our figure, the angle R Pr being, by reason of the small inclination of the planes SPR and R Sr, necessarily much smaller than the angle RS r. In proportion as the planes of the orbits are brought to coincidence, a very trifling angular movement of P p about P S as an axis will make a great variation in the situation of the point r, where its prolongation intersects the ground plane.

(635.) Referring to the figure of art. 622., we perceive that although the motion of the node is retrograde whenever the momentary disturbed arc P Q lies between the planes CA and C G A of the two orbits, and vice versâ, indifferently whether P be in the act of receding from the plane C A, as in the quadrant C G, or of approaching to it, as in G A, yet the same identity as to the character of the change does not subsist in respect of the inclination. The inclination of the disturbed orbit (i. e. of its momentary element) Pq or Pq', is measured by the spherical angle Pr H or Pr' H. Now in the quadrant CG, Pr H is less, and Pr' H greater than PCH; but in G A, the converse. Hence this rule: 1st., If the disturbing force urge P towards the plane of M's orbit, and the undisturbed motion of P carry it also towards that plane; and 2dly, if the disturbing force urge P from that plane, while P's undisturbed motion also carries it from it, in either case the inclination momentarily increases; but if, 3dly, the disturbing force act to, and P's motion carry it from-or if the force act from, and the motion carry it to, that plane, the inclination momentarily diminishes. Or (including all the cases under one alternative) if the action of the disturbing force and the undisturbed motion of P with reference to the plane of M's orbit be of the same character, the inclination increases; if of contrary characters, it diminishes.

(636.) To pass from the momentary changes which take

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