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

CHAP. XI.

PRECESSION OF THE EQUINOXES.

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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

those within the arcs c T, d V, less so than they would otherwise be. 2dly, The nodes of this ring, regarded as a whole, without respect to its change of figure, would retreat upon that plane.

(520.) But suppose this ring, instead of consisting of discrete molecules free to move independently, to be rigid and incapable of such flexure, like the hoop we have supposed in art. 507., then it is evident that the effort of those parts of it which tend to become more inclined will act through the medium of the ring itself (as a mechanical engine or lever) to counteract the effort of those which have at the same instant a contrary tendency. In so far only, then, as there exists an excess on the one or the other side will the inclination change, an average being struck at every moment of the ring's motion; just as was shown to happen in the view we have taken of the inclinations, in every complete revolution of a single disturbed body, under the influence of a fixed disturbing one.

(521.) Meanwhile, however, the nodes of the rigid ring will retrograde, the general or average tendency of the nodes of every molecule being to do so. Here, as in the other case, a struggle will take place by the counteracting efforts of the molecules contrarily disposed, propagated through the solid substance of the ring; and thus, at every instant of time, an average will be struck, which average being identical in its nature with that effected in the complete revolution of a single disturbed body, will, in every case, be in favour of a recess of the node, save only when the disturbing body, be it sun or moon, is situated in the plane of the earth's equator, or in the case of the fig. art. 510.

(522.) This reasoning is evidently independent of any

consideration of the cause which maintains the rotation of the ring; whether the particles be small satellites retained in circular orbits under the equilibrated action of attractive and centrifugal forces, or whether they be small masses conceived as attached to a set of imaginary spokes as of a wheel, centering in S, and free only to

CHAP. XI. PRECESSION OF THE EQUINOXES.

331

shift their planes by a motion of those spokes perpendicular to the plane of the wheel. This makes no difference in the general effect; though the different velocities of rotation, which may be impressed on such a system, may and will have a very great influence both on the absolute and relative magnitudes of the two effects in question-the motion of the nodes and change of inclination. This will be easily understood, if we suppose the ring without a rotatory motion, in which extreme case it is obvious, that so long as M remained fixed there would take place no recess of nodes at all, but only a tendency of the ring to tilt its plane round a diameter perpendicular to the position of M, bringing it towards the line S M.

(523.) The motion of such a ring, then, as we have been considering, would imitate, so far as the recess of the nodes goes, the precession of the equinoxes, only that its nodes would retrograde far more rapidly than the observed precession, which is excessively slow. But now conceive this ring to be loaded with a spherical mass enormously heavier than itself, placed concentrically within it, and cohering firmly to it, but indifferent, or very nearly so, to any such cause of motion; and suppose, moreover, that instead of one such ring there are a vast multitude heaped together around the equator of such a globe, so as to form an elliptical protuberance, enveloping it like a shell on all sides, but whose mass, taken together, should form but a very minute fraction of the whole spheroid. We have now before us a tolerable representation of the case of nature*; and

*That a perfect sphere would be so inert and indifferent as to a revolution of the nodes of its equator under the influence of a distant attracting body appears from this, that the direction of the resultant attraction of such a body, or of that single force which, opposed, would neutralize and destroy its whole action, is necessarily in a line passing through the center of the sphere, and, therefore, can have no tendency to turn the sphere one way or other. It may be objected by the reader, that the whole sphere may be conceived as consisting of rings parallel to its equator, of every possible diameter, and that, therefore, its rodes should retrograde even without a protuberant equator. The inference is incorrect, but our limits will not allow us to go into an exposition of the fallacy. We should, however, caution him, generally, that no dynamical subject is open to more mistakes of this kind, which nothing but the closest attention, in every varied point of view, will detect.

it is evident that the rings, having to drag round with them in their nodal revolution this great inert mass, will have their velocity of retrogradation proportionally diminished. Thus, then, it is easy to conceive how a motion, similar to the precession of the equinoxes, and, like it, characterized by extreme slowness, will arise from the causes in action.

(524.) Now a recess of the node of the earth's equator, upon a given plane, corresponds to a conical motion of its axis round a perpendicular to that plane. But in the case before us, that plane is not the ecliptic, but the moon's orbit for the time being; and it may be asked how we are to reconcile this with what is stated in art. 266. respecting the nature of the motion in question. To this we reply, that the nodes of the lunar orbit, being in a state of continual and rapid retrogradation, while its inclination is preserved nearly invariable, the point in the sphere of the heavens round which the pole of the earth's axis revolves (with that extreme slowness characteristic of the precession) is itself in a state of continual circulation round the pole of the ecliptic, with that much more rapid motion which belongs to the lunar node. A glance at the annexed figure will explain this better than words. P is the pole of the ecliptic, A the pole of the moon's orbit, moving round the small circle A B C D in 19 years; a the pole of the earth's equator, which at each moment of its progress has a direction perpendicular to the varying position of the line A a, and a velocity depending on the varying intensity of the acting causes during the period of the nodes. This velocity, however, being extremely small, when A comes to B, C, D, E, the line A a will have taken up the positions B b, Cc, Dd, Ee, and the earth's pole a will thus, in one tropical revolution of the

P

AE

a

b c d

B