place in the relations of nature to the accumulated effects produced in considerable lapses of time by the continued action of the same causes, under circumstances varied by these very effects, is the business of the integral calculus. Without going into any calculations, however, it will be easy for us to demonstrate from the principles above laid down, the leading features of this part of the planetary theory, viz. the periodic nature of the change of the inclinations of two orbits to each other, the re-establishment of their original values, and the consequent oscillation of each plane about a certain mean position. As in explaining the motion of the nodes, we will commence, as the simplest case, with that of an exterior planet disturbed by an interior one at less than half its distance from the central body. Let A CA' be the great circle of the heavens into which M's orbit seen from S is projected, extended into a straight line, and Ag Ch A' the corresponding projection of the orbit of P so seen. Let M occupy some fixed situation, suppose in the semicircle A C, and let P describe a complete revolution from A through g Ch to A'. Then while it is between A and g or in its first quadrant, its motion is from the plane of M's orbit, and at the same time the orthogonal force acts from that plane: the inclination, therefore, (art. 635.) increases. In the second quadrant the motion of P is to, but the force continues to act from, the plane, and the inclination again decreases. A similar alternation takes place in its course through the quadrants Ch and h A. Thus the plane of P's orbit oscillates to and fro about its mean position twice in each revolution of P. During this process if M held a fixed position at G, the forces being symmetrically alike on either side, the extent of these oscillations would be exactly equal, and the inclination at the end of one revolution of P would revert precisely to its original value. But if M be elsewhere, this will not be the case, and in a single revolution of P, only a partial compensation will be operated, and an overplus on the side, suppose of diminution, will remain outstanding. But when M comes to M', a point equidistant from G on the other side, this effect will be precisely reversed (supposing the orbits circular). On the average of both situations, therefore, the effect will be the same as if M were divided into two equal portions, one placed at M and the other at M', which will annihilate the preponderance in question and effect a perfect restoration. And on an average of all possible situations of M, the effect will in like manner be the same as if its mass were distributed over the whole circumference of its orbit, forming a ring, each portion of which will exactly destroy the effect of that similarly situated on the opposite side of the line of nodes. (637.) The reasoning is precisely similar for the more complicated cases of arts. (625.) and (627.). Suppose that owing either to the proximity of the two orbits, (in the case of an exterior disturbed planet) or to the disturbed orbit being interior to the disturbing one, there were a larger or less portion, de, of P's orbit in which these relations were reversed. Let M be the position of M' corresponding to de, then taking G M'G M, there will be a similar portion d'e' bearing precisely the same reversed relation to M', and therefore, the actions of M' M, will equally neutralize each other in this as in the former state of things. (638.) To operate a complete and rigorous compensation, however, it is necessary that M should be presented to P ir every possible configuration, not only with respect to P itself, but to the line of nodes, to the position of which line the whole reasoning bears reference. In the case of the moon for example, the disturbed body (the moon) revolves in 27d.322, the disturbing (the sun) in 365d-256, and the line of nodes in 67934-391, numbers in proportion to each other about as 1 to 13 and 249 respectively. Now in 13 revolutions of P, and one of M, if the node remained fixed, P would have been presented to M so nearly in every configuration as to 13 1 operate an almost exact compensation. But in 1 revolution of M, or 13 of P, the node itself has shifted or about 1 of a revolution, in a direction opposite to the revolutions of M and P, so that although P has been brought back to the same configuration with respect to M, both are of a revolution in advance of the same configuration as respects the node. The compensation, therefore, will not be exact, and to make it so, this process must be gone through 19 times, at the end of which both the bodies will be restored to the same relative position, not only with respect to each other, but to the node. The fractional parts of entire revolutions, which in this explanation have been neglected, are evidently no farther influential than as rendering the compensation thus operated in a revolution of the node. slightly inexact, and thus giving rise to a compound period of greater duration, at the end of which a compensation almost mathematically rigorous, will have been effected. (639.) It is clear then, that if the orbits be circles, the lapse of a very moderate number of revolutions of the bodies will very nearly, and that of a revolution of the node almost exactly, bring about a perfect restoration of the inclinations. If, however, we suppose the orbits excentric, it is no less evident, owing to the want of symmetry in the distribution of the forces, that a perfect compensation will not be effected either in one or in any number of revolutions of P and M, independent of the motion of the node itself, as there will always be some configuration more favourable to either an increase of inclination than its opposite is unfavourable. Thus will arise a change of inclination which, were the nodes and apsides of the orbits fixed, would be always progressive in one direction until the planes were brought to coincidence. But, 1st, half a revolution of the nodes would of itself reverse the direction of this progression by making the position in question favour the opposite movement of inclination; and, 2dly, the planetary apsides are themselves in motion with unequal velocities, and thus the configuration whose influence destroys the balance, is, itself, always shifting its place on the orbits. The variations of inclination dependent on the excentricities are therefore, like those independent of them, periodical, and being, moreover, of an order more minute (by reason of the smallness of the excentricities) than the latter, it is evident that the total variation of the planetary inclinations must fluctuate within very narrow limits. 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. (640.) 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 plancts,) 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'. (641.) 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, and undistinguishable from it, except in theory. This lastmentioned phænomenon is, however, due to another cause, analagous, 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. (642.) The precession of the equinoxes, as we have shown in art. 312., 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 phanomenon 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. (643.) Suppose in our figure (art. 611.) 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 |