and which, under the form of mathematical theorems of great beauty, simplicity, and elegance, involve the history of the past and future state of the planetary orbits during ages, of which, contemplating the subject in this point of view, we neither perceive the beginning nor the end.

(492.) Were there no other bodies in the universe but the sun and one planet, the latter would describe an exact ellipse about the former (or both round their common centers of gravity), and continue to perform its revolutions in one and the same orbit for ever; but the moment we add to our combination a third body, the attraction of this will draw both the former bodies out of their mutual orbits, and, by acting on them unequally, will disturb their relation to each other, and put an end to the rigorous and mathematical exactness of their elliptic motions, either about one another or about a fixed point in space. From this way of propounding the subject, we see that it is not the whole attraction of the newly introduced body which produces perturbation, but the difference of its attractions on the two originally present.

(493.) Compared to the sun, all the planets are of extreme minuteness; the mass of Jupiter, the greatest of them all, being not more than one 1300th part that of the sun. Their attractions on each other, therefore, are all very feeble, compared with the presiding central power, and the effects of their disturbing forces are proportionally minute. In the case of the secondaries, the chief agent by which their motions are deranged, is the sun itself, whose mass is indeed great, but whose disturbing influence is immensely diminished by their near proximity to their primaries, compared to their distances from the sun, which renders the difference of attractions on both extremely small, compared to the whole amount. In this case, the greatest part of the sun's attraction, viz. that which is common to both, is exerted to retain both primary and secondary in their

common orbit about itself, and prevent their parting company. The small overplus of force only acts as a disturbing power. The mean value of this overplus, in the case of the moon disturbed by the sun, is calculated by Newton to amount to no higher a fraction than 3000 of gravity at the earth's surface, or of the principal force which retains the moon in its orbit.

(494.) From this extreme minuteness of the intensities of the disturbing, compared to the principal forces, and the consequent smallness of their momentary effects, it happens that we can estimate each of these effects separately, as if the others did not take place, without fear of inducing error in our conclusions beyond the limits necessarily incident to a first approximation. It is a principle in mechanics, immediately flowing from the primary relations between forces and the motions they produce, that when a number of very minute forces act at once on a system, their joint effect is the sum or aggregate of their separate effects, at least within such limits, that the original relation of the parts of the system shall not have been materially changed by their action. Such effects supervening on the greater movements due to the action of the primary forces may be compared to the small ripplings caused by a thousand varying breezes on the broad and regular swell of a deep and rolling ocean, which run on as if the surface were a plane, and cross in all directions, without interfering, each as if the other had no existence. It is only when their effects become accumulated in lapse of time, so as to alter the primary relations or data of the system that it becomes necessary to have especial regard to the changes respondingly introduced into the estimation of their momentary efficiency, by which the rate of the subsequent changes is affected, and periods or cycles of immense length take their origin. From this consideration arise some of the most curious theories of physical astronomy.

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(495.) Hence it is evident, that in estimating the disturbing influence of several bodies forming a system, in which one has a remarkable preponderance over all the rest, we need not embarrass ourselves with combinations of the disturbing powers one among another, unless where immensely long periods are concerned; such as consist of many thousands of revolutions of the bodies in question about their common centers. So that, in effect, the problem of the investigation of the perturbations of a system, however numerous, constituted as ours is, reduces itself to that of a system of three bodies: a predominant central body, a disturbing, and a disturbed; the two latter of which may exchange denominations, according as the motions of the one or the other are the subject of enquiry.

sun.

(496.) The intensity of the disturbing force is continually varying, according to the relative situation of the disturbing and disturbed body with respect to the If the attraction of the disturbing body M, on the central body S, and the disturbed body P, (by which designations, for brevity, we shall hereafter indicate them,) were equal, and acted in parallel lines, whatever might otherwise be its law of variation, there would be no deviation caused in the elliptic motion of P about S, or of each about the other. The case would be strictly that of art. 385.; the attraction of M, so circumstanced, being at every moment exactly analogous in its effects to terrestrial gravity, which acts in parallel lines, and is equally intense on all bodies, great and small. But this is not the case of nature. Whatever is stated in the subsequent article to that last cited, of the disturbing effect of the sun and moon, is, mutatis mutandis, applicable to every case of perturbation ; and it must be now our business to enter, somewhat more in detail, into the general heads of the subject there merely hinted at.

(497.) We shall begin with that part of the disturbing force which tends to draw the disturbed body

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, A P N 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 Pp 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 Pp, but an arc Pq, lying either above or below Pp, according to the preponderance of the forces exerted by M on P and S.

L

R

M

K

(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

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 PK, is a pulling force; and as P K, being parallel to SM, 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 Pq, 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 R S 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 rin 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