view of the matter. Among the fundamental relations of dynamics, relations which presuppose no particular law of force like that of gravitation, but which express in general terms the results of the action of force on matter during time, to produce or change velocity, is one usually cited as the "Principle of the conservation of the vis viva," which applies directly to the case before us. This principle (or rather this theorem) declares that if a body subjected at every instant of its motion to the action of forces directed to fixed centers (no matter how numerous), and having their intensity dependent only on the distances from their respective centers of action, travel from one point of space to another, the velocity which it has on its arrival at the latter point will differ from that which it had on setting out from the former, by a quantity depending only on the different relative situations of these two points in space, without the least reference to the form of the curve in which it may have moved in passing from one point to the other, whether that curve have been described freely under the simple influence of the central forces, or the body have been compelled to glide upon it, as a bead upon a smooth wire. Among the forces thus acting may be included any constant forces, acting in parallel directions, which may be regarded as directed to fixed centers infinitely distant. It follows from this theorem, that, if the body return to the point P from which it set out, its velocity of arrival will be the same with that of its departure; a conclusion which (for the purpose we have in view) sets us free from the necessity of entering into any consideration of the laws of the disturbing force, the change which its action may have induced in the form of the orbit of P, or the successive steps by which velocity generated at one point of its intermediate path is destroyed at another, by the reversed action of the tangential force. Now to apply this theorem to the case in question, let M be supposed to retain a fixed position during one whole revolution of P. P then is acted on, during that revolution, by three forces 1st. by the central attraction of S directed always to S; 2nd. by that to M, always directed to M; 3rd. by a force equal to M's attraction on S; but in the direction M S, which

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therefore is a constant force, acting always in parallel directions. On completing its revolution, then, P's velocity, and therefore the major axis of its orbit, will be found unaltered, at least neglecting that excessively minute difference which will result from the non-arrival after a revolution at the exact point of its departure by reason of the perturbations in the orbit produced in the interim by the disturbing force, which for the present we may neglect.

(664.) Now suppose M to revolve, and it will appear, by a reasoning precisely similar to that of art. 662., that whatever uncompensated variation of the velocity arises in successive revolutions of P during M's recess from S will be destroyed by contrary uncompensated variations arising during its approach. Or, more simply and generally thus: whatever M's situation may be, for every place which P can have, there must exist some other place of P (as P'), in which the action of M shall be precisely reversed. Now if the periods be incommensurable, in an indefinite number of revolutions of both bodies, for every possible combination of situations (M, P) there will occur, at some time or other, the combination (M, P') which neutralizes the effect of the other, when carried to the general account; so that ultimately, and when very long periods of time are embraced, a complete compensation will be found to be worked out.

(665.) This supposcs, however, that in such long periods the orbit of M is not so altered as to render the occurrence of the compensating situation (M, P') impossible. This would be the case if M's orbit were to dilate or contract indefinitely by a variation in its axis. But the same reasoning which applies to P, applies also to M. P retaining a fixed situation, M's velocity, and therefore the axis of its orbit, would be exactly restored at the end of a revolution of M; so that for every position PM there exists a compensating position PM'. Thus M's orbit is maintained of the same magnitude, and the possibility of the occurrence of the compensating situation (M, P') is secured.

(666.) To demonstrate as a rigorous mathematical truth the complete and absolute ultimate compensation of the va

riations in question, it would be requisite to show that the minute outstanding changes due to the non-arrivals of P and M at the same exact points at the end of each revolution, cannot accumulate in the course of infinite ages in one direction. Now it will appear in the subsequent part of this chapter, that the effect of perturbation on the excentricities and apsides of the orbits is to cause the former to undergo only periodical variations, and the latter to revolve and take up in succession every possible situation. Hence in the course of infinite ages, the points of arrival of P and M at fixed lines of direction, SP, S M, in successive revolutions, though at one time they will approach S, at another will recede from it, fluctuating to and fro about mean points from which they never greatly depart. And if the arrival of either of them at P, at a point nearer S, at the end of a complete revolution, cause an excess of velocity, its arrival at a more distant point will cause a deficiency, and thus, as the fluctuations of distance to and fro ultimately balance each other, so will also the excesses and defects of velocity, though in periods of enormous length, being no less than that of a complete revolution of P's apsides for the one cause of inequality, and of a complete restoration of its excentricity for the other.

(667.) The dynamical proposition on which this reasoning is based is general, and applies equally well to cases wherein the forces act in one plane, or are directed to centers anywhere situated in space. Hence, if we take into consideration the inclination of P's orbit to that of M, the same reasoning will apply. Only that in this case, upon a complete revolution of P, the variation of inclination and the motion of the nodes of P's orbit will prevent its returning to a point in the exact plane of its original orbit, as that of the excentricity and perihelion prevent its arrival at the same exact distance from S. But since it has been shown that the inclination fluctuates round a mean state from which it never departs much, and since the node revolves and makes a complete circuit, it is obvious that in a complete period of the latter the points of arrival of P at the same longitude

will deviate as often and by the same quantities above as below its original point of departure from exact coincidence; and, therefore, that on the average of an infinite number of revolutions, the effect of this cause of non-compensation will also be destroyed.

(668.) It is evident, also, that the dynamical proposition in question being general, and applying equally to any number of fixed centers, as well as to any distribution of them in space, the conclusion would be precisely the same whatever be the number of disturbing bodies, only that the periods of compensation would become more intricately involved. We are, therefore, conducted to this most remarkable and important conclusion, viz. that the major axes of the planetary (and lunar) orbits, and, consequently, also their mean motions and periodic times, are subject to none but periodical changes; that the length of the year, for example, in the lapse of infinite ages, has no preponderating tendency either to increase or diminution, that the planets will neither recede indefinitely from the sun, nor fall into it, but continue, so far as their mutual perturbations at least are concerned, to revolve for ever in orbits of very nearly the same dimensions as at present.

(669.) This theorem (the Magna Charta of our system), the discovery of which is due to Lagrange, is justly regarded as the most important, as a single result, of any which have hitherto rewarded the researches of mathematicians in this application of their science; and it is especially worthy of remark, and follows evidently from the view here taken of it, that it would not be true but for the influence of the perturbing forces on other elements of the orbit, viz. the perihelion and excentricity, and the inclination and nodes; since we have seen that the revolution of the apsides and nodes, and the periodical increase and diminution of the excentricities and inclinations, are both essential towards operating that final and complete compensation which gives it a character of mathematical exactness. We have here an instance of a perturbation of one kind operating on a perturbation of another to annihilate an effect which would otherwise

accumulate to the destruction of the system. It must, however, be borne in mind, that it is the smallness of the excentricities of the more influential planets, which gives this theorem its practical importance, and distinguishes it from a mere barren speculative result. Within the limits of ultimate restoration, it is this alone which keeps the periodical fluctuations of the axis to and fro about a mean value within moderate and reasonable limits. Although the earth might not fall into the sun, or recede from it beyond the present limits of our system, any considerable increase or diminution of its mean distance, to the extent, for instance, of a tenth of its actual amount, would not fail to subvert the conditions on which the existence of the present race of animated beings depends. Constituted as our system is, however, changes to anything like this extent are utterly precluded. The greatest departure from the mean value of the axis of any planetary orbit yet recognized by theory or observation (that of the orbit of Saturn disturbed by Jupiter), does not amount to a thousandth part of its length. The effects of these fluctuations, however, are very sensible, and manifest themselves in alternate accelerations and retardations in the angular motions of the disturbed about the central body, which cause it alternately to outrun and to lag behind its elliptic place in its orbit, giving rise to what are called equations in its motion, some of the chief instances of which will be hereafter specified when we come to trace more particularly in detail the effects of the tangential force in various configurations of the disturbed and disturbing bodies, and to explain the consequences of a near approach to commensurability in their periodic times. An exact commensurability in this respect, such, for instance, as would bring both planets round to the same configuration in two or three revolutions of one of them, would appear at first sight to destroy one of the essential elements of our demonstration. But even supposing such an exact adjustment to subsist at any epoch, it could

Greater deviations will probably be found to exist in the orbits of the small extra-tropical planets. But these are too insignificant members of our system to need special notice in a work of this nature.