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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 riplings 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 correspondingly 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.
(608.) 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 hundreds of revolutions of the bodies in question about their common center. So that, in effect, so far as we propose to go into its consideration, 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.
(609.) Both the intensity and direction of the disturbing force are continually varying, according to the relative situation of the disturbing and disturbed body with respect to the sun. 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. 454.; 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.
(610.) To obtain clear ideas of the manner in which the disturbing force produces its various effects, we must ascertain at any given moment, and in any relative situations of the three bodies, its direction and intensity as compared with the gravitation of P towards S, in virtue of which latter force alone P would describe an ellipse about S regarded as fixed, or rather P and S about their common center of gravity in virtue of their mutual gravitation to each other. In the treatment of the problem of three bodies, it is convenient, and tends to clearness of apprehension, to regard one of them as fixed, and refer the motions of the others to it as to a relative center. In the case of two planets disturbing each other's motions, the sun is naturally chosen as this fixed center; but in that of satellites disturbing each other, or disturbed by the sun, the center of their primary is taken as their point of reference, and the sun itself is regarded in the light of a very distant and massive satellite revolving about the primary in a relative orbit, equal and similar to that which the primary describes absolutely round the sun. Thus the generality of our language is preserved, and when, referring to any particular central body, we speak of an exterior and an interior planet, we include the cases in which the former is the sun and the latter a satellite; as, for example, in the Lunar theory. It is a principle in dynamics, that the relative motions of a system of bodies inter se are no way altered by impressing on all of them a common motion or motions, or a common force or forces accelerating or retarding them all equally in common directions, i. e. in parallel lines. Suppose, therefore, we apply to all the three bodies, S, P, and M, alike, forces equal to those with which M and P attract S, but in opposite directions. Then will the relative motions both of M and P about S be unaltered; but S, being now urged by equal and opposite forces to and from both M and P, will remain at rest. Let us now consider how either of the other bodies, as P, stands affected by these newly-introduced forces, in addition to those which before acted on it. It is clear that now P will be simultaneously acted on by four forces; firstly, the attraction of S in the direction P S; secondly, an additional force, in the same direction, equal to its attraction on S; thirdly, the attraction of M in the direction P M; and fourthly, a force parallel to M S, and equal to M's attraction on S. Of these, the two first, following the same law of the inverse square of the distance S P, may be regarded as one force, precisely as if the sum of the masses of S and P were collected in S; and in virtue of their joint action, P will describe an ellipse about S, except in so far as that elliptic motion is disturbed by the other two forces. Thus we see that in this view of the subject the relative disturbing force acting on P is no longer the mere single attraction of M, but a force resulting from the composition of that attraction with M's attraction on S transferred to P in a contrary direction.
(611.) Let C P A be part of the relative orbit of the disturbed, and M B of the disturbing body, their planes intersecting in the line of nodes SAB, and having to each other the inclination expressed by the spherical angle P Aa. In M P, produced if required, take M N : M S:: M SJ : M PJ. Then, if S M * be taken to represent, in quantity and direction, the accelerative attraction of M on S, MS will represent
• The reader will be careful to observe the order of the letters, where forces are represented by lines. M S represents a force acting from M towards 8,811 from S towards M.
in quantity and direction the new force applied to P, parallel to that line, and N M will represent on the same scale the accelerative attraction of M on P. Consequently, the disturbing force acting on P will be the resultant of two forces applied at P, represented respectively by N M and M S,
which by the laws of dynamics are equivalent to a single force represented in quantity and direction by N S, but having P for its point of application.
(612.) The line NS, is easily calculated by trigonometry, when the relative situations and real distances of the bodies are known; and the force expressed by that line is directly comparable with the attractive forces of S on P by the following proportions, in which M, S, represent the masses of those bodies which are supposed to be known, and to which, at equal distances, their attractions are proportional: — Disturbing force : M's attraction on S:: N S : S M; M's attraction on S : S's attraction on M:: M : S;
S's attraction on M : S's attraction on P:: S Ps : S Ma: by compounding which proportions we collect as follows: —
Disturbing force : S's attraction on P:: M. N S. S P2: S.SM8.
A few numerical examples are subjoined, exhibiting the results of this calculation in particular cases, chosen so as to exemplify its application under very various circumstances,
throughout the planetary system. In each case the numbers set down express the proportion in which the central force retaining the disturbed body in its elliptic orbit exceeds the disturbing force, to the nearest whole number. The calculation is made for three positions of the disturbing body — viz. at its greatest, its least, and its mean distance from the disturbed.
(613.) If the orbit of the disturbing body be circular, S M is invariable. In this case, N S will continue to represent the disturbing force on the same invariable scale, whatever may be the configuration of the three bodies with respect to each other. If the orbit of M be but little elliptic, the same will be nearly the case. In what follows throughout this chapter, except where the contrary is expressly mentioned, we shall neglect the excentricity of the disturbing orbit
(614.) If P be nearer to M than S is, M N is greater than M P, and N lies in M P prolonged, and therefore on the opposite side of the plane of P's orbit from that on which M is situated. The force N S therefore urges P towards M's plane, and towards a point X, situated between S and M, in the line S M. If the distance M P be equal to M S as when P is situated, suppose, at D or E, M N is also equal to M P or M S, so that N coincides with P, and therefore X with S, the disturbing forces being in these cases directed towards the central body. But if M P be greater than M S, M N is less than M P, and N lies between M and P, or on the same side of the plane of P's orbit that M is situated on. The force N S, therefore, applied at P, urges P towards the contrary side of that plane towards a point in the line M S pro