sured from L towards T. The latter body, then, in consequence m of the attraction of L equal to and towards L, and of this last teraction of equal forces, be at rest; and L, by their co-operation, will be urged towards T by the sum of attractions. This principle and its consequences are not confined to the case in which L and T should be in motion solely from the agency of their mutual attraction. Should they possess any rotatory motion round any point, such as their centre of gravity, we may still consider T to be at rest, by hypothetically imparting to it and to L, a motion equal to its own, and contrary to its direction. The results then of the preceding Chapters would hold true in a system of two bodies, if the symbol (see p. 6.) represented the sum of their masses. And we may now see why Kepler's Law is not exactly true, and what quantity of correction it requires (see p. 30.). In order to draw the conditions of our mathematical theory nearer to the true conditions of nature, we have substituted attractive masses instead of points, and modified the results. The modification is a very slight one. But the next departure from the simple mathematical system of two bodies will lead us into investigations of very considerable intricacy. For, in nature, there is no heavenly body that is acted on solely by one body: it is always subjected to the action of a third; and, in fact, to the action of every body in the universe. Now, if we take the system of three bodies, and wish to determine the laws of the motion of one of these subjected to the action of the other two, we shall find the exact solution of the problem impossible; that is, beyond the powers, in their present state, of all the methods of calculation, whether they be analytical or geometrical. On this account, we must be contented with an approximate solution: and this we shall be able to obtain; since, in every case which nature presents to us of three or more bodies, the actions of the third and fourth body, are, by reason of their remoteness, very small in disturbing the motion of the second round the first considered as the centre. This circumstance of the minuteness of the third body's action, explains why the investigation of the case of two bodies is made to precede that of three or more. The first investigation is preparatory to the latter; and its results, with slight modifications, belong to it. An exact ellipse is described when there are two bodies, and a curve differing little from an ellipse is described, when there are three or more bodies. The coincidence of the curve described in the first instance with a simple curve of known properties, has been the cause why the elliptical motion of a planet is considered as its natural and proper motion, and, accordingly, (that there might be no incongruity in language,) why a third is called a disturbing body: for its action obstructs the operation of the laws of elliptical motion. This, however, as it is plain, is mere mathematical fiction and contrivance: an ellipse is not the curve that is really described, but that curve is described, the equation of which is assigned by the solution of the problem of the three bodies. To arrive at that solution we make a stage at the problem of two bodies, not because it is necessary, but because it is mathematically convenient. In the next Chapter we will considèr the general effects of the disturbing force of a third external body: and, investigate an expression for its value. The way will thus be, in some degree, prepared for the solution of the problem of the three bodies. For, such is the title attached, for distinction's sake, to the investigation, in which it is required to find the laws of motion, the form of the orbit, &c. of a body in motion round one attracting body and disturbed, relatively to that motion, by the attraction of a third remote body and it was under such form that Clairaut first stated the Lunar Theory, (see Mem. Acad. Paris, 1745. p. 329: and Theorie de la Lune, p. 3. ed. 2.) *. * Here properly ends the elliptic theory. CHAP. V. THEORY OF PERTURBATIONS. A third attracting Body introduced into the System of two Bodies. Its Effects in disturbing the Laws of Motion and the Elements of that System. Expressions of the Values of the resolved Parts of the disturbing Force; the Ablatitious; the Addititious: the Force in the direction of the Radius Vector; the Tangential Force; Effects of these Forces in altering Kepler's Laws, &c. Approximate Values of the Forces when the disturbing Body is very remote. Expressions for the Forces, in the Problem of the three Bodies, by means of the Partial Differentials of a Function of the Body's Parallax, Longitude and Latitude. In the preceding cases, we have supposed L to revolve round T, by virtue of some projectile motion oblique, in its direction, Ꮮ to LT, and of a centripetal force always urging L towards the point T. The centripetal force was supposed to arise from an attraction resident in the component particles of the bodies L and T, and to be proportional to the number of such particles; or, in other words, to the masses of the bodies L and T. m and M, therefore, expounding those masses, and the law of the force being according to the inverse square of the distance (r), the centripetal force M+ m was represented by If we illustrate these reasonings by reference to the Phenomena of Nature, and suppose L to represent the Moon, and T the Earth, then, in consequence of their mutual attraction, the Moon, as has been shewn, will describe an ellipse round the Earth, and areas proportional to the time, if we suppose all foreign agency, or the attraction of the other bodies and planets of the system, to be abstracted. In like manner, areas exactly proportional to the time, and an exact ellipse lying in the same plane, will be described, if L should represent any one of the planets, and T the Sun: or if L should represent a satellite, and T its primary: this essential condition being, in all cases, observed, namely, that the attraction of no other planet or satellite should interfere with the centripetal force. But, it is plain, this condition can never be observed. Every particle of matter is supposed to be endowed with an attractive quality; consequently, if L and T should represent the Moon and Earth, S representing the Sun, or Venus, or any other planet, would attract both L and T, and be attracted by them: which attraction, from the circumstances under which it acts, is called a Disturbing Force. Let m' represent the mass of S, then in this system of three bodies, L, T, and S, The four latter forces disturb the elliptical motion of L, not with their whole quantities, but with their difference. Suppose, for instance, that S should be so distant from L and T, that the difference of their distances relative to the distances themselves, might be neglected, and that finite parts of lines drawn, from L and Ttowards S might be esteemed parallel: then, L and T would be equally urged towards S and in parallel directions: and consequently, the relative motion of L round Twould not be disturbed: for the force of the attraction of S under these circumstances, would be the same as that of the communication of two equal impulses to L and T (see p. 43.). This circumstance of an evanescent, or very minute difference of the forces by which L and T are drawn towards S, takes place in nature. If L and T should represent the Moon and Earth, and S should represent either Jupiter or Saturn, then, by reason of the great distances of those planets, L and T would, equally and in parallel directions, be urged towards S, and no perturbation, or, at most, a very slight one, of the laws of elliptical motion, and of the equable description of areas, would ensue. Such a case as we have now instanced, does not mathematically belong to the problem of the three bodies, that is, does not require any special methods, or methods of solution beyond those that have been already used. It may, however, for the sake of extending a classification, be made to belong to it, and be considered as its most simple and limiting case. Since a third body, so distant as Jupiter is relatively to the Moon and Earth, would produce no alteration in Kepler's Laws, we must enter on the real problem of the three bodies, by assuming a case in which the third body S although very, should not be so excessively, distant, but that lines drawn from it to L and T should, in some positions, be unequal in length, and inclined the one to the other. Such a case, amongst many others, is that in which the Sun should be the central, Saturn the revolving, and Jupiter the disturbing body *. For when Saturn is in opposition, or Jupiter is Actio quidem Jovis in Saturnum non est omnino contemnenda. Nam, &c. et hinc oritur perturbatio orbis Saturni in singulis planeta hujus cum Jove conjunctionibus adeo sensibilis ut ad eandem Astronomi hæreant. Newton, p. 409. ed. 3. 1726. |