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the rejection of terms that are diminished beyond a certain minuteness, the quantities P, T and S might immediately be exhibited.

The values of P, T, S, being thus obtained, the place of the body may be determined, as we have already observed (p. 65.) by the solution of the equations (4), (5), (6). But these involve dt: this quantity, therefore, must be eliminated, as in the former case, (see p. 22.) by a process of the same principle and kind, but, by reason of the additional conditions, more operose.

The preceding equations differ from those which obtain (see p. 6.) ΦΩ

in a system of two bodies, solely by the last terms &c. This mode > dx

of representing the differential equations was first used by Lagrange, (see Acad. Berlin, 1776. p. 210. 1781. p. 214.) and it has been adopted by Laplace and other foreign mathematicians (see Mec. Cel. p. 254. also Mem. Inst. tom. IX. p. 12, &c.).

lf, besides S, a fourth body (S') should disturb the motion of L round T, it would be necessary merely to add to the former value of

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m", x", y", r", &c. being its mass, co-ordinates, distance, &c. and then the preceding equations would obtain.

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and consequently, instead of the three preceding equations, we shall

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Before we proceed to that operation (which will be one of the subjects of the ensuing Chapters), we wish to state the form under which the mathematicians, that succeeded Newton, considered the problem of the Moon's Orbit.

If we take no account of the disturbing force that acts perpendicularly to the plane of the body's orbit, then the forces acting on the body are two, P and T, and the integration of the equations would be the solution of a problem, in which it should be required to find the curve described by a body acted on by two forces, one in the direction of the radius, the other in a direction perpendicular to the radius. It was under the terms of this latter statement that Clairaut first proposed (see Mem. Acad. des Sciences, 1745, p. 342. and Theorie de la Lune, p. 3.) the question of the lunar orbit. The force in the direction of the radius in part arose from the mutual attraction of the Moon and Earth, and, in part, from the Sun's disturbing force. The other force that acted perpendicularly to the radius originated wholly from the Sun. Dalembert, (see Recherches sur differens points, &c.) although he derived his differential equations by a process different from that which Clairaut followed, yet, like him, reduced to a similar statement, the lunar theory. And Thomas Simpson, our countryman, notwithstanding some peculiarity in those Propositions which are preparatory to the main one, yet reduced it en demiere Analyse' to a form similar to that which his illustrious contemporaries had arrived at.

The shortest and most direct way of proceeding would be immediately to deduce and integrate the differential equations above-mentioned: that is, in other words, to solve the Problem of the Three Bodies. But the difficulties of that problem are such, that it is desirable, and especially in an Elementary Treatise like the present, to lessen them by all possible means. We shall therefore consider whether there are any obvious inferences that present themselves on the introduction of a third disturbing body into the system of two bodies. The third body, as we have already seen, destroys the equable description of areas and the elliptical form of the orbit. The disturbing force of the Sun, for instance, prevents the Moon from describing an ellipse round the Earth, and the disturbing force of the Moon prevents the Earth from describing an ellipse round the Sun. Neither the Solar

nor Lunar theories, therefore, can be exactly constructed according to Kepler's Laws. But, may not the point called the Centre of Gravity, which, in the doctrines of Equilibrium and Dynamics, possesses several curious properties, possess some here? May it not, notwithstanding the agency of disturbing forces on the bodies of the system, observe, either exactly, or nearly so, Kepler's Laws? The enquiries of mathematicians, in the rise of Physical Astronomy, would be naturally directed to this subject. Newton considered it in the first Propositions of the 11th Section. And, as it will soon appear, the laws of the motion of the centre of gravity, and the form of the curve described by it, belong to an investigation more difficult than the preceding, but less so than those that follow. We shall find ourselves, indeed, at a problem intermediate to the problems of two and of three bodies; and, although its solution is not essential and might be dispensed with, yet it is a convenient solution, will illustrate the subject matter of enquiry, and will furnish, in some instances, very easy methods of computing the effects of disturbing forces. For these reasons we shall briefly consider it in the following Chapter.


The Motion of the Centre of Gravity of two or more Bodies not affected by their mutual Action: their Centre of Gravity attracted by a distant External body (the System revolving round it) by a Force nearly as the Inverse Square of the Distance: it describes therefore an Ellipse, nearly, round that Body. The Centre of Gravity of the Earth and Moon, the Centres of Gravity of Jupiter and his Satellites, of Saturn and his, all describe, very nearly, Ellipses round the Sun, and Areas proportional to the Times. Values of the Disturbing Forces that prevent the exact Description. The Moon's Menstrual Motion: Values of the Perturbations of her Parallax and Longitude by the Earth's Action: Value of the Menstrual Parallax.

IF S represent a central attracting body, and a a body revolving round it, then, if the law of attraction should be inversely as the square of the distance, and no disturbing force should






operate, a (as it has been shewn in pp. 27, &c.) would revolve either in a circle, or in an ellipse round S, and would describe areas proportional to the time.

If b be introduced as a third, and as (see pp. 47. 49, &c.) a disturbing body, and if the distance (r) of a from S, be much greater than that (r) of b from a, and also if m', the mass of the disturbing body, be very small relatively to the masses (M and m) of S and a,

then, as we have seen in p. 60. the two forces acting on a

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and, under the influence of these forces, the body a will no longer describe an ellipse round S, nor areas proportional to the time.

The preceding conditions and inferences will hold good, if S represent the Sun, a the Earth, and b the Moon: for in this instance

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of gravity of a and b: and the first point to be considered is this: how will the quiescence or motion of G be affected by the mutual action (supposing such action to be carried on according to the laws of attraction) of the bodies a and b? For, the curve described by G and the laws of its motion will depend partly on the motion which it has independently of S, and partly on the force by which it is urged to S. We will consider the first point, and instead of two bodies a and b, we will take more; so that the results may extend, beyond the case of the Earth and Moon, to those of Jupiter and Saturn and their systems of satellites.

For the sake of simplicity, suppose the bodies a, b, c, and whose masses are respectively m, m', m", to be in the same plane and let their respective rectangular co-ordinates, with reference to a point S and a line SO, be x, y, x', y', x", y",


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