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between that planet and the Sun, his disturbing force is to the force of the Sun, (or the centripetal force by which Saturn is y's mass urged towards the Sun) as 1 to 211; since

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In the preceding position of Jupiter, the effect of his disturbing force would be merely to increase the centripetal force of the Sun, and consequently, the equable description of areas would not be disturbed. But this would not happen in other positions of Jupiter. In fact, whenever the third body is so near as either to diminish or augment, in any position, the central force by which the revolving body is urged, it must, in other positions, produce a

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disturbing force oblique in its direction to the radius, and consequently (see p. 19.) tending to interrupt the equable description of areas. For, if S be the third body, and m' its mass, its disturbing force on L, when L is at A, is

m'

m'

SA2 ST

; and,

consequently, can have none effect, except (see p. 48.) the difference of SA and ST, bear, (when it is, according to the conditions of the case, numerically expounded) some sensible proportion to SA and ST. But if that be the case, then, at points, such as L, intermediate between A and B, the difference of ST and SL will be of some sensible magnitude, and consequently the difference of the forces by which L and T are urged to S, and on which the disturbing force depends, will be of some magnitude. That dif

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ference may be represented by a line Lt either parallel to ST, or nearly so, and consequently will always (except at four points) admit of being resolved into two other lines representing forces, one in the direction of the radius LT, the other in a direction perpendicular to LT. It is this last force (see p. 19.) which prevents the operation of Kepler's Law of the equable Description of Areas.

This is not the sole effect of a disturbing force; for since, in some positions (see p. 49.), it augments, and in others diminishes* the centripetal force, and not according to the law of its variation, the law of the resulting force, by which L is urged, is not that which it was supposed to be, when (see p. 27.) the orbit described by L was proved to be an ellipse.

The line of the apsides, which in an ellipse is its major axis, will not, as we shall hereafter see, remain fixed; and, besides these changes which take place in the plane of the body's orbit, there are others that will affect the position of the plane itself. For, if the plane of the disturbing body's orbit be not coincident with. that of the revolving body, the disturbing force, if represented by a line (analogous to Lt, see fig. p. 49.) is represented by a line inclined to the plane of the orbit of L: and consequently, the force represented by this line may be resolved into two others; one in the plane of L's orbit, the other perpendicular to it. This last force will have a tendency to change the plane's inclination, and also (when combined with the body's motion) to change the position of its intersection with another plane (such as the ecliptic), or, which is the same thing, to change (see Astronomy, p. 40.) the longitude of the nodes.

Such are, on general grounds, the discernible effects of a disturbing force; they will, perhaps, be more distinctly perceived from its mathematical expression, which we shall now proceed to investigate and first, for the sake of simplicity, we will suppose the planes of the orbits of the revolving and disturbing bodies to be coincident.

* If Saturn be in conjunction, then Jupiter attracting the Sun more than it does Saturn has the effect of diminishing the centripetal force.

Let, LT r, ST=r', SL = y, and the angle STL = w, let also M, m, m', be, respectively, the masses of the bodies placed

m

at T, L, and S; T (the Earth) being considered to be the central body round which L (the Moon), disturbed by the action of S (the Sun), is supposed to revolve; then, the force by which

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This latter force always acts in the direction LT, and, since it increases the centripetal, which is reckoned the chief force, it is technically called the Addititicus Force. The centripetal force in this case, since T is supposed to be fixed, must be represented by the sum of the attractions of L to T and of T to L (see p. 43.), and therefore, by

M+ m

consequently, the compounded force of L to T (which is in fact, a centripetal force) is, now,

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This, however, is not the whole force in the direction of LT;

*There are several methods, besides the one in the text, for resolving the disturbing force. In that we supposed (for illustration), S, L, and T to represent the positions of the Sun, Moon and Earth. Suppose now S still to represent the Sun, but L and T Jupiter and

Saturn, then S is the central, L may be the revolving, and T the disturbing body; or, T may be the revolving, and L the disturbing: take the former case, and let the symbols for these bodies be used to denote their masses: let S1, St be denoted by x, x', LI, Tt, by y, y', SL, ST, by r, r', Tt, Ll being perpendicular to St: then the force by which ⚫ L is drawn to S, in the direction of Sl, by the mutual attraction of L and S, is (see p. 51.)

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and the forces by which T draws S and L in the same direction are, respectively,

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but L is disturbed in this direction by the difference only of these latter forces, consequently by

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and accordingly the whole force acting on L in the direction parallel to

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there still remains to be added to it, a resolved part of that other

force, which acts in a direction parallel to ST. >

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The addititious force, since it acts in the direction of a line

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expresses the force in a direction parallel to x by which Saturn is acted on, when it revolves round the Sun, and is disturbed by Jupiter and these two expressions are the same as what M. Laplace uses in his Theory of Jupiter and Saturn. (See Memoirs of the Academy of Sciences, 1785. pp. 38, &c.)

The foregoing, as we have said, is, very nearly, Laplace's method of valuing the forces. For the purpose of farther illustration, and solely for such purpose, Euler's, that which he gave in the 7th Vol. of the Prix de l'Academie des Sciences, 1769. is subjoined.

Let O be the place of the Sun (O), M of Jupiter (Y), N of Saturn (↳), let MR be perpendicular to OM, NS to MN, and let the symbols

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O, 4, h, denote the masses of the bodies they are meant to signify, then the forces attracting the Sun, Jupiter, Saturn,

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Now, as before, if the Sun be supposed fixed, and Jupiter be the

body

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