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Z

ST

The difference (D) of forces by which L)

and Tare urged towards S, in a direction.
parallel to ST

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We must now resolve the centripetal force () into the

directions TI, LI.

The resolved part of, in the direction TI,=.

μ P =

co
cos.$ 4,

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Hence, collecting the several parts of the forces,

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which expressions are the same as what Mayer has deduced in his Theoria Luna, p. 4.

Of these forces, the whole of T is a disturbing force, and of P and S, the parts that arise from the disturbing force, are

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These expressions for P, T, and S being substituted in the equations (4), (5), (6), of p. 10. and the equations being solved, every thing relating to the parallax, latitude and longitude, (see Astronomy, Chap. XXXIV.) of the disturbed body, or, to speak without reference to the elliptical theory, of a body agitated by the above forces, would become known. But, as we have already remarked (p. 59.), we must avail ourselves of every means to facilitate the solution; and, accordingly, if

should be con

siderably greater than r, we ought to substitute, instead of

1

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approximate expression; thus, substituting for as before (see

y3

p. 59.), and supposing, from the smallness of the inclination, r

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These expressions result by rejecting, (see p. 59.) from the

expansion of the terms that involve higher powers of than

the cube. If we go a step farther, and retain the terms involving

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(See Mem. des Sçavans Etrangers, Tom. VII. p. 14. also, Simpson's Tracts, p. 176.)

When, besides the plane of the body's orbit, another plane like that of the ecliptic is considered, the forces P, T, and S will consist of terms involving, besides constant quantities, p, v,

and s; and, (yet this is a matter of mere analytical convenience,) they may be expressed by means of partial differential coefficients (see Prin. of Anal. Calc. p. 79,) of a function of those quantities: thus, if v and v' be the longitudes of L and S, the angle STI, or E12- υ,

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and we may obtain expressions still more simple by assuming

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* These expressions are the same as Laplace's Mec. Cel. Liv. II. p. 151. and Lib. VII. Theorie de la Lune, p. 181.

This mode of representing the forces, although simple and convenient, is not an obvious one, when P, T, S are to be expressed in terms of u, v and s. But it would be easily suggested, if the forces and equations were expressed in terms of the rectangular co-ordinates x, y, z. This is another advantage (see p. 40.) of these symmetrical equations, which we have not deduced before or elsewhere, for fear of interrupting the course of investigation.

Let, (see the figure of p. 63.) the co-ordinates of L reckoned from T, be x,y, z; of S, x', y', z';

==

then SL (A) =

[(x' − x)2 + (3′ − y)2 + (2′ — 2)2],

and, on the same principles as before, (see p. 64.)

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the force of S on L, in a direction parallel to x, =

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of S on T, in the same direction, = x.

m'

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Hence, since the centripetal force in the direction parallel to is

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we have the whole force in that direction thus ex

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Y and Z will be similarly expressed: Hence, substituting their values

in the equations of p. 8. there result

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The advantage of thus expressing the forces consists in this: that, if Q should be expressed by part of an expanded function, in other words, by that part of the series which remains after

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Now it is obvious, that the third terms are

coefficients of, and the fourth or last terms of

let, therefore,

partial differential

x x' + y y + z z'

;

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If we suppose, as in the Note to p. 52. L to be Jupiter and T Saturn, the three preceding equations will be those which we must use for determining Jupiter's motion: and, for determining Saturn's, we shall have, as it is plain from pp. 53, &c.

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or, to render the two sets of equations more alike, we may thus express them

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