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(k being an arbitrary quantity introduced for the purposes of cor

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If we substitute these values in the expression for dv, and then reduce the expression by the ordinary methods, there will result

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dvis, in this expression, the variation or inequality in longitude arising from the disturbing force: but nt expounds the mean motion: no term then of nt can possibly enter into the expression for dv and accordingly, we must have

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If, which is nearly the case, we make N2n, and besides

write m instead of 2, (see p. 140.)

n'

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m

- m

2 m2

1

- m

(1-3m)] sin. (n't − n t + é−c),

and, if according to the numerical values of a, d', m, as given in

*

p. 265. we compute the coefficient of the preceding term, we shall have

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2

............

log. (1-m)......... 1.96623

log. 2 m2...... 2. 04883.log. (1-2m)...... .. T.92972

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Since, the eccentricity being supposed nothing, the mean and true motions are the same.

The above value of v agrees with that which, derived from different principles, was given in p. 85.

If, in the preceding expression, we write, as in p. 180o + U instead of v', we shall have

dv8". 9. sin. (180o + U – v)

= 8". 9. sin. (U

v),

which, very nearly, is Laplace's expression: (see Mec. Cel. tom. III. P. 108.

The perturbations then in the Solar parallax and longitude are, after the establishment of the equations of pp. 258. 268., very easily deduced. One cause of the facility of deduction is the abstraction of the condition of the eccentricity, which abstraction is arbitrary or hypothetical: another, and which must always exist, is the minuteness of the radius of the Lunar Orbit, compared with that of the Solar: a minuteness such as to render unnecessary, as a compendium of computation, that formula (see p. 262.) by which R is expressed in a series of terms involving the cosines of multiple arcs.

The deduction, in the present Chapter, of the perturbations of the Solar Orbit by the Moon's action is intended principally to illustrate the use of the newly derived differential equations: but the results serve, besides, to confirm, or are confirmed by, those results, which, in Chap. VI. were obtained by the method of

*This method of determining the perturbation of a primary by the action of its satellite (for such is the case of the Moon disturbing the Earth's motion) originated with Dalembert (see Recherches sur differens points dans le systême du monde. tom. II. pp. 20. 47, &c.). In the same treatise, however, that acute writer shews that we ought to prefer, in investigating the perturbation of the planets, a systematic integration of the differential equations: or, in other words, a direct solution of the Problem of the Three Bodies.

the centre of gravity: a method, (if we look to its use in the Theory of Perturbations) partial, and restricted, almost completely, to the case to which it was applied.

The uses of the differential equations of pp. 258. 268. are not sufficiently illustrated by the preceding case. They will be more adequately illustrated by the research of the Earth's perturbations from the action of Jupiter, and especially, if we retain in it, the condition of the eccentricity of the Solar Orbit. This case will serve too, more fully than the preceding, to shew the utility of developing R into a series of terms involving the cosines of multiple arcs and, will, accordingly, illustrate one ground of distinction (see p. 254.) between the Lunar and Planetary Theories. But it will not serve as a characteristic illustration of this latter point.

CHAP. XVII.

On the Development of R in terms of the Cosines of the Mean Motions of the disturbed and disturbing Planets. On the Method of Computing the Coefficients of the Development, when the Radius of the Orbit of the Disturbed Body differs considerably from that of the Disturbing: Application of the Formula to the Case of Jupiter disturbing the Earth. New Formulæ necessary when the Radii of the Orbits of the two Bodies are nearly Equal.

Ir we revert to p. 66, we shall find that, when the inclination of the orbits of the disturbing and disturbed bodies is neglected,

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In the instance given in the preceding Chapter, great facility was afforded to the computation, by assuming the orbits, both of the revolving and of the disturbing body, devoid of eccentricity. In consequence of which assumption, we had

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This was one source of facility: another (and, in an Elementary Treatise, we cannot well insist too much on the important points) was the minuteness of. By reason of that minuteness the

series for 1 (see p. 261.) converged so rapidly, that it was suf

MM

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