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They have (and this is one of the usual effects of progressive Science) considered the variations of the elements under a general point of view, and reduced the expressions of their values to six similar differential formulæ. From one of these it results that the major-axis of a planet's orbit is subject to no secular inequality and consequently that the planets, notwithstanding their mutual action, will constantly preserve the same distances from the Sun.

This is one of the points of the permanence or stability of the Planetary System; a subject of considerable importance and interest: and, the periodical inequalities of parallax, longitude, and latitude, having been investigated, this might now seem to be the proper place to consider the changes produced by disturbing forces in the dimensions and position of a planet's orbit. And so indeed it would be, were the preceding solution of the Problem of the Three Bodies immediately applicable to the case of any planet revolving round the Sun and disturbed by another planet. But the fact is otherwise. The instance, indeed, of Venus revolving round the Sun and disturbed by the Earth resembles, in its general character, that of the Moon revolving round the Earth and disturbed by the Sun. To each case belongs the same differential equation, and the same method of integration. There is, however, a difference which is to be found in the detail, and which is entirely mathematical. We will explain in what it consists.

The value of P which is given in page 60, and which was used in the succeeding series of solutions was derived from the general value of page 57. by expanding (see p. 59.) and by rejecting, in its development, all terms after the second. Now this rejection is founded on the minuteness of, and in the Lunar

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ys

but in the case of Venus disturbed by the

2

3

.723332: therefore, (1)*, (;-)',

&c. cannot be

rejected and consequently the analytical expression for P cannot remain the same as it was in the Lunar Theory. The pro

cesses and results, therefore, of Chapters IX, X, &c. will at least require some modification, or, as we shall soon see, the invention of new methods.

In the less simple expression for P, then, the planetary theory seems more complicated than the Lunar; but in other respects it is much less so. The chief cause is, the smallness of the disturbing force of any planet compared with the Sun's disturbing force. The Lunar perturbations require thirty equations, but three are sufficient to express the inequalities of Venus produced by Jupiter's action. For the simple cases, then, that occur in the Planetary Theory, the apparatus of formulæ and processes, that has been used in determining the Moon's place, is too cumbrous and complicated. The formulæ will serve indeed, as Clairaut in the Memoirs of the Academy of Sciences for 1754. pp. 521, &c. made them to serve, for determining the Earth's place disturbed by the Moon, Jupiter and Venus. But the method is not an expeditious one: and the first result, the expression of the mean anomaly in terms of the true, is not the main object of investigation. That object is, the true longitude in terms of the mean: and in order to obtain it, the Reversion of Series (see pp. 228, &c.), an operation of some difficulty, must be used.

A shorter method of solution, for the more simple cases, has been obtained by abandoning the equations [a], [b] of p. 95. for equations expressing r and its differentials, v and its differentials, in terms of nt, n d t, and other quantities. The equations of solution, when obtained, are, indeed, more easy of application than the former; but they are deduced by less obvious pro

cesses.

We have stated then two points of distinction between the Lunar and Planetary Theories, and which entitle the latter to a separate discussion.

The case that bears the strictest analogy to the theory of the Lunar perturbations is that of a Satellite of Jupiter disturbed by the Solar attraction. The problems in every particular are precisely the same. The Satellite's mean longitude, in order that its true may be found, must be corrected by the three equations

of the Variation, the Evection, and the Annual Equation. But these are very small corrections, and the other equations that correspond to the Lunar exist only theoretically, and are insignificant when numerically expounded.

The instances that resemble, but less closely, the Moon disturbed by the Sun are, as it has been already stated, Venus disturbed by the Earth, or by Mars, or by Jupiter; or, Mars disturbed by Jupiter; or, one of Jupiter's Satellites disturbed by another more distant Satellite. To these cases, that solution of the Problem of the Three Bodies which was used in the Lunar Theory does not immediately (see p. 254.) apply. We may presume also that it will not, without some modification, apply to the pertur bation of a planet disturbed by another, the orbit of which is interior to that of the former. For still less closely than either of the two preceding instances, does that of the Earth disturbed by the Moon, or by Venus, resemble the Moon disturbed by the Sun. It would be a loss of time to attempt to describe, in general terms, in what the difference consists. We must descend into the details and view it nearly.

The succeeding Chapters then, will be specially appropriated to the Planetary Theory, which, in many respects, is less complicated than the Lunar; and, in some of its instances, we shall find ourselves thrown back on the most simple cases of the Problem of the Three Bodies. The Planetary Theory, however, is not without its peculiar difficulties.

СНАР. XVI.

ON THE PLANETARY THEORY.

Differential Equation for determining the Radius Vector: Expression for R: its development into a Series of Cosines of Multiple Arcs. Conditions on which the Convergency of such Series depends. Application of the Differential Equation to the Investigation of the Perturbations in the Radius Vector and Longitude of the Earth by the Moon's Action.

THE first object in this Chapter is to obtain differential equations from which the radius vector and longitude may be obtained more concisely than from the equations [a], [b], of p. 95. that have been employed in the Lunar Theory.

If in the equations [4], [5], of p. 92. we suppose the body's latitude to be nothing, and substitute instead of P and T, their values such as are given in p. 66, namely,

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Multiply [1] by dr, and [2] by rdv, and add the results,

then

=

d

dr.d2r+r dr.dv2 + r2dv. d2 v + μ = d e3)

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R

Since, Mar
Mar dr+ dRdv) = SdR.*

For the purpose of eliminating dv from this equation [3], substitute, instead of r2d v2, that value which the equation [1] multiplied by r will give, then

dR
dr

d12 [(dr)2 + rd3 r] - " + " + 2 ƒ d R + r

+2fd + = 0.

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and if or be made to signify that variation of r which arises from the disturbing force, then, on neglecting (or) and the products of m' (see p. 66.) and dr, there will result

1

ròr

dR

2de [d2 (m2)
; [d2 (r2) + d2 (2 r ò r)] − 2 +
d " uror + 2 + 2 fd R + rd R

= 0.

But, when there is no disturbing force, and r, accordingly,

has its elliptical value,

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da that variation of a which is due to the disturbing force), we

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We must now consider whether it is possible to express

* Anal. Calc. p. 78.

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