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But another cause operates; the Lunisolar precession (that which is caused by the action of the Sun and Moon on the protuberant equatoreal parts of the Earth) varies as the cosine of the obliquity. The obliquity then decreasing, the precession must be increased, and it will now be about 0".09 greater than it was at the commencement of our Æra: the true precession therefore of 1800 will now be greater than the true precession of the year 1 by 0.303 +0".09, or, 0".393. The Sun, (assuming its mean motion in 24 hours to be 59' 8"), would describe this space (0.393) in about 9".3, which is the computed excess of the tropical year at the beginning of the Era above the present tropical year.

It is the computed excess; being merely a result from theory. Antient observations are inaccurate far beyond 9 seconds, and, consequently, we can only say that the progression of the equinoctial points from the disturbing forces of the planets is a probable result. The point, however, may be settled in future times, if observations should then be made as accurately as they are at present*

*.

Besides the progression of the equinoctial points, there are other inequalities, discussed in this Chapter, that, at present, ought to be viewed as mere results of theory. Such are, for instance, the variations of the inclinations of the planes of the orbits of planets. These, hitherto, have not been determined by observation: they are too minute, and antient observations are too inaccurate; if the former are as minute as theory shews them to be, it is hopeless to expect to determine them by the latter. This is another point reserved for future Astronomers.

* The matter can never be altogether free from uncertainty. If, by observations, made 500 years hence, compared with modern, the tropical year should then appear to be less, the fact might be accounted for by supposing the Lunisolar precession to be less diminished by the direct motion of the equinoctial points: the mean quantity of the Lunisolar precession itself being always supposed the same. And it would be accounted for, with a high degree of probability, if the computed progres. sion (see pp. 452, &c.) agreed with the difference of the lengths of the tropical years as made out from the comparison of observations.

There is, however, one exception to what has been just said. The diminution of the obliquity, which is a consequence of a change of inclination in the Earth's orbit, may now be considered as established by observations, although 70 years ago there were Astronomers who asserted that it was constant, and, which is more strange, denied that it could vary on Newton's Principles.

We have in the present Chapter deduced and exemplified the expressions for the secular inequalities of the elements of a planet's orbit. We have also, on restricted conditions indeed, established some very curious properties concerning the limits within which both the variations of the eccentricities and the inclinations are confined. In such and like properties consists the stability of the Planetary System: which, of all the results furnished us by Physical Astronomy, is, perhaps, the most interesting. It merits then some farther consideration; and, in the next Chapter, we will endeavour to render more general those which are to be considered (see pp. 422, 435.) as its essential theorems.

CHAP. XXIII.
XXIII.

Stability of the Planetary System with regard to the Mean Distances. The Mean Distances subject only to Periodical Inequalities and not to Secular. Stability of the Planetary System with regard to the Eccentricities and Inclinations. Theorems which express the Conditions to which their Variations are subject.

THE

HE constant parts of the development of R, (so it appears by p. 414,) do not contain the quantity and since

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it was thence inferred that, with regard to such constant parts,

d R

de

was 0 in other words, that the axis major was subject to no secular variation.

This, which is an important point, may be considered under another point of view.

The arguments of terms in the value of R (see pp. 279, 281, &c.) independent of the eccentricities, are

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of terms involving the first powers of the eccentricities, the argu

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and (p+1)n't - pnt + (p + 1) ε — pe — π'.

The arguments of terms involving the squares of the eccentricities will be

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so that, it is plain, we may generally represent a term in the development of R, by

P. cos. (p'n't- pnt + A),

in which p', p, will be integers having their difference

(p'-p),

connected with the powers or products of the eccentricities that are involved in the coefficient P.

Now dR is the differential of R, when those quantities are made to vary which determine the place of the body m (be they co-ordinates, or radius vector and longitude): but these quantities being expressed, by means of the variable quantity nt and of certain constant quantities, the differential of R corresponding to the term

P. cos. (p'n't- pnt + A),

must be obtained by making n t vary; accordingly,
d R = Ppndt.sin. (p'n' t − p nt + A),

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a2. Ppndt. sin. (p'n' t−pnt + A).

Now p', p are integers, and p' - p may = 0, or ± 1, or ±2, or &c. and if p' and p could be taken such that

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then there would result, in the above expression, at least one term in the variation of a equal to

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A being constant; and, accordingly, there would result in

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μ

a term Pput.sin. A increasing with the time, altering and continuing to alter the mean distance. But, so it happens, the mean motions, n and n', of the disturbed and disturbing planets are such that p'n' can never equal pn. If the Earth be the planet disturbed by the actions of all the others, its mean motion (see Table of Periods, Astronomy, p. 283.) is not commensurable with the mean motion of any other planet. Its mean distance, therefore, suffers no secular change from the disturbing forces of the planets. The same holds good of the mean distance of every other planet, and for the same reason. The mean motion of Jupiter, for instance, is not to the mean motion of any other planet as number is to number. Twice Jupiter's mean motion is indeed, as we have seen in Chapter XIX, nearly equal to five times Saturn's; the consequence of which is, that their motions are affected with inequalities of a very long period: so long, indeed, that the inequa lities are of the nature of secular inequalities, and become blended with the mean motions; and this latter is a result deducible from the preceding expression: for, make p′ = 5, and p=2, and then 4 Pndt sin. (5n't — 2nt + A),

δα

μ

which expression will, for a great length of time, continue of the same sign; since, 5 n' - 2 n being very small, t must be very great before 5 n't 2nt from 0 can become 180°. But, the mean distance continuing either to increase or decrease during a long period, the mean motion will continue to decrease or to increase during the same period.

By whatever method, then, we examine the effect of the disturbing forces of the system on the mean distances of the planets, it appears that those distances are subject to no secular change. They vary periodically, that is, they increase for a time by small quantities, and again, having reached a certain limit, by like degrees decrease: for, such is the nature of the change indicated by the term

Ppndt.sin. (p'n't- pnt + A).

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