E

as deduced from observation. The values of m and of

M'

a'

a

E-M and therefore, of being pretty accurately known, E+M will be determined, that is, the ratio of the sun's parallax to that of the moon: but the moon's parallax is well known; therefore, also, that of the sun can be calculated. The value so obtained for the sun's parallax is 8.63221", while those given by the two last transits of Venus fall between 8.5" and 8.7".*

Secular Acceleration.

99. Halley, by the comparison of ancient and modern. eclipses, found that the moon's mean revolution is now performed in a shorter time than at the epoch of the recorded Chaldean and Babylonian eclipses. The explanation of this phenomenon, called the secular acceleration of the moon's mean motion, was for a long time unknown: it was at last satisfactorily given by Laplace.

The value of p, Art. (50), on which the length of the mean period depends, is found, when the approximation is carried to a higher order, to contain the quantity e' the eccentricity of the earth's orbit. Now, this eccentricity is undergoing a slow but continual change from the action of the planets, and therefore p, as deduced from observations made in different centuries, will have different values.

The value of p is at present increasing, or the mean motion is being accelerated, and it will continue thus to increase for a period of immense, but not infinite duration; for, as shewn by Lagrange, the actions of the planets on the eccentricity of the earth's orbit will be ultimately re

* Pontécoulant, Système du Monde, vol iv. p. 606,

versed, e' will cease to diminish and begin to increase, and consequently p will begin to decrease, and the secular acceleration will become a secular retardation.

It is worthy of remark that the action of the planets on the moon, thus transmitted through the earth's orbit, is more considerable than their direct action.

Inequalities depending on the Figure of the Earth.

100. The earth, not being a perfect sphere, will not attract as if the whole of its mass were collected at its centre: hence, some correction must be introduced to take into account this want of sphericity, and some relation must exist between the oblateness and the disturbance it produces. Laplace in examining its effect found that it satisfactorily explained the introduction of a term in the longitude of the moon, which Mayer had discovered by observation, and the argument of which is the true longitude of the moon's ascending node.

By a comparison of the observed and theoretical values of the coefficient of this term, we may determine the oblateness of the earth with as great accuracy as by actual measures on the surface.

101. By pursuing his investigations, with reference to the oblateness, in the expression for the moon's latitude, Laplace found that it would there give rise to a term in which the argument was the true longitude of the moon.

This term, which was unsuspected before, will also serve to determine the earth's oblateness, and the agreement with the result of the preceding is almost perfect, giving the compression,* which is about a mean between the different values obtained by other methods.

* Pontécoulant, Système du Monde, vol. iv.

Perturbations due to Venus.

102. After the expression for the moon's longitude had been obtained by theory, it was found that there was still a slight deviation between her calculated and observed places, and Bürg, who discovered it by a discussion of the observations of Lahire, Flamsteed, Bradley, and Maskelyne, thought it could be represented by an inequality whose period would be 184 years and coefficient 15". This was entirely conjectural, and though several attempts were made, it was not accounted for by theory.

About 1848, Professor Hansen, of Seeberg, in Gotha, having commenced a revision of the Lunar Theory, found two terms, which had hitherto been neglected, due to the action of Venus. One of them is direct and arises from a 'remarkable numerical relation between the anomalistic 'motions of the moon and the sidereal motions of Venus ' and the earth; the other is an indirect effect of an inequality 'of long period in the motions of Venus and the earth, ' which was discovered some years ago by the Astronomer 'Royal.'*

The periods of these two inequalities are extremely long, one being 273 and the other 239 years, and their coefficients are respectively 27.4" and 23.2". These are considerable 'quantities in comparison with some of the inequalities already recognised in the moon's motion, and, when applied, they are found to account for the chief, indeed the only re'maining, empirical portion of the moon's motion in longitude 'of any consequence; so that their discovery may be con'sidered as a practical completion of the Lunar Theory,

* Report to the Annual General Meeting of the Royal Astronomical Society, Feb. 11, 1848.

at least for the present astronomical age, and as establishing 'the entire dominion of the Newtonian Theory and its ana'lytical application over that refractory satellite.**

Motion of the Ecliptic.

103. We have seen, Art. (14), that our plane of reference is not a fixed plane, but its change of position is so slow that we have been able to neglect it, and it is only when the approximation is carried to a higher order, that the necessity arises for taking account of its motion.

It has been found to have an angular velocity, about an axis in its own plane, of 48′′ in a century, and the correction thus introduced produces in the latitude of the moon a term

- cw cos (0-4),

where ∞ is the angular velocity of the ecliptic, the angular

с

velocity with which the ascending node of the moon's orbit recedes from the instantaneous axis about which the ecliptic rotates, the longitude of this axis at time t, and the longitude of the moon at the same instant.

Let Am be the position of the ecliptic at time t, A the point about which it is turning, VA = 4, MN the moon's orbit, M the moon, and Mm a perpendicular to the ecliptic; m = 0; Mm=lat.=ß. ¿ the inclination of the orbit, and N the longitude of the node.

* Address of Sir John Herschel to the Meeting of the Royal Astronomical Society.

Let APN'm' be the ecliptic after a time St.

Any point whose longitude is L may be considered as moving perpendicularly to the ecliptic with a velocity

w sin(L− 4).

Hence, the point N will move in the direction NP with a velocity o sin (N-4). And N' will move along PN' with a velocity o sin (N− p) coti;

therefore

dN dt

= ∞ sin (N−4) coti.

Again, the point of the ecliptic 90° in advance of N, will move towards the moon's orbit with a velocity

=ca {cosi sin sin (N−4) – cosy cos(N−4)},

=

but cosi.sin cosẞ sin (0-N) and cosy = cos cos(0–N): therefore

SB=-cw cos(0 − p).

The discovery of this term is due to Professor Hansen; its coefficient is extremely small, about 1.5"; but, being of a totally different nature from those due to successive ap