To find the value of g to the third order.

95. This is to be obtained in a very similar manner from the d's

equation +8= &c. We shall, in the argument, write g for

de

3m2a's

{1 + cos(2 - 2m)},

m3k {1+ cos(2 - 2m)} {sin(g) + §m sin(2 − 2m − g)}

96. Hence, to the third order of approximation,

mean motion of apse 1 mean motion of node

=

с

3 m2 + 22,5 m3

8+ 75m

=

=

PARALLACTIC INEQUALITY.

79

and since my nearly, we see that the moon's apse progredes nearly twice as fast as the node regredes.

In the case of one of Jupiter's satellites, m is extremely small, for the periodic time round Jupiter is only a few of our days, and the periodic time of Jupiter round the sun is 12 of our years, and therefore m, the ratio of these periods, is very small.

Hence, the apse of one of Jupiter's satellites progredes along Jupiter's ecliptic, with pretty nearly the same velocity as the node regredes, assuming these motions to be due to the sun's disturbing force; they are, however, principally due to the oblateness of the planet.

Parallactic Inequality.

97. In carrying on the approximations to a higher order, it is found, as we stated Art. (55), that the expressions for

Since

=

a

E+ Ma

α

bo, nearly, is of the second order, these terms are

of the fourth order, but the coefficient of being near unity, they will become important in u, and therefore in 0, Art. (27). We can easily obtain the terms to which they give rise in the values of u and 0,

Substituting in the differential equation for u, we get

98. The corresponding term in the value of will also be of the third order,

T

h2u3

de is of the fourth order and will not rise in t;

therefore

dt

=

SECULAR ACCELERATION.

81

This term, whose argument is the angular distance of the sun and moon, is called the parallactic inequality on account of its use in the determination of the sun's parallax, to which purpose it was first applied by Mayer by comparing the analytical expression of this coefficient with its value as deduced

E

from observation. The values of m and of and therefore, of

M'

being pretty accurately known, will be determined,

α
a

E M E+M 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

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

G

Lagrange, the actions of the planets on the eccentricity of the earth's orbit will be ultimately reversed, 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, if not greater, accuracy than 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 compreswhich is about a mean between the different values obtained by other methods.

sion

* Pontécoulant, Système du Monde, vol. iv.