740. The sine of the horizontal parallax of the moon is

R being the terrestrial radius, but as this arc is extremely small, it may be taken for its sine; hence, if

— { 1 + e2+} y2+e (1+e2) cos (cv —w) —¡y2 cos (2gv—20)}+du

a

R

be put for u, and quantities of the order e rejected, the parallax

will be

R

a

a

(1+e3){1+e [1−‡y2+}y® cos (2gv=20)] cos(cv—w)+adu-sds}.

In the untroubled orbit of the moon the radius vector, and, consequently, the parallax, varies according to a fixed law through every point of the ellipse. Its mean value, or the constant part of the

horizontal parallax, is

R

19
a

to which the rest of the series is applied

as corrections arising both from the ellipticity of the orbit and the periodic inequalities to which it is subject.

741. In order to compute the constant part of the parallax, let o be the space described by falling bodies in a second in the latitude, the square of whose sine is, I and R' the corresponding lengths of the pendulum and terrestrial radius, the ratio of the semicircumference

to the radius, E and m the masses of the earth and moon; then, supE+ m=1,

posing

T being the number of seconds in a sidereal revolution of the moon ; and by article 735

Now the length of the pendulum, independent of the centrifugal force,

this value augmented by 3.74, to reduce it to the equator, is 3427′′.9; hence the equatorial parallax of the moon in functions of its true longitude is

The greatest value of the parallax is 1° 1' 29".32, which happens. when the moon is in perigee and opposition; the least, 58' 29".93, happens when the moon is in apogee and conjunction.

E

742. With m =

Mr. Damoiseau finds the constant part of the

74

equatorial parallax equal to 3431".73.

743. The lunar parallax being known, that of the sun may be determined by comparing the coefficients of the inequality

in the moon's mean longitude with the same derived from observation. In the tables of Burg, reduced from the true to the mean longitude, this coefficient is 122".378; hence

i+1=

122".378
122".014

=1".00298, and 1
a,

a

1".00298
400

which is the mean parallax of the sun in the parallel of latitude, the square of whose sine is .

Burckhardt's tables give 122".97 for the value of the coefficent, whence the solar parallax is 8".637, differing very little from the value deduced from the transit of Venus. This remarkable coincidence proves that the action of the sun upon the moon is very nearly equal to his action on the earth, not differing more than the three millionth part.

744. The constant part of the lunar parallax is 3432".04, by the observations of Dr. Maskelyne, consequently the equation

of whose sine is; if R', the mean radius of the earth, be assumed as unity, the mean distance of the moon from the earth is 60.4193 terrestrial radii, or about 247583 English miles.

745. As theory combined with observations with the pendulum, and the mensuration of the degrees of the meridian, give a value of the lunar parallax nearly corresponding with that derived from astronomical observations, we may reciprocally determine the magnitude of the earth from these observations; for if the radius of the

earth be assumed as the unknown quantity in the expression in article 646, it will give its value equal to 20897500 English feet.

'Thus,' says La Place,' an astronomer, without going out of his observatory, can now determine with precision the magnitude and distance of the earth from the sun and moon, by a comparison of observations with analysis alone; which in former times it required long voyages in both hemispheres to accomplish.'

746. The apparent diameter of the moon varies with its parallax, for if P be the horizontal parallax, R' the terrestrial radius, r the radius vector of the moon, D her real, and A her apparent diameters; then

a ratio that is constant if the earth be a sphere. It is also constant at the same point of the earth's surface, whatever the figure of the may be.

earth

thus if be multiplied by the moon's apparent semidiameter, the corresponding horizontal parallax will be obtained.

Secular Inequalities in the Moon's Motions.

747. It has been shown, that the action of the planets is the cause of a secular variation in the eccentricity of the earth's orbit, which variation produces analogous inequalities in the mean motion of the moon, in the motion of her perigee and in that of her nodes.

The Acceleration..

748. The secular variation in the mean motion of the moon denominated the Acceleration, was discovered by Halley; but La Place first showed that it was occasioned by the variation in the eccentricity in the earth's orbit. The acceleration in the mean mo