is the eccentricity of the earth's orbit at any time t, whence the acceleration is

10."1816213. T2+0".018538444 . T3,

T being any number of centuries before or after 1801.

In consequence of the acceleration, the mean motion of the moon is 7' 30" greater in a century now than it was 2548 years ago.

Motion of the Moon's Perigee.

749. In the first determination of the motion of the lunar perigee, the approximation had not been carried far enough, by which the motion deduced from theory was only one half of that obtained by observation; this led Clairaut to suppose that the law of gravitation was more complicated than the inverse ratio of the squares of the distance; but Buffon opposed him on the principle that, the primordial laws of nature being the most simple, could only depend on one principle, and therefore their expression could only consist of one term. Although such reasoning is not always conclusive, Buffon was right in this instance, for, upon carrying the approximation to the squares of the disturbing force, the law of gravitation gives the motion of the lunar perigee exactly conformable to observation, for e' being the eccentricity of the terrestrial orbit at the epoch, the equation c = √1-p-p'e2 when reduced to numbers is c = 0.991567, consequently (1 c) v the motion of the lunar perigee is 0.008433 . v ; and with the value of c in article 735 given by observation, it is 0.008452. v, which only differs from the preceding by 0.000019. In Damoiseau's theory it is 0.008453. v, which does not differ much from that of La Place. The terms depending on the squares of the disturbing force have a very great influence on the secular variation in the motion of the lunar perigee; they make its value three times as great as that of the acceleration for the secular inequality in the lunar perigee is

p'

2√1+P

or, when the coefficient is computed, it is

3.00052 m2 (e12 – e2)ndt,

and has a contrary sign to the secular equation in the mean motion.

The motion of the perigee becomes slower from century to century, and is now 8'.2 slower than in the time of Hipparchus.

Motion of the Nodes of the Lunar Orbit.

750. The sidereal motion of the node on the true ecliptic as determined by theory, does not differ from that given by observation by a 350th part; for the expression in article 727 gives the retrograde motion of the node equal to 0.0040105v, and by observation

The secular inequality in the motion of the node depends on the variation in the eccentricity of the terrestrial orbit, and has a contrary sign to the acceleration. Its analytical expression gives

q

2√√1+9

(e12-e2) dv = 0.735452 m3 (e-ē3)dv.

As the motion of the nodes is retrograde, this inequality tends to augment their longitudes posterior to the epoch.

751. It appears from the signs of these three secular inequalities, as well as from observation, that the motion of the perigee and nodes become slower, whilst that of the moon is accelerated; and that their inequalities are always in the ratio of the numbers 0.735452, 3.00052, and 1.

752. The mean longitude of the moon estimated from the first point of Aries is only affected by its own secular inequality; but the mean anomaly estimated from the perigee is affected both by the secular variation of the mean longitude, and by that of the perigee; it is therefore subject to the secular inequality-4.00052 m2ƒ (e12 – e2) dv more than four times that of the mean longitude. From the preceding values it is evident that the secular motion of the moon with regard to the sun, her nodes, and her perigee, are as the numbers 1; 0.265; and 4; nearly.

753. At some future time, these inequalities will produce variations equal to a fortieth part of the circumference in the secular motion of the moon; and in the motion of the perigee, they will amount to

no less than a thirteenth part of the circumference. They will not always increase: depending on the variation of the eccentricity of the terrestrial orbit they are periodic, but they will not run through their periods for millions of years. In process of time, they will alter all those periods which depend on the position of the moon with regard to the sun, to her perigee, and nodes; hence the tropical, synodic, and sidereal revolutions of the moon will differ in different centuries, which renders it vain to attempt to attain correct values of them for any length of time.

Imperfect as the early observations of the moon may be, they serve to confirm the results that have been detailed, which is surprising, when it is considered that the variation of the eccentricity of the earth's orbit is still in some degree uncertain, because the values of the masses of Venus and Mars are not ascertained with precision; and it is worthy of remark, that in process of time the developement of the secular inequalities of the moon will furnish the most accurate data for the determination of the masses of these two planets.

754. The diminution of the eccentricity of the earth's orbit has a greater effect on the moon's motions than on those of the earth. This diminution, which has not altered the equation of the centre of the sun by more than 8'.1 from the time of the most ancient eclipse on record, has produced a variation of 1° 8' in the longitude of the moon, and of 7°.2 in her mean anomaly.

Thus the action of the sun, by transmitting to the moon the inequalities produced by the planets on the earth's orbit, renders this indirect action of the planets on the moon more considerable than their direct action.

755. The mean action of the sun on the moon contains the inclination of the lunar orbit on the plane of the ecliptic; and as the position of the ecliptic is subject to a secular variation, from the action of the planets, it might be expected to produce a secular variation in the inclination of the moon's orbit. This, however, is not the case, for the action of the sun retains the lunar orbit at the same inclination on the orbit of the earth; and thus in the secular motion of the ecliptic, the orbit of the earth carries the orbit of the moon along with it, as it will be demonstrated, the change in the ecliptic affecting only the declination of the moon. No perceptible

change has been observed in the inclination of the lunar orbit since the time of Ptolemy, which confirms the result of theory.

756. Although the inclination of the orbit does not vary from the change in the plane of the ecliptic; yet, as the expressions which determine the inclination and eccentricity of the lunar orbit, the parallax of the moon, and generally the coefficients of all the moon's inequalities, contain the eccentricity of the terrestrial orbit, they are all subject to secular inequalities corresponding to the secular variation of that quantity. Hitherto they have been insensible, but in the course of time will increase to an estimable quantity. Even now, it is necessary to include the effects of this variation in the inequality called the annual equation, when computing ancient eclipses.

757. The three co-ordinates of the moon have been determined in functions of the true longitudes, because the series converge better, but these quantities may be found in functions of the mean longitudes by reversion of series. For if nt, w, 0, and e, represent the mean motion of the moon, the longitudes of her perigee, ascending node and epoch, at the origin of the time, together with their secular equations for any time t, equation (240) becomes

And if Q' be the sum of the coefficients arising from the square of the series S, and depending on the angle v + y'; Q" the sum of the coefficients arising from the cube of S, and depending on the angle Cv, &c. &c., the general term of the new series, which gives the true longitude of the moon in functions of her mean longitude, is

1

− { Q + 3 C. Q′ − } 62. Q′′ – 14 63. Q′′''+&c.} . sin (6(nt + e) + 4)

24

La Place does not give this transformation, but Damoiseau has computed the coefficients for the epoch of January 1st, 1801, and has found that the true longitude of the moon in functions of its mean longitude nte is

This is only the transformation of La Place's equation (240), but Damoiseau carries the approximation much farther.

758. The first term of this series is the mean longitude of the moon, including its secular variation.

The second term

22639".7 sin {ch — w)

is the equation of the centre, which is a maximum when

that is, when the mean anomaly of the moon is either 90° or 270°. Thus, when the moon is in quadrature, the equation of the centre is 6° 17′ 19′′.7. double the eccentricity of the orbit. In syzigies it

is zero.

759. The most remarkable of the periodic inequalities next to the equation of the centre, is the evection