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

v - (nt + €): =

or to abridge

— {C。.e. sin (cv — w) + C,. e2 sin 2 (cv — ☎) + C. e. sin 3(cv) + &c. }

v (nt + e) = S.

The general term of the series is

Q. sin (6v+4).

And if Q' be the sum of the coefficients arising from the square of the series S, and depending on the angle v + '; Q' the sum of the coefficients arising from the cube of S, and depending on the angle 6v + 4, &c. &c., the general term of the new series, which gives the true longitude of the moon in functions of her mean longitude, is

- { Q + £ 6. Q' − } 62. Q′′ — — 63. Q''+ &c.} . sin (C(nt + e) + ¥) 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.

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

which is at its maximum and 4589". 61, when 21-2mλ = cλ+w is either 90° or 270°, and it is zero when that angle is either 0° or 180°. Its period is found by computing the value of its argument in a given time, and then finding by proportion the time required to describe 360°, or a whole circumference. The synodic motion of the moon in 100 Julian years is

and

445267°.1167992 = X

ηλ

890534°.2335984 = 2 {λ – mλ}

years.

is double the distance of the sun from the moon in 100 Julian If 477198°.839799 the anomalistic motion of the moon in the same period be subtracted, the difference 413335°.3937994 will be the angle 21 2ηλ cλ+, or the argument of the evection in 100

Julian

years: whence

413335°.3937994: 360° :: 365.25: 314.811939 =

the period of the evection. If t be any time elapsed from a given period, as for example, when the evection is zero, the evection may be represented for a short time by

This inequality is a variation in the equation of the centre, depending on the position of the apsides of the lunar orbit. When the

moon be in conjunction at m, the sun draws her from the earth; and if she be in opposition in m', the sun draws the earth from her; in both cases increasing the moon's distance from the earth, and thereby the eccentricity or equation of the centre. When the moon is in any other point of her orbit, the action of the sun may be resolved into two, one in the direction of the tangent, and the other according to the radius vector. The latter increases the moon's gravitation to the earth, and is at its maximum when the moon is in quadratures; as it tends to diminish the distance QE, it makes the

ellipse still more eccentric, which increases the equation of the centre. This increase is the evection. Again, if the line of apsides be at

orbit, thereby making it approach the circular form, which diminishes the eccentricity. If the moon be in quadratures, the increase in the moon's gravitation diminishes her distance from the earth, which also diminishes the eccentricity, and consequently the equation of the centre. This diminution is the evection. Were the changes in the evection always the same, it would depend on the angular distances of the sun and moon, but its true value varies with the distance of the moon from the perigee of her orbit. The evection was discovered by Ptolemy, in the first century after Christ, but Newton showed on what it depends.

760. The variation is an inequality in the moon's longitude, which increases her velocity before conjunction, and retards her velocity

direction of mE, which produces the evection, and the other in the direction of mT, tangent to the lunar orbit. The latter produces the variation which is expressed by

This inequality depends on the angular distance of the sun from the moon, and as she runs through her period whilst that distance increases 90°, it must be proportional to the sine of twice the angular distance. Its maximum happens in the octants when λ – mλ = 45°, it is zero when the angular distance of the moon from the sun is either zero, or when the moon is in quadratures. Thus the vari

ation vanishes in syzigies and quadratures, and is a maximum in the octants.

The angular distance of the moon from the sun depends on its

Thus the period of the variation is equal to half the moon's synodic revolution. The variation was discovered by Tycho Brahe, and was first determined by Newton.

761. The annual equation

is another remarkable periodic inequality in the moon's longitude. The action of the sun which produces this inequality is similar to that which causes the acceleration of the moon's mean motion. The annual equation is occasioned by a variation in the sun's distance from the earth, it consequently arises from the eccentricity of the terrestrial orbit. When the sun is in perigee his action is greatest, and he dilates the lunar orbit, so that the angular motion of the moon is diminished; but as the sun approaches the apogee the orbit contracts, and the moon's angular motion is accelerated. This change in the moon's angular velocity is the annual equation. It is a periodic inequality similar to the equation of the centre in the sun's orbit, which retards the motion of the moon when that of the sun increases, and accelerates the motion of the moon when the motion of the sun diminishes, so that the two inequalities have contrary signs.

The period of the annual equation is an anomalistic year. It was discovered by Tycho Brahe by computing the places of the moon for various seasons of the year, and comparing them with observation. He found the observed motion to be slower than the mean motion in the six months employed by the sun in going from perigee to apogee, and the contrary in the other six months. It is evident that as the action of the sun on the moon varies with his distance, and