which is at its maximum and ±4589", 61, when 2λ - 2mλ - cλ+@ 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

445267°,1167992 = λ

ηλ

and

890534°.2335984 = 2 {λ = mλ}

is double the distance of the sun from the moon in 100 Julian years. If 477198°.839799 the anomalistic motion of the moon in the same period be subtracted, the difference 413335°.3937994 will be the angle 22mλ c+, or the argument of the evection in 100

Julian years: whence

413335°.3937994: 360° :: 365.25 : 31d.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 = 45o, 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.

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

therefore depends on the eccentricity of the earth's orbit, whatever affects the eccentricity must influence all the motions of the moon.

762. The variation has been ascribed to the effect of that part of the sun's force that acts in the direction of the tangent; and the evection to the effect of the part which acts in the direction of the radius vector, and alters the ratio of the perigean and apogean gravities of the moon from that of the inverse squares of the distance. The annual equation does not arise from the direct effect of either, but from an alteration in the mean effect of the sun's disturbing force in the direction of the radius vector which lessens the gravity of the moon to the earth.

763. Although the causes of the lesser inequalities are not so easily traced as those of the four that have been analysed, yet some idea of the sources from whence they arise may be formed by considering that when the moon is in her nodes, she is in the plane of the ecliptic, and the action of the sun being in that plane is resolved into two forces only; one in the direction of the moon's radius vector, and the other in that of the tangent to her orbit. When the moon is in any other part of her orbit, she is either above or below the plane of the ecliptic, and the line joining the sun and moon, which is the direction of the sun's disturbing force, being out of that plane, the sun's forces resolved into three component forces; one in the direction of the moon's radius vector, another in the tangent to her orbit, and the third perpendicular to the plane of her orbit, which affects her latitude. If then the absolute action of the sun be the same in these two positions of the moon, the component forces in the radius vector and tangent must be less than when the moon is in her nodes by the whole action in latitude. Hence any inequality like the evection, whose argument does not depend on the place of the nodes, will be different in these two positions of the moon, and will require a correction, the argument of which should depend on the position of the nodes. This circumstance introduces the inequality

54.83. sin (2gλ 2λ + 2mλ - 20)

in the moon's longitude. The same cause introduces other inequalities in the moon's longitude, which are the corrections of the variation and annual equation. But the annual equation requires a cor

rection from another cause which will introduce other terms in the perturbations of the moon in longitude; for since it arises from a change in the mean effect of the sun's disturbing force, which diminishes the moon's gravity, its coefficient is computed for a certain value of the moon's gravity, consequently for a given distance of the moon from the earth; hence, when she has a different distance, the annual equation must be corrected to suit that distance.

764. In general, the numerical coefficients of the principal inequalities are computed for particular values of the sun's disturbing force, and of the moon's gravitation; as these are perpetually changing, new inequalities are introduced, which are corrections to the inequalities computed in the first hypothesis. Thus the perturbations are a series of corrections. How far that system is to be carried, depends on the perfection of astronomical instruments, since it is needless to compute quantities that fall within the limits of the errors of observation.

765. When La Place had determined all the inequalities in the moon's longitude of any magnitude arising from every source of disturbance, he was surprised to find that the mean longitude computed from the tables in Lalande's astronomy for different epochs did not correspond with the mean longitudes computed for the same epochs from the tables of Lahere and Bradley, the difference being as follows:

Whence it was to be presumed that some inequality of a very long period affected the moon's mean motion, which induced him to revise the whole theory of the moon. At last he found that the series which determines the mean longitude contains the term

sin (3v-3mv+3cmv-2gv-cv+20+w-3w'}

= yee's. e.