t 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 force s 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 2x + 2mλ 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 a sin (3v-3mv+3cmv-2gv-cv+20+w-3w'} .. {3-3m+3c'm-2g-c} a a' {3-3m+3c'm-2g-c} depending on the disturbing action of the sun, that appeared to be the cause of these errors. The coefficient of this inequality is so small that its effect only becomes sensible in consequence of the divisor acquired from the double integration. Its maximum, deduced from the observations of more than a century, is 15".4. Its argument is twice the longitude of the ascending node of the lunar orbit, plus the longitude of the perigee, minus three times the longitude of the sun's perigee, whence its period may be found to be about 184 years. The discovery of this inequality made it necessary to correct the whole lunar tables. 766. By reversion of series the moon's latitude in functions of her mean motion is found to be 767. The only inequality in the moon's latitude that was discovered by observation is Tycho Brahe observed, in comparing the greatest latitude of the moon in different positions with regard to her nodes, that it was not always the same, but oscillated about its mean value of 5° 9', and as the greatest latitude is the measure of the inclination of the orbit, it was evident that the inclination varied periodically. Its period is a semi-revolution of the sun with regard to the moon's nodes. 768. By reversion of series it will be found that the lunar parallax at the equator in terms of the mean motions is 769. The planets are at so great a distance from the sun, and from one another, that their form has no perceptible effect on their mutual motions; and, considered as spheres, their action is the same as if their mass were united in their centre of gravity: but the satellites are so near their respective planets that the ellipticity of the latter has a considerable influence on the motions of the former. This is particularly evident in the moon, whose motions are troubled by the spheroidal form of the earth. 474 CHAPTER III. INEQUALITIES FROM THE FORM OF THE EARTH. 770. THE attraction of the disturbing matter is equal to the sum of all the molecules in the excess of the terrestrial spheroid above a sphere whose radius is half the axis of rotation, each molecule being divided by its distance from the moon; and the finite values of this action, after it has been resolved in the direction of the three coordinates of the moon, are the perturbations in longitude, latitude, and distance, caused by the non-sphericity of the earth. In the determination of these inequalities, therefore, results must be anticipated that can only be obtained from the theory of the attraction of spheroids. By that theory it is found that if p be the ellipticity of the earth, R its mean radius, the ratio of the centrifugal force at the equator to gravity, and the sine of the moon's declination, the attraction of the redundant matter at the terrestrial equator is ע (bø − p) — (v − }) the sum of the masses of the earth and moon being equal to unity. Hence the quantity R which expresses the disturbing forces of the moon in equation (208) must be augmented by the preceding expression. 771. By spherical trigonometry v, the sine of the moon's declination in functions of her latitude and longitude, is v = sin w√√1 - s2 sin fv + s cos w, in which w is the obliquity of the ecliptic, s the tangent of the moon's latitude, and fv her true longitude, estimated from the equinox of spring. The part of the disturbing force R that depends on the action of the sun, has the form Qre when the terms depending on the solar parallax are rejected. Hence |