Now C- A is a very small fraction; n', the Sun's mean motion, is very small in comparison of n; and 8 varies very slowly; we shall, therefore, in integrating these equations, take no account of its variation. Thus, eliminating w,, where A and B are constants of integration. This term is independent of the disturbing force; but having for its period 2π p which does not differ much from a day, it must be re jected, since, as we have already remarked in Art. 26, such terms are altogether insensible. Hence we conclude that powers of n' above the second being neglected. We see, then, that the effect of the Sun's attraction is to cause the Earth's axis to travel in a plane perpendicular to that of the disturbing couple which it produces, and with an angular velocity proportional to its moment. This velocity, though so small as to be insensible to the most delicate observations, yet leads to values of and which can by no means be neglected. 34. We proceed to determine the motion of the axes in space. Substituting in the equations of Art. 5, we have Now from the spherical triangle formed by the intersection of the equator, the ecliptic and a declination circle through the Sun, it is easily seen that 35. It appears from these formulæ that the motion of the Earth's axis is of two kinds; partly secular, partly periodic. It is convenient to consider these separately. The former affects the equinoxes alone, and, being proportional to the time, indicates uniform motion; this motion is one of regression, since has been measured in a direction contrary to that of the apparent motion of the Sun. Considered with reference to the apparent diurnal motion of the stars, the effect is to place the equinoxes in advance of the position they would occupy if fixed: hence it obtained the name of the Solar Precession of the Equinoxes. The latter furnishes corrections both on and 0, which go through all their changes in half a year: it is called the Solar Nutation of the Earth's axis; the correction on forming the Nutation in Longitude, that on the Nutation in Latitude. 36. We will now examine the effect produced by the Moon's action on the motion of the Earth's axis. Since the investigations which have been given of the effect of the Sun's disturbing force contain nothing to restrict their generality except the special assumptions made with regard to small quantities, we shall first consider what modifications are required in order that they may be applied in the case of the Moon. Let y', ', n", I' denote relatively to the plane of the Moon's orbit the same quantities which relatively to the ecliptic have been denoted by y, 0, n', I; also let m, m” be the masses of the Earth and Moon: then, as in the case of the Sun, we have approximately, but we cannot in this case neglect m, as it is in fact much larger than m': retaining it, we have if mλm". Also n" is small, though not so small as n' in comparison of n. Hence to determine the motion of the Earth's axis with reference to the plane of the Moon's orbit, we have The periodical terms in these equations go through all their values in half a month, and are so small that they are usually neglected. Thus we may consider the inclination of the plane of the Earth's orbit to that of the Moon as constant and equal to its mean value, and the precession as uniform and given by These results, deduced from the corresponding formulæ in the case of the Sun, suppose the plane of the Moon's orbit fixed. Now the line of nodes moves too rapidly to allow of this hypothesis, but the only effect of considering its motion dy would be to add to a term depending upon its velocity dt and not upon the disturbing force of the Moon upon the Earth. Since our object is to trace the effects of this force only, such terms must be omitted. 37. It now remains to determine the motion of the Earth's axis with reference to the ecliptic. Let be the obliquity of the ecliptic, the longitude of the equinox, a the longitude of the node of the Moon's orbit measured from the same origin as , i the inclination of the Moon's orbit to the ecliptic, I' its inclination to the equator. Then supposing measured from the node of the Moon's orbit on the ecliptic, we have by Spherical Trigonometry (as in Art. 7) (3). cos I' = cos cos i + sin @ sin i cos (ỵ — a) ..... We must now differentiate these equations in order to in terms of d. In so doing we shall dt |