dx di omit the terms involving and since these quantities dt dt' depend not upon the disturbing force, but upon the motion of the Moon's orbit. Thus = (cos e cosi+sin sin i cos ) (cos i sin i cot cos ) 1 - sin'i – cot 20 sin 2i cos — — sin'i cos 20), 2 1 Letv be the mean retrograde velocity of the Moon's nodes, the value of at the epoch, then we may write Q = vt+; also for we may write its mean value I; and since i is a very small angle we may neglect sin3¿ when multiplied by a periodical term. Thus 38. These equations determine the motion of the Earth's axis due to the attraction of the Moon. They are similar to the equations expressing the Sun's action, and the remarks made in Art. 35 might, mutatis mutandis, be repeated here. The Lunar Precession is 3n" 112 C-A 1 t. 2n (1+x) cos I (cos'i - sin2i) e. 2 If we add to this the Solar Precession (Art. 32), we find for the whole permanent effect of the Sun and Moon upon the equinoxes This is called the Luni-solar Precession, to distinguish it from the Precession due to the secular motion of the ecliptic in consequence of the attractions of the Planets, and which is called Planetary Precession. Adding together the effects of Lunar and Solar Nutation, we find for the whole Nutation in Longitude where and denote respectively the mean longitude of the ascending node of the Moon's orbit and of the Sun. Similarly, the whole Nutation in Latitude is In each of these expressions the first term, due to the action of the Moon, is the most important, since n" is larger than both n' and v. 39. If we consider only the motion of the Earth's axis due to Precession, it appears from the preceding formulæ that it maintains a constant inclination to the pole of the ecliptic, and describes a right circular cone about it with uniform velocity. An axis possessing this motion exactly we shall term the mean axis of the Earth. The motion of the axis due to Lunar Nutation can now be exhibited as follows: then, taking unity as the radius of the celestial sphere, x and y are the small linear spaces traversed by the intersection of the Earth's axis with the circumference in two directions at right angles. Eliminating from these equations, we have Hence, in consequence of Lunar Nutation, the extremity of the axis may be considered to move in an ellipse whose semi-axes are in the ratio of cos 21 to cos I. The centre of this ellipse is at the point of intersection of this mean axis with the celestial sphere, and its plane a tangent to the sphere at that point. By supposing the mean axis to describe the cone uniformly, while the true axis describes this ellipse about it, the real motion will be represented. This conception is due to Bradley, who arrived at it by observation. 40. The annual value of the Luni-solar precession (see Art. 38) = 112 n n 3n C-4 cos I (cos'i- sin'i) + n'. 2n' (n2 (1 + x) n 2 Observation gives about 50".1 as the numerical value of this expression. Hence, by substituting the known values of n, n', v, I, i, we have a relation between C-A and λ, by means of which either may be determined when the other - is known. 41. We have taken no account in our calculations of the Precession caused by the attractions of the planets on the Earth, since it is too trifling to be appreciable. THE END. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS. |