CHAPTER VII. APPENDIX. In this chapter will be found collected a few propositions intimately connected with the results or the processes of the Lunar Theory as explained in the previous pages. Reference has been made to some of them in the course of the work, and the interest and importance of the others are sufficient to justify their introduction here. 91. The moon is retained in her orbit by the force of gravity, that is, by the same force which acts on bodies at the surface of the earth. The proof of this is merely a numerical verification; the data required from observation are, the space fallen through from rest in 1" by bodies at the earth's surface the radius of the earth ...... the periodic time of the moon =16.1 feet, =4000 miles, = = 271 days, the distance of the moon from the earth's centre=60×4000 miles. The force of the earth's attraction 1 (dist.)2* Therefore, the space fallen through in 1" at distance of moon by a body moving from rest under the earth's action = 16.1 feet ='00447 feet. G But the moon in one second describes an angle during which the approach to the earth 2π 273.24.600, Therefore, the space through which the moon is deflected in one second from her straight path, is just the quantity through which she would fall towards the earth, supposing her to be subject to the earth's attraction, and we may, therefore, conclude that she is retained in her orbit by the force of gravity. When first Newton, in 1666, attempted to verify this result, he found a difference between the two values equal to one-sixth of the less: the reason of his failure was the incorrect measures of the earth, which he made use of in his computation; and it was not till about 16 years later that he was led to the true result, by using the more correct value of the earth's radius obtained by Picart. Principia, lib. III., prop. 4. 92. The moon's orbit is everywhere concave to the sun. Let S, E, and M be the centres of the sun, earth, and moon. We must bring the sun to rest by applying to each body forces equal and opposite S M E to those which act on the sun; but these are so small that we may neglect them and consider the moon as moving round the sun fixed, and disturbed by the earth alone. This last must be resolved into two, one in MS, the other perpendicular to it. Therefore, the whole central force on the moon in MS and the proposition will be proved if we shew that this force but the least value of the central force corresponds to m' cos M-1, and is then E SM EM It is, therefore, always positive, or the path always concave to the sun. At new moon the force with which the moon tends to the sun is, therefore, greater than that with which she tends to the earth: the earth being itself in motion in the same direction, and, at that instant, with greater velocity, will easily explain how, notwithstanding this, the moon still revolves about it. Central and Tangential Disturbing Forces. 93. We have hitherto considered the effects of the central and tangential disturbing forces in combination; but it will be interesting to determine to which of them the several inequalities principally owe their existence. (1) To determine the effect of the central disturbing force. u =α [ 1— ‡ k2 — { m2 + § m2e cos(c0—a)—§m2 cos{(2—2m)0—2ẞ} (1 − 3 k2 — 1m2 + e cos(c)—a)+1m2 cos {(2—2m)0—2ẞ} +me cos{(2−2m − c) 0 − 2B + a} — 4k2 cos2 (g0—y)—§m2e' cos(m0+B−5). If we compare this with the value of u found Art. (48), we see that the elliptic inequality, the reduction, and the annual equation are due to the central or radial force, as also one half of the variation and about a third of the evection. It would perhaps be proper to separate the absolute central force from the central disturbing force; the terms due to the latter are those which contain m; therefore, the elliptic inequality and the reduction are the effects of the former, except that in the elliptic inequality the introduction of C, the motion of the apse, is due to the disturbing force. or (2) To determine the effect of the tangential disturbing force. Let the central disturbing force be zero; whence u= a { h2u3 do 2 de (1 — §m2 cos {(2 – 2m) 0 — 2ß} { +21m2e cos {(2- 2m - c) 0-2ẞ+a}; (1+e cos(c0 − a) + 1m2 cos {(2 − 2m) 0 — 2ß} +2¿me cos {(2-2m − c) 0 − 2B + a}. We have here the remaining half of the variation and rather more than two-thirds of the evection as the effects of the tangential disturbance. Also c=1, or, to the second order, the tangential force has no effect on the motion of the apse. The inequalities in the longitude could be easily obtained from the relation but they would lead to the very same conclusions as the discussion of the values of u. |