2 2 (1+m) 2-2m -c 2.(1-m) 2-2m+c and the value off, which is required in the second formula of the equation (see p. 138.) is immediately had by multiplying the preceding value into h2 = a. If we use this latter form of the differential equation, then, from the assumed value of u, we must find (see p. 138.), and, since it is intended to neglect all terms that involve e2 and the higher powers of e, we shall have If we now collect and arrange the terms of the differential If we substitute for the coefficients of the cosines of the arcs cv, 20-2mv+ cv, the letters A, B, C, we shall have, by integrating according to the process of P. 100. + 3 Ke B C 2a ̧.(c2—1). (2—2 m)2-1 (2-2m-c)2-1 (2-2m+c)2 — 1 i). If we compare this with the former instance, we shall find that the introduction of the condition of a small eccentricity (so small that all terms involving its square, &c. are neglected) increases the value of u in the first approximation, by two new terms, corresponding to equations of which the arguments (see Astron. p. 324.) are 2 v 2 m v cv, and 2 v 2 mv + co. The coefficients A, B, C, involve the disturbing force, and therefore, according to the hypothesis, are very small if they were not, the value of u, which results from the approximate integration of the differential equation, could not agree with the assumed value, namely + cos. cv. It can never agree exactly with it. But if we suppose them nearly equal, then, by the comparison of like terms, we shall be able to determine the value of c, which hitherto has been supposed an arbitrary quantity. And if we determine c, we shall know 1-c, which denotes the pro a a T gression of the apogee, and thence obtain (if we concede the calculation to have been rightly conducted) a test of the truth of Newton's Law of Gravitation. According to this method, Clairaut (see Mem. Acad. 1745.) Xfirst proceeded and reasoned. The assumed equation corresponding to the assumption of the body's place in the periphery of a moveable ellipse, was nearly equal to the resulting value of u; that which represented, or nearly so, the inverse radius vector of the real orbit. Thence the following equations, arising from the comparison of terms, would be nearly true: Now, a (see p. 133.) is what the Moon's mean distance would have been, had there been no disturbing force. But, since that force always acts, a is no real quantity, such as can be found by observation. It must be determined by calculation: and the equation of p. 133. is sufficient for that purpose; thence pute the Moon's mean motion, we have (see p. 95.) Now, (see p. 131.) the constant part of u is to be represented (when the plane of the Moon's orbit is supposed to be not inclined to that of the ecliptic by = (1 - e2): and since (see p. 131.) in the same case, h = √[a, (1 − e2)], we have the constant part of 1 hu2 equal , nearly and consequently, the part of the ex- pansion of dt not involving cosines or sines, and therefore not a2 periodic, would be ..do: Now (see Astronomy, p. 308, and p. 132*.). Hence, the progression of the apogee, whilst the Moon describes the angle v, is (1-c) v, which is equal to .0042.v, nearly, and consequently the progression in a whole revolution = .0042 × 360° = 1° 30′ 43′′, nearly, a quantity about half of that (3o 2′ 22′′), which is determined by the most accurate observations. This is a brief notice and description of that notorious error, which, on its first appearance, caused (if we may so express ourselves) so great a sensation in the mathematical world. In one of the most remarkable of the heavenly phenomena, the progressions of the aphelia of the planetary orbits, theory and calculation were erroneous to the amount of half the real quantity. So erroneous a defalcation seemed to portend to Newton's System, that fate which, not long before, Descartes's had experienced. *In p. 132. for .01748013, read .0748013. But it may be said, that the preceding solution, with regard to the Moon, must be inexact. For, in the first place, no account is made of those terms which involve the square and higher powers of e: secondly, the plane of the Moon's orbit is supposed, (contrary to the fact) to be coincident with the plane of the equator: and thirdly, the solar orbit is supposed to be circular, whereas its eccentricity (e) is equal (see Astron. Chap. XVIII.) .016814: these are obvious causes of incorrectness, which, by the mere labour of calculation, may be removed: and amongst the results of that calculation it would appear, whether their removal would cause the error of the computed quantity of the apogee to disappear also. By simply following then the natural course of successive corrections, we shall arrive at that point. It will be seen that the erroneous determination of the progression does not depend on any of the causes just enumerated. Its cause will be detected in that method of approximation, to which the imperfection of analytical science obliges us to have recourse. By the expression, natural course of successive corrections, is meant, the addition of small quantities to terms already computed, as corrections due to those terms on account of conditions before omitted and now supplied; or, of conditions previously simplified and now more nearly restored to their true state. For instance, the solar orbit being nearly elliptical, and having been (see p. 128. 137.) supposed circular, its eccentricity is a condition omitted. To correct the error arising from this omission, we must compute several small terms for the purpose of augmenting the component parts of II (see pp. 128. 137.). Again, the plane of the Moon's orbit, having been (see the same pages) supposed coincident with the ecliptic, its inclination is a condition omitted: and the restoring of this condition will increase II by several small terms. The rejection, however, of all terms that involve the square and higher powers of the eccentricity (e) is to suppose one condition of the problem more simple than it ought to be assumed: for, in the Lunar Theory, e = .0548729; and even Clairaut, Dalembert, and Thomas Simpson who first treated of it, have extended their approximations beyond the first powers of the eccentricity. In the next Chapter the approximation will be extended, so |