2fd R + rd R by a series of cosines such as A cos. qnt: for then we should be able immediately to integrate the equation by the method of pp. 97, &c. The value of R (see p. 66.) on making s = 0, and p=r is thus expressed, m'r cos. (v' — v) — m' √[rrr.cos. (v′ — v)+r2] Now, supposing e, é to denote the epochs at which the bodies are in the perihelia of their orbits, we have (see p. 32.) let us begin with a simple case, and suppose the orbits to be so nearly circular that the terms involving e, é, may be neglected, then which may always, whatever are the relative magnitudes of a, a', be expanded into a series of terms such as A) + A(1). cos. (n't — nt+e' — e)+A2). cos. 2 (n' t − n t + e − e) + &c. 2s.(2s+1) Now (1 - ax)2 = 1 + 2s.ax + 1.2 *The coefficients of this product may, and rather more regularly, be thus expressed: (see Mec. Anal. 2de Partie, Sect. VII.). M,N, 000 (M + N. cos. w + 0. cos. 2 w + &c.) &c. representing the coefficients of 1, x + + ,&c. in the preceding product (p. 260.) H On account of the importance of the formula*, we have deduced its most general expression in which s may designate any number. In the instance that gave rise to the investigation, and if this fraction be substituted, we shall have the same series as Laplace has given in p. 272. of his Mec. Celeste. It appears then, if we regard merely the analytical expression and not the convergency of the series as dependent on the value of a, that R can always be expressed by a series such as • The development of › during his researches on the perturbation of the planets, deduced by Lagrange and by the aid of impossible quantities. In this method he has been followed by Laplace and other authors (see Mem Berlin, 1781. p. 257. Mec. Anal. p. 142. Acad. des Sciences, 1785, p. 68. Mec. Celeste, tom. I. pp. 271, 272. Vince's Astron. vol. II. p. 191.) The demonstration in the text is obtained with an expedition quite equal to that which the use of imaginary symbols is able to confer. or of (r"-2rr' cos. w +r3) was, Ao + A11) cos. (n' t − n t + é' − e) + A . cos. 2 (n't −nt+é' − e)+&c. and consequently, 2fdR + r dR , or, in this case, 2ƒdR+a. dR da can always be expressed by means of terms involving the cosines of arcs the multiples of n't-nt+-e; in which case (see pp. 259, &c.) the equation of p. 258. can be integrated by the method of pp. 97, &c. We must now consider whether the method is an easy practical one: and that must depend, as it is plain, on the convergency of the series that expresses the value of R. If, or be a small fraction, the series will quickly converge; and a few of its terms will, with sufficient exactness, represent its sum. in 400 If the fraction, should be as small as it is the Lunar Theory, the series would converge so quickly that it would be sufficiently exact to retain its two first terms. There is no reason why we should not use the series when it has been once invented but otherwise it would have been an useless refinement to have invented it for so simple a case. The binomial theorem see p. 59. immediately affords the proper result. In the case last alluded to, a', which expresses the radius of the orbit of the disturbing planet, was the radius of the Sun's orbit: and,, being very small, the series converged very rapidly but we may still obtain a converging series, if a', continuing to represent the radius of the orbit of the disturbing body, should be less than a: in other words, if the orbit of the disturbing body should be interior to that of the disturbed: for, since y√(a-2 a a' cos. w + a2), and consequently, the same form of development would belong to each case and, if the first converged, a' being greater than a, the latter would converge, and ought to be used, a' being less than a and, as it is plain, the convergency would be the same, if As far then as depends on the facility of computing R (and consequently of computing P and T, see pp. 66.) by the convergency of the series expressing it, the problem of Jupiter's perturbations by the action of Mars is equally easy with that of the perturbation of Mars by Jupiter. Hence too, that application of the series, which in the case of the Earth, Moon and Sun, we stated (p. 262.) to be most simple, the Sun being the disturbing body, will be equally so, when the Moon becomes the disturbing body: and, since this is a case the most simple of any in the Theory of Perturbations, whether the orbit of the third body be without or within that of the disturbed, we will apply to it the formulæ of this Chapter. The results may then be compared with those which have already, on different principles, been previously (see Chap. VI.) obtained. If we make 4 s= 1, 2s will = 1 1 2 (see p. 260.) will = - a and if we reject, which we may do in this case, since |