171, 173 Mr. CAYLEY: On the Secular Acceleration of the Moon's Mean Motion. hand, and advanced treatises on the other hand; to be attractive to the general reader, useful to the amateur, and "handy" also as an occasional book of reference to the professional astronomer. Pains have been taken to incorporate the most recent discoveries in all branches of the science, and the Work appears to be very complete in its range of subjects; and in general to be fully 173, 174 entitled to take up the position claimed for it by the author. The chronological tables of Astronomers and of Discoveries in Book 9, and the tables of Eclipses, Comets, Star Catalogues, &c., contained in the Appendices, will probably be found useful for reference. VOL. XXII. Supplemental Notice. No. 5 bis. On the Secular Acceleration of the Moon's Mean Motion.* The present Memoir exhibits a new method of taking account, in the Lunar Theory, of the Variation of the Excentricity of the Sun's Orbit. The approximation is carried to the same extent as in Prof. Adams' Memoir "On the Secular Variation of the Moon's Mean Motion" (Phil. Trans., vol. cxliii. (1853), pp. 397-406); and I obtain results agreeing precisely with his, viz., besides his new periodic terms in the longitude and radius vector, I obtain in the longitude the secular term (− 2 m2 + 3771 m2) S (e2 – E'2 ) ndt, 64 which is, in fact, as will be shown, included implicitly in the results given in Professor Adams' Memoir. In quoting the foregoing results, I have written e2-E2 in the place of (e' +f't)2 — e12 = 2 e' f't, which in the notation of the present Memoir it should have been; and I purposely refrain from here explaining the precise signification of the symbols: this is carefully done in the sequel. The method appears to me a very simple one in principle; and it possesses the advantage that it is not incorporated step by step with a lunar theory in which the excentricity of the Sun's orbit is treated as constant; but it is added on to such a lunar theory, giving in the Moon's coordinates the supplementary terms which arise from the variation of the solar excentricity, and thus serving as a verification of any process employed for taking account of such variation. I have given the details of the work in a series of Annexes, 1 to 23: this appears to me the best course for presenting the investigation in a readable form. I. The inclination and excentricity of the Moon's orbit, and, à fortiori, the variation of the position of the Ecliptic, and the Sun's latitude, are neglected; and the longitudes are measured from a fixed point in the Ecliptic. I write n, the actual mean motion of the Moon at a given epoch; *This Memoir, an abstract of which appeared in the December number, p. 32, has been ordered by the Council to be printed in extenso.-ED. viz., it is assumed that the mean longitude at the time tis $+nt+n2t2+ &c. where i, n, n,, &c. are absolute constants; and, moreover, a, the calculated mean distance of the Moon; that is, n2 a3 is the sum of the masses of the Earth and Moon; a is therefore an absolute constant; and, in like manner, n', the actual mean motion of the Sun at the same epoch, a', the calculated mean distance of the Sun; 2 2 that is, if it were necessary to pay attention to the secular variation of the mean motion of the Sun, the assumption would be that the mean longitude was ' + n' t + n'2 t2 + &c., i', n', n', &c. being absolute constants, and n'2 a'3 the sum of the masses of the Sun and Earth; a' would thus also be an absolute constant. But for the purpose of the present investigation the secular variation of the mean longitude of the Sun is neglected, or it is assumed that the mean longitude of the Sun is '+n't ', n' being absolute constants; and that n' a'3 is the sum of the masses of the Sun and Earth, a' being thus also an absolute constant. be The Sun is considered as moving in an elliptic orbit, the excentricity whereof is e'+de' or e'+ft, e and f' being abso lute constants; the longitude of the Sun's perigee may taken to be +(1-c') n't; so that the mean anomaly g'is =n't + - ['-(1-c') n't] = c'n' t + '-'; c', ', being absolute constants; but c' is in fact treated as being =1. Hence, if r', v' are the radius vector and longitude of the Sun, we have 175, 176 Mr. CAYLEY: on the Secular Acceleration of the Moon's Mean Motion. then, taking the usual approximate expression of the Disturbing | The second of these equations gives Function, the equations of motion are dd v dt dv dt 176, 177 2 n2 + = m2 n2 P, dov m2 n2 Q, + co 2 3 a-(-in-2), Q = In these equations C is determined, as above, by the condition may contain no constant term; the values of g', v', de, d' are of course given by the theory of elliptic motion, and those of e, v are given by the ordinary lunar theory, in which the excentricity of the solar orbit is treated as a constant; and then, de, v being obtained by integrating the equations, the radius vector and longitude of the Moon are a (e+g) and v+dv respectively. dP ୧ de бет d P dP |