BEING THE ANNUAL HALF-VOLUME OF THE MEMOIRS AND PROCEEDINGS OF THE ROYAL ASTRONOMICAL SOCIETY. LONDON: PRINTED BY GEORGE BARCLAY, CASTLE STREET, LEICESTER SQUARE. 1853. ROYAL ASTRONOMICAL SOCIETY. VOL. XIII. Nov. 12, 1852. No. I. J. C. ADAMS, Esq., President, in the Chair. Rev. W. C. Williams, North London Collegiate School; and Richard Lee, Esq., Wheatley, Oxon, were balloted for and duly elected Fellows of the Society. DISCOVERY OF THALIA. On the evening of December 15th, at 6h 30m mean time, Mr. Hind detected, at Mr. Bishop's Observatory, another small planet, belonging to the group between Mars and Jupiter. Like Calliope, this planet has been brought to light by means of the ecliptical star-maps, which are now in course of publication from the above Observatory. The following are the early observations of the planet, which has been named Thalia by Mr. Bishop. It shines as a star of between the tenth and eleventh magnitudes, with a pale watery light. By M. George Rümker, from the Observations at Paris, Nov. 18; and at Hamburg, Nov. 28 and Dec. 3. An ephemeris, from Dec. 6 to Jan. 3, was sent at the same time, which has been communicated to the principal observers. MASSILIA. Elements. By M. G. Rümker. Computed from the Hamburg Observations, Sept. 30, Oct. 26, and Nov. 19. An ephemeris based on these elements was sent by M. G. Rümker at the same time this has been forwarded to our principal observers. The observations of the solar eclipse of July 28, 1851, which form Part I. of Vol. XXI. of the Memoirs, are now published. There are six plates and 120 pages of letterpress. The price to Fellows is 5s., and 10s. to the public. There is a separate titlepage for the purchasers of the part who may not want the volume. On a New Method of Computing the Perturbatior: of Planets. By M. Encke, Director of the Royal Observatory of Berlin. Translated into English and Illustrated with Notes by G. B. Airy, Esq., Astronomer Royal. (Appendix to the Nautical Almanac for 1856.) The main object of the author is to supply an easy and accurate method of computing the perturbations of the various small planets revolving between Mars and Jupiter, the number of which is constantly increasing from fresh discoveries. The method, however, is applicable to any of the other bodies of the planetary system. Let t denote the time from which it is proposed to compute the action of the disturbing body. By a slight transformation the differential equations of the motion of the disturbed planet are reduced to the following form : x°, yo, 2o, being the elliptical co-ordinates of the disturbed planet, and its radius vector, as deducible from the elements of the orbit in which it was revolving at the instant t; X, Y, Z, the resolved values of the disturbing force in the directions of corresponding rectangular axes; %, n,, the perturbations of x, y, z, counted from t; dr, the perturbation of the radius vector; and k a constant quantity. These equations are integrated by mechanical quadratures, the successive numerical values of the quantity under the sign of the double integral in each equation being corrected by a peculiar process, founded on the theory of finite differences. dt dtdt' The first step is to divide the period for which the perturbations are to be computed, into a number of sufficiently small intervals of time, each equal to 7, and to calculate the values of X, Y, Z, for the times t―, t + 1⁄2 ï, t + 1⁄2 ï, &c., using for this purpose the elliptical co-ordinates x, y, z, of the disturbed planet. As , n, S, dě dn dr are assumed severally equal to zero at the commencement of the period for which the perturbations are to be computed, the first three values of the quantities under the sign of the double d #dn d* integral (or ), are readily found with sufficient accuracy d t2 d t d t2 by an approximative process, and from them the corresponding values of 1⁄2, 1, 3, are deduced. For all the subsequent values of %, n, Z, the computation is uniform. The mode of procedure is this:-From the three last values of an approximate value of the same quantity for the following argument is readily deduced, and a provisional value of is then found by means of the formula of inted2% d d2% gration="f+"f being a numerical quantity computed with the utmost facility by the aid of results already obtained. The corresponding values of and being found by a similar process, the three resulting numbers are employed in computing the second part of the expression under the integral sign in each of the fundamental equations. Finally, the numbers thus found, when applied to X, Y, Z, give the corrected values of d2 % d2 n d2 which in their turn lead by double summation to the dt2' d t d t2' true values of %, n, C. By a similar process the values of E, n,, for the next argument are ascertained, and so on in succession, the operation being rapidly carried on throughout the entire period for which the perturbations are to be computed. In order to test the practical utility of this method, M. Encke compared its results with those obtained by the method of the variation of arbitrary constants. For this purpose the perturbations of Vesta by Jupiter from 1854, May 24, to 1855, October 7, were subjected to a double computation. The following table exhibits the values of the polar co-ordinates of the disturbed planet at the end of this period, as resulting from the use of both methods: : |