Epoch, Mean Time. ELEMENTS OF THE MINOR PLANETS, OR ASTEROIDS. Mean Longitude Longitude of the at the Epoch. Perihelion. 1848, Jan. 1, (B) 68° 48′ 47.5 33° 0' Longitude of the Inclination to the Sid. Revolution Mean Distance 23./2 110° 18/ 0 26 31. 4 235 30 32. 2 103 23 14 29 59. 9 259 2.5 86 4 53 32 26 57 17 26 27 18 49 55 55 55 54 48 43 36 36 39 38 35 3."8 5° 53' 54. 4 18. 7 1 36 31. 6 48. 7 7 8 5. 721 34 8. 7 14 46 50. 2 3 5 44. 7 4 36 55. 3 9 46. 6 16 32 9. 3 11 43 54. 6 45. 6 13 3 1. 1 6."2 1193.222 2. 2 6. 8 1269.633 29. 7 1325.147 1345.315 16. 0 1346.111 54. 2 1352.166 1365.150 4. 5 59. 5 1379.457 1387.100 1400.803 1408.038 1425.857 1511.370 1512.106 1517.455 1554.212 1566.768 42. 2 59. 1 Eccentricity. 2.201653 0.1565408 2.334935 0.2181896 2.361081 0.0895694 0.2317452 0.1233821 0.2507330 0.1457463 0.2020361 0.1624455 2.625877 2.640004 2.384977 2.385917 2.393066 2.408360 2.425160 2.434106 2.450113 0.0998142 2.479240 0.1308650 2.577402 0.1887517 2.578238 0.0853696 2.584312 0.1683269 0.2353947 0.1874427 2.652790 0.0861356 2.670837 0.2548847 2.768051 0.0766523 2.772858 0.2398150 2.911710 0.1036109 0.1376471 2.922984 Tropical revolution Sidereal 66 66 Synodic Nodal 66 Sidereal revolutions of the satellites, and their mean distances from the planets about which they revolve. The distances are expressed in terms of the equatorial radius of the planet. Different Revolutions of the Moon. JUPITER. Mean Distance. SATURN. Tethys .... Dione ...... Rhea.. Titan Hyperion Days. 27.3215255 27.3215830 29.5305885 Mean Distance. 27.5545704 27.2122222 6.04853... 26.5 64.359.... Sider. Revolution. Days. 1.7691378 Days. 3.351...... .... 0.94271 4.300.... 1.37024 5.284...... 1.88780 6.819............ 2.73948 ... 4.51749 22.081......... .21.18 Sider. Revolution. Sun Saturn.. Jupiter Mars Earth Venus. Mercury... URANUS. Mean Distance. 13.120.. 17.022.. 19.845....... 10.9611 22.752..... .... 13.4559 45.507............ 38.0750 Masses and densities of the sun and planets, the mass of the sun and density of the earth being each assumed = 1. Sider. Revolution. Days. 5.8926 8.7068 1 359551 401839 1 4865751 Densities. unknown. 0.242 0.138 0.238 0.948 1.000 0.923 1.12 Denoting the earth's mass by a unit, the moon's mass is about , and her density about 0.615. Remark. The masses of the planets given above, except that of Neptune, are taken from a table in the Astr. Nach. No. 443. That of Mercury has been very recently obtained by Prof. Encke, from the effects of this planet in disturbing the motion of the comet which bears his name. A NUMBER of the formulæ included in the following collection are used in the present work. The demonstrations may be found in any good work on Trigonometry.* They are introduced here, and numbered in order to facilitate the references. 1. sin2 a 2. sin a = 3. sin a = APPENDIX TO PART I. 4. cos a From a single arc or angle a, the radius being 1. 7. sin a = cos2 a = 1 tang a cos a or, sin 2 a 8. cos a tang a 9. cos a = 10. tang a 11. tang 1 a 12. tanga TRIGONOMETRICAL FORMULA. 5. tang a √ (1 + tang” a) sin a 6. cot. a = cos a 1 tang a cos a sin a 2 sina cosa 2 sin a cos a 2 sina a 1 sin a 1 + cos a cos a 1 2 cos2 1 sin a 1 cos a 1 + cos a For two arcs a and b of which a is supposed to be the greater. 13. sin (a + b) sin a cos b + cos a sin b 15. tang (a+b) I tang a tang b ✯ 16. sin a cos b sin (a + b) + 1⁄2 sin (a 18. sin a sin b * The best treatise upon that subject, in our language, is that by Chauvenet, recently published. 34. cos a — b) – b) 2 sin 1 (a + b) cos ž (a b) sin (a+b) cos a cos b sin (a - b) cos a cos b tang b 28. tang (a + b) = 37. cot A 36. cot a = sin (a+b) sin a sin b sin (a - b) sin a sin b sin a cos b sin a+ sin b cos a + cos b sin a sin b cos a + cos b sin b For a Spherical Triangle, in which A, B, C, are the angles, and a, b, and c, the opposite sides, as in Fig. 58. 33. sin A sin b sin B sin a cos a sin a + sin b sin asin b 38. sin A = √ 39. tang (ba) tang c 1 + 40. tang 1⁄2 (b − a) 41. tang (BA) cot C tang c cos 1⁄2 (B A) (BA) A) cos sin(B cos 1⁄2 (b + a) |