Masses and Densities of the Sun and Planets, the mass of the Sun and density of the earth, being each assumed = 1. 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, 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. APPENDIX TO PART I. TRIGONOMETRICAL FORMULA. 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. For a single arc or angle a, the radius being = 1. 16. sin a cos b = sin a sin b tang atang b I tang a tang b 1 sin (a + b) + 1⁄2 sin (a — b) sin (a + b) — — sin ( a — - b) cos (a + b) 1⁄2 cos (a + b) 17. cos a sin b = cos (a—b) + - b) Most of them are contained in the Trigonometry attached to Legendre's Geometry ; and those not given in that work are easily deduced from others that are there given. 22. cos b + cos a = 2 cos(a + b) cos i (a —b) 2 sin ( a + b) sin (a — b) For a Spherical Triangle, in which A, B, and C, are the angles, and a, b, and c, the opposite sides, as in Fig. 58. For a right angled spherical triangle in which C is the right angle, and the opposite side c, the hypothenuse, as in Fig. 59. 45. cos c = cos a cos b 48. tang asin b tang A 51. If any small arc or angle a, not exceeding, or not much exceeding a degree, be expresed in seconds, and if w = 206264".8, we have, For the sine of a small arc is very nearly equal to the length of the arc itself; and to obtain the length of an arc, expressed in seconds, we have this proportion. As the number of seconds in the whole circumference is to the seconds in the arc, so is the length of the circumference to the length of the arc. Hence, 1296000" a 6.2831853: length of a, As the circumference of a circle, divided by 6.283 &c., gives the radius, it is evident that 206264′′.8 are the seconds in the radius. Cor. The number of seconds in an arc is equal to the product of w by the length of the arc; the radius being unity. INVESTIGATIONS OF VARIOUS FORMULE. 52. To find formula for the values of the rectangular co-ordinates CD and AD, of a place A, on the earth's surface, Fig. 4, and its distance AC from the centre, in terms of the latitude and eccentricity. = Put AC, the equatorial radius Cq being assumed = 1, e earth's eccentricity. Then, we have, CD = ? cos p', and AD = sin '. Let the semicircle eBq be described on eq, and let DA be produced to meet it in B. Then, by Conic Sections, Cp: Cq: DA: DB, Adding together the squares of the equations (A) and (B) we have, 53. To find the times of longest and shortest twilight at a given place. Let HZG, Fig. 32, be the meridian of the place, Z its zenith, HR its horizon, FG, parallel to HR, 18° below, P the elevated pole, AB the part of sun's diurnal path included between HR and FG, PA and PB arcs of declination circles, and ZA and ZB arcs of verticle circles. Put L = PH latitude of the place, D = sun's declination, and 2a = 18°. (App. 34), we have, = Then, |