(38.) Solution of triangles by series. [1] Given the sides a, b, and the angle C of a plane triangle : [2] Given the sides a, b, and the angle C of a spherical tri log, cos c=log. (cosa.cos 16) + tana.tan 1b.cos C log,c=log, a — § (2) ? } (A + B) = m 90° — † C + S„(−1)−1( cot a m sin m C m m 'tan bm cos m C m m tan ∞ cosm C 0 log.cosc=log. (cosa.cos 1b)+(-1)-1(tana.tan) m (39.) Correction for a compound base. posed of two straight lines a and b, 180°-a", the correction is If the base be cominclined at an angle ab x2 0,00000000001175. a+b (40.) Reduction of an oblique to a horizontal angle. Let a, ẞ, be the angles of elevation of two objects; C, the horizontal angle: then [1] C−y={(a + ß)2 tan Ly− (a — ß)2 cot1y}. [2] (sin¦ C)2 = sin § (a+a— ß). sin § (a — a — ß). cos a. cos B ; log SinC={log Sin (a + a-ẞ) + log Sin (a-a-B) (41.) Reduction of a spherical to a plane triangle. [1] Let C be the spherical angle, C- the corresponding plane angle; then 1 α= {(a + b) tan C - (a —b)2 cot C}. [2] Legendre's Theorem: 00= area of triangle 3r2 = {(A + B + C − 180°). The value of a in seconds is area of triangle 3 x 0,000004848 If the area be found in feet, and a degree on the Earth's surface=365155 feet, then log10 divisor = 9,803894. Reduction of a plane to a spherical triangle. Let γ be the plane angle contained by the sides a, ß; C the corresponding spherical angle; then (L. 293-313; C. ch. xx; W. ch. xii; Enc. Met. V. 1. p. 698,9.) FORMULE FOR THE CONSTRUCTION OF TABLES. (42.) cos 30° = (1 + 1) ]* = c1 ; cos 1 30° = 1 (1 + c,) ]* = c2 ; cos + 30° = ↓ (1 + c2) |3 ] = c3 ; = $1 sin 15° = (1+1) ↓ (1 − 1 ) 1 — s1 ; sin 15° = sin 15° = &c. &c. By continuing this process we obtain sin 15° and thence which equation may be solved by approximation. sin (a + b) = sin a {sin a ― sin (a - b)} — 4 sin a. (sin 6)2: Putting b=1°, we obtain Delambre's formula, sin (a+1°)=sin a+ {sin a― sin (a — 1o)} — 4 sin a. (sin 30′)2. sin (60° + a) = sin a + sin (60o — a). tan (45° + a) = 2 tan 2a + tan (45° — a). cosec a=cot a + tana. (43.) Sines of arcs expressed by surds. (W. ch. iv ; C. ch. vi.) |