And in the second member of the equation putting for y this its first approximate value, we have for a second approximation, and the two terms of this expression having been separately determined by means of logarithms, the number of seconds in y is known. Hymers' Astronomy, Art. 148. Ex. 2. In the same manner it may be shewn from =-n. n y 2 - - . (cos m -. cotz) - (1 - 2. cota), nearly; 2 n whence, neglecting cot ≈ with respect to cos m, and 2. 2 with respect to 1, we get for a first approximation, cot ≈ the terms of this expression are, as in Ex. 1, determined separately by means of logarithms. 131. Hymers' Astronomy, Art. 124. To expand, independently of Demoivre's Theorem, Sin 0 and Cose in series ascending by powers of 0, assuming If sin can be expanded in a series ascending by integral powers of 0, such a series cannot possibly contain any but odd powers of 0. For since sin = - sin (-0), it appears that when O is changed into - 0, the sine of continues of the same magnitude, but changes its sign,-a result which cannot take place if the series contain either constant terms, or terms involving even powers of the angle. Again, since cos cos (-0), it follows that the series representing the value of cannot involve odd powers of 0: since if the series did involve such powers of 0, it would not continue of the same magnitude and sign, when was changed into - 0. Let ... Cos = a + ɑ2 Ø2 + α1 Ø1 + And if o be written for 0, this equation becomes, 1 = a; Cos e = 1 + a2 93 + a1 01 +...... Adding these values of Sin and Cos 9, we have 1 + 0 + a¿02 + a ̧Ð3 + a ̧01 + ... = cos 0 + sin 0 ; writing + for 0, 1 + (0 + $) + a2 (0 + $)3 + a3 (0 + $)3 + • = cos (0 + $) + sin (0 + $) ... = cos. {cos + sin 0} + sin . {cos - sin 0} 2 = {1 + a, p2 + ... } · { 1 + 0 + а„Ø ̈ + а„Ø ̈ + ... } 3 - + { $ + a‚ ̧Ø3 + ... }. {1 − 0 + a ̧02 – az03 +...}, |