708 PHILOSOPHICAL TRANSACTIONS. [anNO J 706. 



the 2d members: now the 1st term of each 2d member contains tlie even 

 power z"", with two factors of which the one A'^'" n^'" is the constant quantity 

 ) . 2 . . . 27?i (§ 1), and therefore does not vanish; then the other factor of this 

 power vanishes. Therefore the sine of an arc is a function of that arc, of the 

 form sin. z = z — az^ + bz^ + cz^ + dz'' + &c. which contains only the odd 

 powers of z. 

 We therefore have, sin. z = z — az^ + bz'^ + cz' + dz^ + &c. 



sin. 2z = 2z — 2'az' + 2*bz^ + 2'cz" + l^jiz^ + &c. 



sin, 3z = 3z — 3^Az' + 3'bz' + 3'cz' + 3'dz'' + &c. 



sin. 4z = 4z — 4^az^ + 4'bz' + 4'cz' + 4^dz'-' + &c. 

 Take now the 3d differences, and we obtain, 



— 2* sin.^ \ z cos. ^z = - A"' (4\. . 1^) az' + a'" (4' . , 1^) bz^ + &c. 



— 2^ sin.^ i z cos. 4- z = — A"' (5* . . 2^) az^ + A'" (5' . . 2') Bz' + &c. 



— 2^ sin.^ 4- z cos. f z = — A"' (6'.. . 3^) az' + A"' (6* . . 2') bz' + &c. 

 Then reducing into series the factors of the first members conformable to § 11, 



the first terms of these members are — z^; and the 1st terms of the 2d mem- 

 bers are (§ 1 .) 



— 1.2. 3A^ Then ; 1 = 1 . 2 . 3a; and a = — ^~ — . 



1.2.3 



Taking successively the 4th and 5th differences, we obtain, 

 2'sin.'-i.zcos.4-z = A''(6\. . 1') bc'* + A"(6^. . l') cz' + A^ (6«.. . 1«)dz«+&c. 

 2' sin.* 4 z cos. ■§- z = A^ (7'.. . 2') bz' + A" (;'' . . 2^ cz' + A' (7^.. . 2«) Dz« + &c. 

 2* sin. '-iz COS. y z = A" (8\. . 3") bz' + A" (8^. . 3') cz' + A^' (8«.. . 3^) dz»+ gcc. 



Reducing into series the factors of the 1st members (^ I ]), the 1st terms of 

 these members are z', and the 1st terms of the 2d members are 1.2.. .3bz', (§1.) 



Then; 1 = 1 . 2. . .5b: and b := . 



1 . 2 ... 5 



Taking likewise successively the 6th and 7th diffs. we obtain, 



— 2" sin.' i z COS. -|- z = A'" (s'. . .1') cz' + A"" (8«. . .1^) dz' + &c. 



— 2' sin.' i 7. COS. V z = A"' (p'. . .2') cz' + A^'' (Q^ . .2^) Dz^ + &c. 



— 2' sin.' 4- z COS. y z = A"'' (lO'. . .3') cz' + A"' (10^ . .3°) Dz'^ -{- &c. 

 Reducing into series the factors of the 1st members, the 1st term of these 



members is z', and the 1st terms of the 2d members are 1 . 2. . .7cc'. There- 

 fore; c = — Y-r^ — 77' 



Taking likewise the 8th and Qth differences, we obtain, 



D = -1 ; E =: ; F =: H ; &C. 



~1.2...y 1.2. ..11 ^ 1.2. ..13' 



Therefore finally sin. z=z — ——^^^ + 777,77^ «* — TT^TTTl^^'^i . 2. . 9 '^^—^^' 

 ^13. The process in the cosines is in the same manner. 



Let COS. 2=1— PlZ -\- B2- -|- cz^ -f ds'' + ei' -\- &c. 

 Therefore cos. 2a = 1 — Ikz + 2'bs- -|- 2^cs^ -\- 2"'ds' + &c. 



