VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. 707 



27)1 + 1 diffs. + 2-" + ' sin.'" + ■-!-« (sin. — ^ — o, sin. — ^— a, sin. — -^ 



a, &c. 



By omitting the factor 2™ sin." J- a, these series, as well as the fundamental 

 or original series, recur upon themselves, or are always different respectively as a 

 is or is not commensurable with the circumference. • These series are also those 

 of the sines or cosines of arcs which follow the arithmetic progression of the 

 natural numbers; only with a different origin. 



^ 11. As the sin. z is nothing, when z is nothing; and as the ratio of equality 

 is the limit of the ratio of an arc to its sine; also that any sine is less than its 

 arc; the sin. z is a function of :: of the form z — az" + bz^ + cz*+dz'-|-&c. 



And as the cosine of an arc is 1 when z is O; the cos. z is a function of z of 

 the form l — az + bz" -\- cz* + dz* + &c. 



§ 12. Let then sin. z = z — az" + bz^ + cz* + dz^ + &c. 



We also have sin. 2z = 2z — 2^Az^ + 2*bz^ + 2^cz'' + 2'dz' + &c. 

 and sin. 3z = 3z — 3^az^ + 3^bz' + 3'cz* + 3'dz' + &c. 

 and sin. 4z = 4z — 4^Az- + 4^bz^ + 4^cz^ + 4^Dz' + &c. 



Take the first differences, by which will be obtained, 

 2 sin. -i- z cos. -| z = z — (2' — 1") az^ + (2' — 1^) bz^ + (2' — 1*) cz* + &c. 

 2 sin. 4- z cos. a z = z — (3^ — 2^) az'^ + (3' — 2^) bz' + (3* — 2^) cz^ + &c. 

 2 sin. i z cos. i z = z - (4^ — 3^) az^ + (4^ - 3^) bz^ + (4* - 3") cz' -{- &c. 



Reducing now into series the factors of the first member. (§ li), and 

 making the multiplications; the first terms of the products are Iz; and the 

 first term of each member is also Iz; therefore we now learn only that the first 

 term of the expression for the sine is also Iz. 



Next by taking the 2d differences, we obtain as follows: 



— 2*sin.-izsin.2z= - A"(3- . . 1=)az- + A"(3^ . . 1")bzHA''(3*. . .l')cz*+&c. 



— 2-'sin.'4-zsin.3z= — A''(4^. . 2-) az' -}■ A'''{4\. . 2^)bz' + A''(4*.. . 2^)cz^+&c. 



— 2'sin.-4-zsin.4z= — A"(3-.. . 3')az-+ A''(5^- • 33)Bz^ + A''(5^. . 3-')cz'+&c. 

 Now, the 1st term of each 1st member developed in a series conformable to 



the expressions of § 11, is z^; and the 1st members do not contain the 2d 

 powers of z; while the coefficient of the 1st term az" of the 2d members, 

 which is the 2d dif. of the squares of the natural numbers, does not vanish. 

 Hence, in the 2d members, the co-efficient a of z" is 0. 



It is demonstrated in the same manner, that, in the series, sin. z = z az' 



+ Bz' + cz^ + &c. the co-efficients of all the powers to the even exponents 

 vanish. Namely, having taken the differences of the even order z", which are 

 + z^" sin. z^" i z sin./jz (§ 10); the first term of the product of the factors 

 developed in series, contain z*" + ', the odd power of z; so that in these pro- 

 ducts the even power z^"' is wanting; therefore it ought also to be wanting in 



4X2 



