94 EXPANSION OF FUNCTIONS. 



' + ! A _(2*_1) 

 ~ 



~ 2.3 

 and so on ; 



therefore e = 1 + ax + + a , * + 



, 



LI . LJ 11 



g(q'-U)(a 2 +3') . 



Since e^"' = 1 + a sin' 1 ^ + r (sin'^) 2 + . , . 



LE 



we have, by equating the coefficients of a in this series, and 

 in the result just found, 



- -i , 1 d , 1.3 a- 5 , 



sin x=x+ -.-- + .- + .. 



as already found. 



Also equating the coefficients of a*, we have 

 2 s 2* 4." 2* 4* fi* 



( _i \o o 4 ^.' . * a A V v 



smV)' = . ; - + -x' + .- + a : N.... 



And equating the coefficients of a 3 we have 



122. Expand sin (m sin" 1 a;) in powers of x. 

 Putting y for the function, we may shew that 



J da? dx 

 Proceeding as in Art. 121, we find that 



(n + 1) (n + 2} A^ = (n* - m 2 ) A n ; and thus 



., N m w(! 2 -m 2 ) m(l s -m 1 )(3 8 -w 1 

 sm(msm 1 o;) = ---a;+ , a a; 3 + - 



i y 



Similarly cos (m sin" 1 ^ 

 m* 



[2 



m* m 8 (2 8 -m') . m 2 (2 8 - m') (4' - m a ) 



~~ ~ 



