PHILOSOPHICAL SOCIETY OF WASHINGTON. 59 



But Gauss, in his "Disquisitiones generales circa serievi infini- 

 tam,^' &c., has given another series, which can be used very con- 

 veniently. 



In order to demonstrate this series we shall put 2 a:; for a: in 

 equation (g), and it then becomes 



1 1.3 



2 a? = sin 2 ic + --- sin ^ 2 x + ----- sin ^ 2 a; -I- &<^- 

 2.3 2.4.5 



(1 13 7 



= sin 2 x i 1 + — - sin ' 2 a; 4- -4 -^ sin ^ 2 a; + &c. [ 

 C ' 2.3 2.4.5 ) 



where the nth term within the bracket is of the form 



c„sin^"'^ 2a; = c,„2''(sin''"-^a:) (1 — sin^a;)"-\ 



and therefore only involves even powers of sin x, and the series 



can consequently be put into the form 



2 a? = sin 2 a; {1 -|- a sin ^ a; -|- 6 sin '* a; -|- c sin ® a; -}- &c. } 



a, b, c, &c. being at present unknown. 



In order to determine them we shall differentiate this series, and 



we have 



2 = 2 cos 2 a; { 1 -|- a sin ^ a: -(- & sin ^ a; -j- c sin ® x -|- &c. } 



-[- sin 2 a? 1 2 a sin a; -(- 4 6 sin '' a; -[- 6 c sin ^ a: -}- &c. \ cos x 



= 2 (1 — 2 sin ^ a:) (1 -]- a sin ^a: -|- ^ sin^a? -f c sin^a; -|- &c,) 



-|- 2 (1 — sin^a?) (2 a sin '-^ a; -f 4 6 sin* a; -|- 6 c sin® a? -f &c-) 



whence equating coefficients we have 



2 

 3 a — 2 =0 whence a =-- 



2.4 



5& — 4a = o " & = „-^ 



1c — 66=0 " c=:-— ^- &c, &c. 



3.5 

 2 4. 

 3.5.7 



therefore 



/ 2 2.4 2.4.6 . „ „ \ 



2a; = sin2x(l -|- sin ^ a; -|- ^— sin * x + — — sm « a: + &c. j 



/ 2 2 4 2.4.6 \ 



or a: = sinxcosx 1 -f -sin ^x -[-^-^ sin^a: -|- -— — sin «a;4- &c. I 



which is a very convenient series. 



1 3 3 



If sin X = ':j=H, cos x = -;jj= and sin x cos x =Jq ; 



1 1 U 



If sinx = -^, cosa: = ;7|^ " sina;cosa; = jj3Q; 



