SECT. VI.] DEVELOPMENTS IN SERIES OF SINES. 181 



We should arrive at the same result, starting from the pre 

 ceding equation, 



-x = sin x ^ sin 2# + ^ sin 3x - -r sin 4# + &c. 



A A 6 *f 



In fact, multiplying each member by dx, and integrating, we 

 have 



C -r cos x ~a cos 2x -f ^ cos & -rs cos 4# -f &c. ; 



4 .Z o 4* 



the value of the constant (7 is 



a series whose sum is known to be ~ -^ . Multiplying by dx the 



two members of the equation 



ITT 2 X* 



2 - -T = co 

 and integrating we have 



ITT 2 X* 1 1 



2 - -T = cos a; - ^2 cos 2x + -^ cos 3# - &c., 



If now we write instead of x its value derived from the 

 equation 



^ # = sin a? TT sin 2# + ^ sin 3# -7 sin 4# + &c., 



we shall obtain the same equation as above, namely, 



7T 2 



We could arrive in the same manner at the development in 

 series of multiple arcs of the powers x 5 , a?, x 9 , &c., and in general 

 every function whose development contains only odd powers of 



the variable. 



5- 



217. Equation (A), (Art. 218), can be put under a simpler 

 form, which we may now indicate. We remark first, that part of 

 the coefficient of sin x is the series 



* (0) + V &quot;(0) + #(0) + r (0) + &c, 



