SECT. VI.] LIMITS OF THE DEVELOPMENTS. 180 



Hence we deduce the equation 



.j TT = sin x + g sin 3# 4- - sin 5# -f = sin 7x + &c., (N t 



which has been found before. 



It must be remarked that when a function &amp;lt;f&amp;gt; (x) has been de 

 veloped in a series of sines of multiple arcs, the value of the series 



a sin x -f & sin 2# + c sin 3x + d sin kx + &c. 



is the same as that of the function $ (#) so long as the variable x 

 is included between and IT ; but this equality ceases in general 

 to hold good when the value of x exceeds the number TT. ~ 



Suppose the function whose development is required to be x, 

 we shall have, by the preceding theorem, 



2 irx = sin x I x sin x dx + sin 2x I x sin 2# dx 



+ sin 3# I x sin 3# dx 4- &c. 



The integral I x sin i#cfa? is equal to f T ; the indices and TT, 

 / z 



which are connected with the sign I , shew the limits of the inte 



gral ; the sign -f must be chosen when i is odd, and the sign 

 when i is even. We have then the following equation, 



^x = sin x = sin 2# + ^ sin 3# -j sin 4# + - sin 5^ &c. 

 25 v 4 o 



223. We can develope also in a series of sines of multiple 

 arcs functions different from those in which only odd powers of 

 the variable enter. To instance by an example which leaves no 

 doubt as to the possibility of this development, we select the 

 function cos x, which contains only even powers of x t and which i t \ 

 may be developed under the following form : 



a sin x + 6 sin 2x + c sin 3# + d sin 4&amp;lt;x + e sin 5# + &c., 



*r 



although in this series only odd powers of the variable enter. 



