SECT. VI.] TRIGONOMETRICAL DEVELOPMENTS. 195 



If a be supposed equal to TT, all the terms of the series vanish, 

 except the first, which becomes - , and whose value is sin x we 

 have then &amp;lt;# 



227. The same analysis could be extended to the case in 

 which the ordinate represented by $(x) was that of a line com 

 posed of different parts, some of which might be arcs of curves 

 and others straight lines. For example, let the value of the func 

 tion, whose development is required in a series of cosines of 



multiple arcs, be \^\ -a? } from x = to x = JTT, and be nothing 



from x = JTT to x = TT. We shall employ the general equation (n), - /* 

 and effecting the integrations within the given limits, we find &quot; 



that the general term 1 I U^J - x 2 cos ixdx is equal to/- 3 when i 

 is even) to 4- ^ when i is the double of an odd number, and to 



?, 



-^ when i is four times an odd number. On the other hand, we 



-I 3 -. ,. 



3 ? for the value of tte first term 9 fa&y&e. We have then 



the following development : 



2 cosa; cos %x cos oas cos 



&amp;lt; = 



cos 2ic cos 4# cos 6# 



~J 2^ 4 2 ~* ^2 & c - 



The second member is represented b} a line composed of para 

 bolic arcs and straight lines. 



228. In the same manner we can find the development of a 

 function of x which expresses the ordinate of the contour of a 

 trapezium. Suppose &amp;lt;f&amp;gt;(x) to be equal to x from x = to x = a, 

 that the function is equal to a from x a. to x IT a, and lastly 

 equal to TT - x, from x = TT - a to x = IT. To reduce it to a series 



? * ^ * *^ tf*l&amp;gt; 



^ n ) ,, 132 



