SECT. III.] PARTICULAR CASES. 1-V7 



value of r, less than J-TT. Now, if the arc x be supposed nothing, 

 we have 



1111 



which is equal to JTT. Hence generally we shall have 

 1 111 



- - 7T = COS X ^ COS 3x + - COS OX = COS 



4 3 o 7 



181. If in this equation we assume x = ~ _ , we find 



-^L_-1 1 _i_ 1 1 JL A J: 

 ~ 3~5&quot;7 + 9 + lI 13 15 ^ C ; 



by giving to the arc x other particular values, we should find 

 other series, which it is useless to set down, several of which 

 have been already published in the works of Euler. If we 

 multiply equation (ft) by dx, and integrate it, we have 



7TX . l-o 1 - r 1 * . fl 



-T- = sm x ^ sin 3^ + ^ sm ^ T^&amp;gt; sm tx + &c. 

 4* o o 7&quot; 



Making in the last equation x = | TT, we find 



a series already known. Particular cases might be enumerated 

 to infinity ; but it agrees better with the object of this work 

 to determine, by following the same process, the values of the 

 different series formed of the sines or cosines of multiple arcs. 



182. Let 

 y = sin x - ^ sin 2x + ^ sin 3# - 7 sin 4# . . . 



1 1 



-i -- - sin [m 1) x -- sin mr, 

 m 1 7?i 



m being any even number. We derive from this equation 



-j- = cos x cos 2# + cos ox cos 4fx . . . + cos (m 1) x cos mx ; 



102 



