190 THEORY OF HEAT. [CHAP. III. 



We have, in fact, by the preceding theorem, 

 - TT cos x sin x I cos x sin x dx + sin 2# I cos x sin 2# dx 



4- sin 3x I cos x sin 3# cfce + &c. 

 The integral I cos x sin ix dx is equal to zero when i is an 



odd number, and to . 2 _\ when i is an even number. Supposing 

 successively i = 2, 4, 6, 8, etc., we have the always convergent 



seres 



T TT cos x = = s s m 2 # + ^ ? sin 4 ^ + K &quot;7 sin 

 4 I . o o . o o . / 



or, 



This result is remarkable in this respect, that it exhibits the 

 development of the cosine in a series of functions, each one of 

 which contains only odd powers. If in the preceding equation x 

 be made equal to JTT, we find 



This series is known (Introd. ad analysin. infiniL cap. x.). 



224. A similar analysis may be employed for the development 

 of any function whatever in a series of cosines of multiple arcs. 



Let &amp;lt;(#) be the function whose development is required, we 

 may write 



&amp;lt; (x) a Q cos Ox + a t cos x + a a cos Zx + a a cos 3x + &c. 



+ a i cosix+&c ........... (m). 



If the two members of this equation be multiplied by cosjx, 

 and each of the terms of the second member integrated from 

 x = to x = TT ; it is easily seen that the value of the integral 

 will be nothing, save only for the term which already contains 

 cosjx. This remark gives immediately the coefficient a,; it is 

 sufficient in general to consider the value of the integral 



Icoajx cos ix dx, 



