204 THEORY OF HEAT. [CHAP. III. 



r+7T 



from to infinity. Thus the integral I &amp;lt; (x) cos ix dx is the same 



J -77 



as the integral 



r+ir r+n 



I bfr (%) + ^ ( X )J cos dx, or I F(x] cos ix dx. 



J &quot;IT J -IT 



r+T 

 It is evident also that the integral I ^(x) smixdx is equal 



J -TT 



/*+ /*+&quot; 



to the integral I F(x] sin ixdx, since the integral I &amp;lt;/&amp;gt;(#) swi 



J -7T J -TT 



vanishes. Thus we obtain the following equation (p), which serves 

 to develope any function whatever in a series formed of sines and 

 cosines of multiple arcs : 



cos x 



\ F[x] cos x dx + cos 2# I F(x] cos 2x dx + &c. 

 + sin x \ F(x] sin x dx + sin 2x I F(x) sin 2x dx + &c. 



234. The function F(x), which enters into this equation, is 

 represented by a line F F FF, of any form whatever. The arc 

 F F FF, which corresponds to the interval from. TT to +TT, is 

 arbitrary ; all the other parts of the line are determinate, and the 

 arc F F FF is repeated in each consecutive interval whose length 

 is 27T. We shall make frequent applications of this theorem, and 

 of the preceding equations (ft) and (i/). 



If it be supposed that the function F(x] in equation (p) is re 

 presented, in the interval from IT to + TT, by a line composed of 

 two equal arcs symmetrically placed, all the terms which contain 

 sines vanish, and we find equation (v). If, on the contrary, the 

 line which represents the given function F(x) is formed of two 

 equal arcs opposed in position, all the terms which do not contain 

 sines disappear, and we find equation (/x). Submitting the func 

 tion F(x) to other conditions, we find other results. 



If in the general equation (p) we write, instead of the variable 

 x, the quantity - , x denoting another variable, and 2r the length 



