182 THEORY OF HEAT. [CHAP. III. 



which represents the quantity -(/&amp;gt;(TT). In fact, we have, in 

 general, 



(0)*|*&quot;(0)+|* 



&c. 



Now, the function &amp;lt;f&amp;gt;(x) containing by hypothesis only odd 

 powers, we must have &amp;lt;(0) = 0, &quot;(0) = 0, &amp;lt;/&amp;gt; iv (0) = 0, and so on. 

 Hence 



&amp;lt;f) (x) = x(j)(Q) + TK &amp;lt;fi&quot; (0) + p V W + &amp;lt; ^ c&amp;lt; j 



a second part of the coefficient of sin x is found by multiplying 

 by Q the series 



&amp;lt;T (0) + n&amp;gt; 3 ^(0) + IF &amp;lt;/&amp;gt; vli (0) + ^ ^ lx () + &c -&amp;gt; 



whose value is - $ (TT}. We can determine in this manner the 



7T r 



different parts of the coefficient of sin#, and the components of 

 the coefficients of sin 2#, sin 3x, sin 4&amp;lt;x, &c. We may employ for 

 this purpose the equations : 



f (0) + * &quot;(0) + &amp;lt;^ V (0) + &c. = 



r (0) + ^(0) + &c. = 



^ (&amp;gt; ^ &c - = - 



O 7T 



