348 THEORY OF HEAT. [CHAP. IX. 



359. The same transformation applies to the general equation 

 TT cf&amp;gt; (w) = sin u Idu &amp;lt;f&amp;gt;(u)smu+ sin 2w Idu $ (u) sin 2w + &c. 



/y / /*\ 



Making w = - , denote $ (w) or $ (-) by /(a?), and introduce into 



ft \%/ 



the analysis a quantity ^ which receives infinitely small incre 

 ments equal to dq, n will be equal to -j- and i to ~ ; substituting 

 these values in the general term 



. ix [dx . fx\ . ix 

 sin I d&amp;gt; ( - sin , 

 n J n r \nj n 



we find dq smqx I dxf (x} sin qx. The integral with respect to u 



is taken from u = to u = TT, hence the integration with respect to 

 x must be taken from x = to x = n?r, or from x nothing to x 

 infinite. 



We thus obtain a general result expressed by the equation 



Too Too 



J /(*)&quot;* I djnnpj dxf(x)smqx (e), 



^o ^o 



for which reason, denoting by Q a function of q such that we have 

 f(u)=ldqQsmqu an equation in which /(it) is a given function, 



2 f 

 we shall have Q = - lduf(u) sinqu, the integral being taken from 



u nothing to u infinite. We have already solved a similar problem 

 (Art. 346) and proved the general equation 



Too /&amp;lt;*&amp;gt; 



^irF(x} \ dqcosqxl dxF(x)cosqx (e), 



*o Jo 



which is analogous to the preceding. 



360. To give an application of these theorems, let us suppose 

 f(x)=x r , the second member of equation (e) by this substitution 



becomes Idq sin qx Idx sin qx of. 

 The integral 



jdx sin qx x* or ^ Iqdx sin qx (qx} r 



