3G6 THEORY OF HEAT. [CHAP. IX. 



From the known equation 



we conclude 



-+00 



N /7r = / dqe~( q+a )\ a being any constant; we have therefore 



- 



i, or 



This equation holds whatever be the value of a. We may de- 

 velope the first member; and by comparison of the terms we shall 



obtain the already known values of the integral ldqe~ q * q n . This 



value is nothing when n is odd, and we find when n is an even 

 number 2w, 



L 



2.2.2.2... 



371. We have employed previously as the integral of the 



du , d?u ,, 

 equation -rr = k^ the expression 



u a^-nW cos n^x + aj3~ n ** kt cos n^x + a a e~ n ** kt cos n B x + &c. ; 

 or this, 



u a^e&quot; n ^ kt sin n^x -h a 2 e~ n ** kt sin n z x + a & e~ n * lkt sin n a x + &c. 



a,, a 2 , a s) &c,, and Wj, w a , n B , &c., being two series of arbitrary 

 constants. It is easy to see that each of these expressions is 

 equivalent to the integral 



(dq e~ q * sin n (x + 2q *Jkt), or Idq e~& cos n 



In fact, to determine the value of the integral 



r* 30 



dq e~^ sin 



J 06 



