34G THEORY OF HEAT. [CHAP. IX. 



shall represent the variable number i by -J- . If in the general 



term . sin (2* + 1) -we put for i and n their values, the term 



2^ + 1 n 



becomes ^sin2&amp;lt;7#. Hence the sum of the series is \ ~sm2qx, 

 2q J $ 



the integral being taken from q = to q = oo ; we have therefore 



the equation \ IT = J I sin 2qx which is always true whatever 



Jo % 



be the positive value of x. Let 2qx = r, r being a new varia 

 ble, we have = and J TT = I - sin r ; this value of the defi 

 nite integral I sin r has been known for some time. If on 



supposing r negative we took the same integral from r = to 

 r = oo , we should evidently have a result of contrary sign -J TT. 



357. The remark which we have just made on the value of 

 the integral I sin r, which is J TT or \ TT, serves to make known 

 the nature of the expression 



2 f^dqsi] 

 *h~^l 



cos qxy 



whose value we have already found (Article 348) to be equal to 

 1 or according as x is or is not included between 1 and 1. 



&quot;We have in fact 



I cos qx sin q = J I sin ^ (x 4- 1) I sin q (x 1) ; 

 the first term is equal to J TT or J TT according as x + 1 is a 

 positive or negative quantity; the second J I sin q (x 1) is equal 



to J TT or J TT, according as x 1 is a positive or negative quantity. 

 Hence the whole integral is nothing if x + 1 and x 1 have the 

 same sign ; for, in this case, the two terms cancel each other. But 

 if these quantities are of different sign, that is to say if we have at 

 the same time 



x -f 1 &amp;gt; and x 1 &amp;lt; 0, 



