148 THEORY OF HEAT. [CHAP. III. 



multiplying by 2 sin x, and replacing each term of the second 

 member by the difference of two sines, we shall have 



2 sin x -T- = sin (x + x) sin (x - x) 



- sin (2a? + x) + sin (2x - a;) 

 + sin (3# + a?) sin (3a? x) 



+ sin {(m 1) a? - a;} sin {(??? -f 1) a? #} 

 - sin (m.r + #) -f sin (ma? - x) ; 

 and, on reduction, 



2 sin a? --,- = sin x + sin w# sin (ma? + x} : 

 dx 



the quantity sin mx - sin (?na; + a?), 



or sin (wa? + J a? - Ja;) - sin (ma? -f 4# + iar), 



is equal to - 2 sin \x cos (wia; + Ja;) ; 



we have therefore 



dn 1 sinA-a? 

 2 



cos mx 



sin a? 



dy _ 1 cos (mx 4- i#) . 

 whence we conclude 



or &amp;lt;to 2 2 cos 



1 f cos (mx -}-fa) 

 ] 2 cos fa 



If we integrate this by parts, distinguishing between the 



factor r- or sec \x, which must be successively differentiated, 



cos^x 



and the factor cos(mx+fa], which is to be integrated several 

 times in succession, we shall form a series in which the powers 



of m + ^ enter into the denominators. As to the constant it 

 is nothing; since the value of y begins with that of x. 



