146 THEORY OF HEAT. , [CHAP. III. 



Each term of the second member being replaced by the 

 difference of two cosines, we conclude that 



- 2 -& sin 2# = cos (- a?) - cos 3# 



cos x + cos 5x 

 -}- cos 3# - cos 7x 



cos 5# + cos 9x 



-f cos (2i 5) a? - cos (2w 1) x 

 cos (2m 3x) -f cos (2m -f 1) #. 

 The second member reduces to 



cos (2m + 1) x cos (2m 1) a-, or 2 sin 2marsiu .r ; 



1 */ sin % 



hence 



180. We shall integrate the second member by parts, dis 

 tinguishing in the integral between the factor smZmxdx which 



must be integrated successively, and the factor or sec x 



COSX 



which must be differentiated successively ; denoting the results 

 of these differentiations by sec x, sec&quot; x, sec &quot; x, ... &c., we shall 

 have 



1 1 



2y = const. ^-- cos 2?H# sec x + - :, sin 2mx sec x 

 2.m 2m 



I 

 4- o~* cos 2m# sec x -f i\&amp;gt;c. ; 



thus the value of y or 



cos x ;r cos 3x + - cos 5x ^ cos 7x + . . . cos (2m 1 ) .r, 



3 o 7 2m - 1 ; 



which is a function of x and m, becomes expressed by an infinite 

 series ; and it is evident that the more the number m increases, 

 the more the value of y tends to become constant. For this 

 reason, when the number m is infinite, the function y has a 

 definite value which is always the same, whatever be the positive 



