152 THEORY OF HEAT. [CHAP. III. 



The total value of tlie integral is then less than the sum of the 

 differentials d (sec 7 a?), taken from x = up to or, and it is greater 

 than this sum taken negatively : for in the first case we replace 

 tlie variable factor cos 2mx by the constant quantity 1, and in 

 the second case we replace this factor by 1 : now the sum of 

 the differentials d (sec&quot; x), or which is the same thing, the integral 

 {d (sec&quot; x), taken from x = 0, is sec&quot; x sec ; sec&quot; x is a certain 

 function of x, and sec&quot;0 is the value of this function, taken on 

 the supposition that the arc x is nothing. 



The integral required is therefore included between 

 + (sec&quot;*e sec&quot; 0) and (sec&quot; x sec&quot; 0) ; 



that is to say, representing by k an unknown fraction positive or 

 negative, we have always 



/ {d (sec&quot; x) cos 2mx] = k (sec&quot; x sec&quot; 0). 

 Thus we obtain the equation 



2u c sec x cos 2mx + - sec x sin Zmx + 3 sec&quot; x cos Imx 

 2m 2m 2ra 8 



in which the quantity ^ 3 (sec&quot; x sec&quot; 0) expresses exactly the 



.- fib 



sum of all the last terms of the infinite series. 



187. If we had investigated two terms only we should have 

 had the equation 



I i j c 



2t/ = c-~ sec x cos Zmx + -^r, sec x sin 2mx + -^ z (sec x- sec O). 



*/// _ y/6 ^ 7/& 



From this it follows that we can develope the value of y in as 

 many terms as we wish, and express exactly the remainder of 

 the series ; we thus find the set of equations 



1 ^k 



*2i/ = c x sec x cos 2mx-^ t (sec x sec 0), 

 9 2 in % m 



2 y c x sec x cos 2mx+ ^ = sec x sin 2mx \ ^7., (sec x sec 0), 

 2??^ 2 m 2 m v 



2 y = c -- sec x cos %mx+ TT 5 sec # sin 2m^ 4- ^ 5 sec&quot; x cos 2m# 

 ^w 2w 2 m 



f &quot; /\\ 



Hr n~s (sec a; sec 0). 



1. /72- 



