200 



THEORY OF HEAT. 



[CHAP. IJI. 



interval equal to STT on the axis of the abscissae. Further, each of 

 these arcs is composed of two symmetrical branches, which cor 

 respond to the halves of the interval equal to 2?r, 



Suppose then that we trace a line of any form whatever &amp;lt;/&amp;gt;&amp;lt;a 

 (see fig. 9.), which corresponds to an interval equal to TT. 



Fig. 9. 



If a series be required of the form 



a + b cos x + c cos 2% + d cos 3x -f &c., 



such that, substituting for x any value whatever X included be 

 tween and TT, we find for the value of the series that of the 

 ordinate X&amp;lt;j&amp;gt;, it is easy to solve the problem : for the coefficients 

 given by the equation (v) are 



if 2 



- l&amp;lt;f&amp;gt;(x) dx, - 



2 r 



, - l(f&amp;gt; (x) cos xdx t &c. 



These integrals, which are taken from x = to x TT, having 

 always measurable values like that of the area Ofon, and the 

 series formed by these coefficients being always convergent, there 

 is no form of the line &amp;lt;&amp;lt;/&amp;gt;a, for which the ordinate X(j&amp;gt; is not 

 exactly represented by the development 



a -f &quot;b cos x -\- c cos 2# + d cos 3# -f e cos 



&c. 



The arc &amp;lt;(/&amp;gt;a is entirely arbitrary ; but the same is not the case 

 with other parts of the line, they are, on the contrary, determinate; 

 thus the arc &amp;lt;a which corresponds to the interval from to TT is 

 the same as the arc &amp;lt;/&amp;gt;a ; and the whole arc a&amp;lt;pa is repeated on 

 consecutive parts of the axis, whose length is 2?r. 



We may vary the limits of the integrals in equation (v). If 

 they are taken from x = ?r to x = TT the result will be doubled : 

 it would also be doubled if the limits of the integrals were 

 and 27r r instead of being and TT. We denote in general by the 



