THEORY OF HEAT. [CHAP. III. 



of multiple arcs. Suppose then that we have reduced to a series of 

 this form a function whose value is constant, when x is included 

 between and a, and all of whose values are nul when x is in 

 cluded between a and IT. We shall employ the general equation 

 (D} y in which the integrals must be taken from x = to x = TT. 

 The values of &amp;lt;(.x) which enter under the integral sign being 

 nothing from x = a to x = TT, it is sufficient to integrate from x 

 to x = a. This done, we find, for the series required, denoting by 

 h the constant value of the function, 



1 f l-cos2a 



~7r&amp;lt;(#) = h &amp;lt;(I cos a) sm x -\ -- ~ - sin 2x 



1 cos 3a . 



_j -- -- sm ^x + &C. 

 o 



If we make /t = JTT, and represent the versed sine of the arc x 

 by versin x, we have 



&amp;lt; (x] = versin a sin a; + ^ versin 2a sin 2# + ^ versin 3 a sin 3# + &C. 1 



This series, always convergent, is such that if we give to x any 

 value whatever included between and a, the sum of its terms 

 will be ^TT ; but if we give to x any value whatever greater than 

 a and less than 4?r, the sum of the terms will be nothing. 



In the following example, which is not less remarkable, the 

 values of $ (x} are equal to sin - for all values of x included 



between and a, and nul for values of as between a and TT. To 

 find what series satisfies this condition, we shall employ equa 

 tion (Z&amp;gt;). 



The integrals must be taken from x = to x = IT ; but it is 

 sufficient, in the case in question, to take these integrals from 

 x = to x = a, since the values of &amp;lt;f&amp;gt; (x) are supposed nul in the 

 rest of the interval. Hence we find 



sin as sin 2a sin Zx sin 3a sin 3x 



+ ~ + ~- + &c 



1 In what cases a function, arbitrary between certain limits, can be developed 

 in a series of cosines, and in what cases in a series of sines, has been shewn by 

 Sir W. Thomson, Cainb. Math. Journal, Vol. n. pp. 258262, in an article 

 signed P. Q. K., On Fourier s Expansions of Functions in Trigonometrical Series. 



