308 THEORY OF HEAT. [CHAP. VI. 



and also lix ^ + ^x^ f = 0, 



or, hty + pfy = ; 

 hence we have 



we can therefore eliminate the quantities -\Jr and ijr&quot; from the 

 integral which is required to be evaluated, and we shall find as the 

 value of the integral sought 



putting for /JL its value, and denoting by U t the value which the 



function u or ^rlx A / y* ) takes when we suppose x = JT. The 

 V V K / 



index i denotes the order of the root m of the definite equa 

 tion which gives an infinity of values of m. If we substitute 



m t or 



\319. It follows from the foregoing analysis that we have the 

 , two equations 



! x f-, fhX\*}X*U* 



b = and 2 -J~ I i 



the first holds whenever the number i and J are different, and the 

 second when these numbers are equal. 



Taking then the equation &amp;lt;j&amp;gt; (x) =a 1 u l + a 2 ii 2 + a 8 u a + &c., in 

 which the coefficients a v a 2 , a 3 , &c. are to be determined, we shall 

 find the coefficient denoted by a. by multiplying the two members 

 of the equation by xu t dx, and integrating from x = to x X ; 

 the second member is reduced by this integration to one term 

 only, and we have the equation 



