SECT. I.] HEATED FINITE BAR. 841 



ing medium is maintained at the constant temperature 0, and that 

 it receives heat from the bar or communicates heat to it through 



Fig. 16*. 



the external surface. It is required to find, after a given time t&amp;gt; 

 what will be the temperature v of a point whose distance from the 

 origin is x. 



We shall consider first the heated bar as having a finite 

 length 2JT, and as being submitted to some external cause which 

 maintains its two ends at the constant temperature 0; we shall 

 then make JT= oc. 



352. We first employ the equation 



r 



and makin v = e~ hf u we have 



_ , 

 dt ~ dx*&amp;gt; 



the general value of u may be expressed as follows : 



u = a i e~ k9iH sin gjc + agr*^ sin gjc + a & e ~ *0& sin g a x -f &c. 



Making then x = X, which ought to make the value of v 

 nothing, we have, to determine the series of exponents g, the 

 condition sin gX= 0, or gX=i7r, i being an integer. 



Hence 



. ^ 



u * =. a^e sin -^ + a 2 e sin =- + &c. 



It remains only to find the series of constants a lt a a , a 3 , &c. 

 Making t = we have 



. . . 



sin -.+ a sin -- + a 3 sin -- + xc. 



