338 THEORY OF HEAT. [CHAP. IX. 



initial temperatures of an infinite prism, of which an intermediate 

 part only is heated. Substituting the value of/(^) in the expres 

 sion for F (x} y we obtain the general equation 



F(x)=\ dqcosqxl dxF(x)cv$qx (e). 



A Jo Jo 



347. If we substitute in the expression for v the value which 

 we have found for the function Q, we have the following integral, 

 which contains the complete solution of the proposed problem, 



-v ^a 



7I = e~ u \ dq cos qx e~ kqH I dx F (x) cos qx. 



. 



The integral, with respect to #, being taken from x nothing 



fcy* to x infinite, the result is a function of q\ and taking then the 

 integral with respect to q from q = to q = oo , we obtain for v a 

 function of x and t, which represents the successive states of the 

 solid. Since the integration with respect to x makes this variable 

 disappear, it may be replaced in the expression of v by any varia 

 ble a, the integral being taken between the same limits, namely 

 from a = to a = oo . We have then 



!L_ _. e -u I fa cos g X e -kq*t I fa 2P( fl ). cos qx, 

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or = e~ ht I dx F(a.) I dq e~ kqZf cos qx cos qy. 



a Jo Jo 



\ 



The integration with respect to q will give a function of x } 



t and a, and taking the integral with respect to a we find a func- 

 ^ tion of x and t only. In the last equation it would be easy to 

 effect the integration with respect to q, and thus the expression 

 of v would be changed. We can in general give different forms 

 to the integral of the equation 



dv , d*v , 



dt =k d J ?~ hv &amp;lt;$&quot; 



they all represent the same function of x and t. 



348. Suppose in the first place that all the initial tempera 

 tures of points included between a and b, from x = 1, to x 1, 

 have the common value 1, and that the temperatures of all the 



