398 THEORY OF HEAT. [CHAP. IX. 



397. The equation (a) being applicable to all values of R, we 

 can suppose in it R infinite ; in which case it gives the solution of 

 the following problem. The initial state of a solid prism of 

 small thickness and of infinite length, being known and expressed 

 by v F(x) t to determine all the subsequent states. Consider the 

 radius E to contain numerically n times the unit radius of the 

 trigonometrical tables. Denoting by q a variable which successively 

 becomes dq, 2dq, 3dq, ... idq, &c., the infinite number n may 



be expressed by -y- , and the variable number i by -|- . Making 

 these substitutions we find 



v = ^- dq I dy. F (a) e~ qH cos q (x a). 



The terms which enter under the sign 2 are differential quan 

 tities, so that the sign becomes that of a definite integral ; and 

 we have 



-j f +ao M-ao 



v = x- doL F (a) I dq e-& cos (qx - qz) (@). 



J-&amp;gt;7T J -oo J - oo 



This equation is a second form of the integral of the equation 

 (QL) ; it expresses the linear movement of heat in a prism of infinite 

 length (Chap. VII., Art. 354). It is an evident consequence of the 

 first integral (a). 



398. We can in equation (/3) effect the definite integration 

 with respect to q- } for we have, according to a known lemma, which 

 we have already proved (Art. 375), 



/. 



I 



J 



+00 



dz e~ z * cos 2hz = e~ h * 



-00 



Making then z* = (ft, we find 



Jt 



Hence the integral (/S) of the preceding Article becomes 



r 



J 



J - 



