364 THEORY OF HEAT. [CHAP. IX. 



The second part of the value of v is 



n~-Tlt .- / ,-- / 



-T^e x \f^ldq e-&amp;lt;? &&* or e V* dr e~* ; 

 making r = q JTti. The integral should be taken from 

 r = oo tor = Jfa -- 7= , 



_ . /% 



or from r = Jht -f j=. to r = co , 



&amp;gt;_ ^y/ Kit 



whence we conclude the following expression : 



3C9. We have obtained (Art. 367) the equation 



to express the law of diffusion of heat in a bar of small thickness, 

 heated uniformly at its middle point between the given limits 



x = a, x + a. 



We had previously solved the same problem by following a 

 different method, and we had arrived, on supposing a = 1, at 

 the equation 



_lcos qx sin ^e- 2 ^, (Art. 348). 



To compare these two results we shall suppose in each x = ; 



denoting again by ^{R} the integral ldre~ rZ taken from r = 

 to r = R, we have 



_ 1 1 /o 



: ~i 3 



\&quot; 1 1 / a y ) 



+ 5 l; - &ft ; 



