SECT. I.] IDENTITY OF DIFFERENT SOLUTIONS. 365 



on the other hand we ought to have 



v = ~ e~ M I sin q e~ q * kf , 

 TT j 



q 



or v = 



[8 



Now the integral Icfo&amp;lt;e~ w2 w 2m taken from u = Q to u = oo has 



a known value, m being any positive integer. We have in 

 general 



Jo 



o -2222 2 2 V* 



The preceding equation gives then, on making q*kt = if, 



T, [2 /, u 2 1 u* 1 \ 



\due~ u 1 1 15 Ti + fr 7T3-- &C. I , 



J V 3/.- o^ ; 





v ii/_j_y 1 



+ :C 



13 ,/fc [2 5 z 



This equation is the same as the preceding when we suppose 

 a. = 1. We see by this that integrals which we have obtained 

 by different processes, lead to the same convergent series, and 

 we arrive thus at two identical results, whatever be the value 

 of x. 



We might, in this problem as in the preceding, compare the 

 quantities of heat which, in a given instant, cross different 

 sections of the heated prism, and the general expression of these 

 quantities contains no sign of integration ; but passing by these 

 remarks, we shall terminate this section by the comparison of 

 the different forms which we have given to the integral of the 

 equation which represents the diffusion of heat in an infinite 

 line. 



r&amp;gt;&amp;gt;-n m , c ., ,. dll ^ d*ll 



3/0. lo satisfy the equation ~r k ^ Z) we may assume 



u = e~ ff e kt , or in general u e~ n ? e n kt , whence we deduce easily 

 (Art. 364) the integral 

 r 

 u = I 



1 Cf. Rieinann, 18. 



