in Conduction of Heat, 165 



Our solution in (7) therefore satisfies the initial condition 

 for the source at the point a?', and the surface conditions 

 were fulfilled by the choice of A in (5). 



We now replace the third integral of (7) by the equivalent 

 integral over the real axis in the «-plane, and obtain our 

 solution for the source in the form 



1 r fo-*') 3 _ (f+f') 2 -. 



l ^Tvml e ikr+e m J 



- g* r %-*» h «« *(* + -) -« sip <* + *o dam (8) 



ttJ a- + h 2 



Since 



e -^COS«frff = — ^72 



Jo ar + /V, 



and 



;o a " 



« 2 + A s 



this may be changed to its final form. 



v= —=\e «« +e "<" -2AJ <r*f.e k 4W dfj (9) 



which agrees with that given by Bryan. 



The solution for the initial distribution v=f (a:) follows in 

 the usual way by integration, and is given by the equation 



- 2A I 6 - A ^ " « / (^)dg^ J . (10) 



The result for the case of constant initial temperature 

 is of interest, as Riemann's original problem was to obtain 

 the distribution in this case after a considerable time had 

 elapsed. 



Put / (a?) = v and integrate : the second integral of (10) 

 simplifies after integration by parts and we obtain 



= — -— 7 \ « Akt dx'-\ e ikt civ' + 2] e~ h *\e m dx' U 

 2^/irktLJo J Jo J 



the form in which Weber gives the solution of this problem. 

 Riemann's result may be at once deduced. 



