[ 162 ] 



XIV. A Problem in Conduction of Heat. By H. S. CARSLAW, 

 M.A V D.Sc. (Lecturer in Mathematics in the University of 



Glasgow) *. 



I^HE problem of the Linear Flow of Heat in a solid 

 extending to infinity on one side of an infinite plane, 

 while radiation takes place across that plane into a medium 

 at zero temperature, was discussed by Riemann (Pa/tielle 

 Differential- Gleicliungen, § 69). His solution depends upon 

 that for the more general problem of the sphere, and is 

 obtained by making the radius of the sphere increase in- 

 definitely. Another solution was given by Bryan f by the 

 synthetical method, and in Weber's new edition of Riemann's 

 book % a third discussion of this problem is to be found. 

 In the following pages another discussion of this problem 



is given, in which use 

 Contour Integration. 



The equations which 

 as follows : — 



~dt ' 



is made of Cauchy's Theorem in 

 the temperature has to satisfy are 



=/g, .>o, . . . . (1)| 





V 



=/(*), 



t=0, 



• 



. . . (2) 







= 0, 



x = 0, 



• 



• • • (3)) 



We begin 



with the solution for a ! 



source in 



an 



infinite solid, 





V= 2 



1 



(X-X')2 



ikt 





. . . (i) 





/ 7 £ 



X irkt 



This may 



be expressed by 



the inte 



gral, 









i r+« 



27r) 



e — ka-t e ia( 



X - X '\U. 







Fig. 1. 



+ 00 



O 

 The path P in a-plane. 



We transform this into the integral over the path (P) of 



* Communicated by Prof. A. Gray, F.R.S. 



t " On an Application of the Method of Images to the Conduction of 

 Heat,'' Proc. Loud. Math. Soc. vol. xxii. 



J Weber-Riemann, Partielle Differential-Gleichwigen, Bd. ii. § 38. 



