604 Prof. H. S. Carslaw on Bronnvicli s Method of 



problems in linear flow the method which Bromwich's work 

 has led me to adopt in the discussion of a lnrge class of pro- 

 blems in the Conduction of Heat. It seems to me easier to 

 build up the required solution by integrating a suitable solu- 

 tion over a certain standard path in the plane of the complex 

 variable. The proper particular solution will be easily ob- 

 tained after a little practice in the method. 



The problems in §§ 2, 3, and 4 have been chosen for their 

 simplicity. They are, of course, elementary and solvable by 

 the ordinary methods. The problem in § 5 is here solved 

 for the first time. The corresponding cases for the semi- 

 infinite rod, and the sphere composed of two different 

 materials, can be treated in the same way. These two 

 questions are referred to by Heaviside * in his discussion 

 of the Age of the Earth, following upon Perry's criticism f 

 of Kelvin's classical treatment of this subject. 



2. Rod of length a. The ends x = and x = a kept at 

 temperatures zero and v respectively. The initial 

 temperature zero. 



Let the temperature at the point x at the time t be v. 

 Then the equations are as follows :— 



~dv ~d 2 v ' 



ST =A: =r— 9, 0<x<a (1) 



v=0, when x = (2) 



v=v , when x = a (3) 



v=0, when £ = (4) 



Fig. 1. 



+ oo 



The path (P) in the a -plane. 

 Consider the value of v given by the integral 

 _ v Q f* sin ax e~ Ka4t 

 V -^)^m^ccc~~ da • ' • • • (5) 

 over the path (P) of fig. 1 in the a-plane. In this path the 

 * Heaviside, ' Electromagnetic Theory,, Vol. II. Chapter v. Cf. §§229 

 t ' Nature,' li. p. 224 et sea. (1895). 



