412 Dr Bromwich, Symbolical methods 



§ 3. Numerical tests of the formulae. 



§ 4 Evaluation of certain symbolical expressions required in 

 ' §§ 2, 3. 



§1. General consideration of a 7nethod for solving 

 Conduction of Heat problems. 



Using the notation explained in §§ 4, 5 of my paper quoted 

 above, it is evident that the typical solution of a Conduction of 

 Heat problem, with a solid originally at zero temperature, and the 

 surface of the sohd maintained at constant temperature v^, is of 

 the form ^ ,-a+c=o ^x , ^, 



V = ^- e>}u-^ . {a > 0) 



Here u is to reduce to unity at the surface and is to satisfy the 

 differential equation ^^^^ - Xu = 



at other points in the solid, where A.^ is Laplace's operator and 

 K = k/c is the fundamental constant of the heat-equation in the 

 solid; k is the conductivity and c the heat capacity (per unit 

 volume). 



To connect this formula with Prof. Carslaw's {I.e. § 2) is easy; 

 if we write A = — kO^ 



u will become U say, where U satisfies the equation 



and also reduces to unity at the surface. The integral for v 

 becomes r ,q 



v= -^ e-"^'^ U -^ , 



where the path of integration is given by 



— /c (^^ — rf) = a, or yf — ^'^ — cc/k, if 6 = ^ + ltj. 



The beginning and end of the path correspond to the two points 

 given by 



— 2K^7y -^ — 00 , — 2k^7] -^ + oo ; 



thus, choosing the upper half of the rectangular hyperbola 



^2 _ 12 = c//C, 



the path will start at infinity in the first quadrant and will end at 

 infinity in the second quadrant as sketched. 



This path agrees with Carslaw's path (P), and so the integral v 

 is identical with his, when U is written out at length. 



To explain the connexion of these complex integrals with 

 Heaviside's symbolical treatment is also easy; Heaviside writes 

 symbohcally o 



dt 



P = ^,- xf^ 



