414 Dr Bromwich, Symbolical methods 



Thus it follows that the poles aj, ag ^''^ purely real; for if a^, a^ 

 are supposed to be conjugate complexes, so also are ^i, j).^\ and 

 thus the product ccfi-j^cji^ is essentially positive, so that the integral 

 fc(f);^<^2 d'^ could not vanish. 



Further we have F = — ^ , and taking the special value ^j 

 for V, we find that 



and (since a^ is not zero) it is evident that a-^ must be negative*. 



The corresponding poles for the functions U will be real (since 

 kB'^ = — A) and will occur in pairs given by ^ = ± v'(~ ^il^)- 



There is no need for the thermal coefficients c and k to be con- 

 stants in these theorems, provided that at any surface of discon- 



tinuity, the functions v and h -^ are continuous, where dfdv impHes 



differentiation along the normal to the surface of discontinuity. 



It will be noticed that the theorem in § 5 of Prof. Carslaw's paper 

 follows at once, because c has one of two constant values; and the 

 ratio of these constants is equal to 



^2 ^ 



2 I ^X ^1 / 1 



Hence I r2</)i</>2 dr + - I r^^j^a^^' = 0, 



Jo o- J a 



which is equivalent to the result obtained by Prof. Carslaw. 



§2. Application of Heaviside's method. 



We can illustrate Heaviside's method conveniently by solving 

 symbolically the problem of a sphere of radius a and heat-con- 

 stants Jc, c, surrounded by a thin shell of thickness I ^ b — a and 

 heat-constants yl'i, c^. The solid so formed is initially at zero 

 temperature, and at a certain instant {t = 0) the outer surface is 

 raised to temperature Vq and is maintained at this temperature 

 afterwards. 



We shall write ^ ^ j^f^^ k.^JcJc^, 



and ^^dt^ ''^' ='<i^i^- 



Then suppose that V reduces to the value A at r =^ a; the 

 differential equations of the problem reduce to 



l^,{rV) = f{rV), ~{rV,)^q,^rV,), 



* It should be noticed that Ave cannot djaw this conckision immediately from 

 the corresponding theorem of § 9 of my L.M.S. paper. 



