in the theory of Conduction o^ Heat 413 



and then solves the equation 



subject to the condition F = 1 at the surface. 



It is evident that the function V will be equivalent to U on 

 writing 6 = iq, or q = — td, and thus Heaviside's forms of the 

 solutions can be translated at once into complex integrals, if 

 desired* ; and it has been proved (see § 4 of my L.M.S. paper) that in 

 general Heaviside's standard method of interpreting his symbolical 

 solutions is equivalent to the evaluation of the original complex 

 integral as a sum of residues (taken for all the poles of the func- 

 tion u/X). 



Before leaving these general considerations it will be con- 

 venient to note briefly the theorems relating to the special differ- 

 ential equation of Diffusion which follow immediately on the lines 

 of §§ 8, 9 in my paper previously quoted. 



The potential energy will be expressible in the form 

 ^ 1 1% (/dvy /dvV /M^) J 



and the dissipation-function will be 



F=lci^^ydr, 



Ml 



both integrals being taken through the volume of the solid; and 

 liere there is no kinetic energy function. 



Thus, if A = «!, ^2 ^■re two distinct poles of the function u, 

 and if </>!, <^2 ^-re the corresponding residues of the function, we 

 find (as in formula (75) in § 9 of my L.M.S. paper) 



jc^^^Jr^O ana Hm-^'-^'i^'tfh^O. 



* Since this relation gives q = r] -i^, it follows that the path selected above 

 •corresponds to one in which the real pari of q is positive; and this agrees with the 

 convention adopted in § 4 below. 



