PEIRCE AND WILLSON. THERMAL CONDUCTIVITIES. 7 



It is usually assiiiued that i« continuous at the surface of separation 

 of two isotropic solids of different conductivities. If Hi and n.^ are nor- 

 mals at a [)oint of such a surface drawn into the first and second conduc- 

 tors respectively, aud if the flow of heat is steady, 



9F,{6) 9 FAQ) ,, ^,. . 9A{e) , 9f,(6) 



or Kj^^ [-K'iT^ — — 0. (14) 



If the temperature differences within a body are comparatively slight, 

 we may often use Fourier's assumption and representy" (6^) approximately 

 by itself. As we shall need to compare the solutions of certain simple 

 problems in the steady flow of heat obtained on this hypothesis with the 

 corresponding solutions obtained ou the assumption that f (&) and 6 are 

 not identical, we may note certain facts in passing. It is easy to prove 

 by an elementary application of Green's Theorem that a function, F, 

 which is harmonic within a given closed surface S, and which upon two 

 given portions, Sy and ASg, of S has the constant values Ci and C2 respect- 

 ively, while at every other part of S its normal derivative is zero, is 

 determined by these conditions. If this function has been found, it is 

 easy to write down the unique function 



F's^^5|-' F+^^i^^^-, (15) 



which is harmonic within S, has the constant value C x on *S'i and the 

 constant value 0\ on S^^ and the normal derivative of which vanishes 

 at all points of S which do not belong to Sy oi' ^'i- T^'^ families of sur- 

 faces defined by the equations, F= constant, V = constant, are identi- 

 cal. If, therefore, two given portions of the surface of a solid isotropic 

 conductor in which there is a steady How of lieat be kept at constant 

 temperatures (Ci and G.) while there is no flow across the rest of its 

 surface, the function F, which on Fourier's hypothesis gives the tem- 

 peratures at all points within the solid, is connected with the function 

 K', which gives /(6') on the assumption that this is not identical with 6 

 itself, by means of the equation 



Gy-C, ^ G,-G, ' ^ ^^ 



and the forms of the isothermal surfaces are inde23endent of the form of 

 the function f. 



