PEIRCE AND WILLSON. — THERMAL CONDUCTIVITIES. 5 



If t) v, t are aiuilytic point fuuctious which define a system of orthog- 

 onal curvilinear co-ordiniites, and h^, h,^, h^ are the gradients of these 

 functions, and if q^, 9,,, q^ are the coraponeuts of the heat flux taken at 

 every point normal to the surfaces of constant ^, 7/, ^ which pass ihrough 

 that point, 



For a given material which would be homogeneous if it were at the 

 same temperature throughout, under given pressure conditions, k is as- 

 sumed to be constant, so tliut k' f (6) is a function of the temperature 

 only. This product is called the specific conductivity of the substance 

 under the given circumstances, and is denoted by F' (0) or by k. We 

 may write, therefore, 



9 6 9F(0) 9F(6) 9F(0) 



. ^ = 7^ — - , V = ^-^^ , w . — 



dx d X d y d z 



{-) 



J 3F{6) . 9 Fid) , 9F{6) ^^, 



If a closed analytic surface, aS", be drawn within the solid and if (4^, n), 

 (ij, n), (^, 11) represent the angles between the exterior normal to /S* at 

 any point on it and the directions at that point in which ^, »/, and l, 

 increase most rapidly, the flux of heat across ^from within outward may 

 be written 



/' 



{q^ cos (^, 11) + '7,, • COS {rj, 71) + q^ . cos {i, n) \d S. (5) 



The surface integral, taken over S, of f^cos (^, «), where 6^^ is any 

 function which, with its space derivatives of the first order, is continuous 

 within and upon S, is equal to the volume integral, extended through the 



U 



9\ J J I 

 space enclosed by S, of /t^ . h^ . h^ . \''r, >'ij , so that the flux across 



9$ 

 S may be expressed by the integral 



-^^^ ( 9^ ^~97, '^'^H ) 



If \j/ (0) is the specific heat per unit volume of the body under the 

 given pressure conditions, we may equate the expression just obtained to 



