Prof. Thomson on the Dynamical Theory of Heat. 445 



for every point of the solid, subject to whatever conditions may 

 be prescribed for the bounding surface, and so to complete the 

 problem of finding the motion of electricity across the body in 



its actual circumstances ; provided the values of -7-, ^-, -r- are 



ax ay az 

 known, as they will be when the distribution of temperature is 

 given. We may certainly, in an electrical problem such as this, 

 suppose the temperature actually given at every point of the solid 

 considered, since we may conceive thermal sources distributed 

 through its interior to make the temperature have an arbitrary 

 value at every point. 



] 77. Yet practically the temperature will, in all ordinary cases, 

 follow by conduction from given thermal circumstances at the 

 surface. The equations of motion of heat, by which, along with 

 those of thermo-electric force, such problems may be solved, are 

 as follows: — (1), three equations, 



(111 ^ A. 1 '^^ II ^^ \ 



dx dy dz) 



. ( ,dt , „dt , dt\ I 



(48) 



to express the components f, ij, ^ of the "flux of heat" at any 

 point of the solid, in terms of the variations of temperature 

 /dt dt dt\ ,.,.,, 

 \^' du' dz) "^"^tiP^^^*^ "^y coefficients k, I, m, k', &c., which 



may be called the nine coefficients of thermal conductivity of the 

 substance ; and (2), the single equation 



d^ dn dd_ 

 dx dy dz 



of which the first member expresses the rate at which heat flows 

 out of any part of the solid per unit of volume ; and the second 

 member, to which it is equated, the resultant thermal agency 

 (positive when there is, on the whole, evolution at xys) produced 

 by the electric currents. 



178. The general treatment of these eleven equations, (45), 

 (46), (47), (48), (49), leads to two non-linear partial differential 



