170 PROFESSOR W. THOMSON ON THE 



y - _ (h — y ( h. i" C H \ 

 \ dm ' dy dz ) 



\ dx dy dz ) 



• (48). 



— — \ m' —^ + m" ^- + m 

 \ da 



dt ., dt dt 



x dy dz 



to express the components £, % 5 of the "flux of heat" at any point of the solid, 



in terms of the variations of temperature r -r- , j=- , -r- ] multiplied by coefficients 



k, I, m, k\ &c, which may be called the nine coefficients of thermal conductivity of 

 the substance ; — and (2) the single equation, 



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 

 x y z 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 equations of the second order and 

 degree for the determination of the functions t and V. 



179. It may be remarked, however, that the second term of the second mem- 

 ber of (49), when the prefixed negative sign is removed, expresses the frictional 

 generation of heat by currents through the solid, and will, therefore, when 

 the electro-motive forces in action are solely thermo-electric, be very small, even 

 in comparison with the reversible generation and absorption of heat in various 

 parts of the circuit, provided the differences of temperature between these different 

 localities are small fractions of the temperature, on the absolute scale from its 

 zero. Excepting then cases in which there are wide ranges (for instance, of 

 50° Cent, or more) of temperature, the second principal term of the second member 

 of (49) may be neglected, and the partial differential equations to which t and V 

 are subject will become linear; so that one of the unknown functions may be 

 readily eliminated, and a linear equation of the fourth order obtained for the de- 

 termination of the other. 



180. Farther, it may be remarked that probably in most, if not in all known 

 cases, the reversible as well as the frictional thermal action of the currents, when 

 excited by thermo-electric force alone, is very small in comparison with that of 

 conduction, perhaps quite insensible. [See above, § 106.] Hence, except when 

 more powerful electro-motive forces than the thermo-electric forces of the solid 

 itself and of its relation to the conductors touching it at any part of its surface, 



