170 PROFESSOR W. THOMSON ON THE 
dt! , at at 
CH (k aa dy We a) 
7 ab dt , at 
n=-(! mt lat 1) pees ee cae 
s=—- (ms Db nl oy nt) 
ro da dy "de 
to express the components {, », 9 of the “flux of heat” at any point of the solid, 
‘dt dt dt 
dx’ dy? dz 
ki, 1, m, k', &e., which may be called the nine coefficients of thermal conductivity of 
the substance ;—and (2) the single equation, 
EE) dime pul ' : 
fete 5 ae HOP HIV) HS MOLI PIIV) + 7 hO'+ig’ sj) | 
3Ea@-$) =H) G64} ao 
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 
ay 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 ¢ 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 ¢ 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, 
in terms of the variations of temperature ( ) multiplied by coefficients 
