132 PROFESSOR W. THOMSON ON THE 
the coefficient A, in the expression—A ‘y+B-+? for the heat developed throughout 
any given conductor, depends. 
110. To express the Second General Law, we must take into account the tem- 
peratures of the different localities of the circuit in which heat is evolved or ab- 
sorbed, when the current is kept so feeble (by the action of the electro-motive force 
P, against the thermo-electric force of the system), as to render the frictional gene- 
ration of heat insensible. Denoting then by a, y the heat absorbed in all parts of the 
circuit which are at the temperature ¢, by the action of a current of infinitely 
small strength Y: so that the term—A ‘y, expressing the whole heat generated not 
frictionally throughout the principal conductor in any case, will be the sum of all 
such terms with their signs changed, or 
AY=24,, 
which gives BQ=A 2. . SENG) 
and, if F denote the value of the electro-motive force Junie to vps the thermo- 
electric tendency, we have 
B= J Diag &: ; - ‘ 3 ; (7). 
The Second General Law, as expressed above in equation (1), applied to the pre- 
sent circumstances, gives immediately, 
i Ac aml 
or, since Y is the same for all terms of the sum, 
s=0 Rebs tha: ela Geen cian ee 
111. Of these equations, (7), and (3) from which it is derived, involve no 
hypothesis whatever, but merely express the application of a great natural law,— 
discovered by JoutE for every case of thermal action whether chemical electrical 
or mechanical,—to the electrical circumstances of a solid linear conductor, having 
in any way the property of experiencing reverse thermal effects from infinitely 
feeble currents in the two directions through it. Equation (9) expresses the 
hypothetical application of the Second General Law discussed above in § 106. 
The two equations, (7) and (9), express all the information that can be derived 
from the General Dynamical Theory of Heat, regarding the special thermal 
and electrical energies brought into action by inequalities of temperature, or 
by the independent excitation of a current, in a solid linear conductor whether 
crystalline or not. The condition that the circuit is to be linear, being merely one 
of convenience in the initial treatment of the subject, may of course be removed 
by supposing linear conductors to be put together, so as to represent the circum- 
stances of a solid conductor of electricity, with any distribution of electric currents 
whatever through it; and we may therefore regard these two equations as the 
Fundamental Equations of the Mechanical Theory of Thermo-electric Currents. 
To work out the theory for crystalline or non-crystalline conductors, it is necessary 
