132 PROFESSOR W. THOMSON ON THE 



the coefficient A, in the expression- A 7 + B7 2 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 t 7 the heat absorbed in all parts of the 

 circuit which are at the temperature t, by the action of a current of infinitely 

 small strength 7 : so that the term- A 7, 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 



A 7= 2 0,7, 

 which gives 2 a,=A ....... (6) ; 



and, if F denote the value of the electro-motive force required to balance the thermo- 

 electric tendency, we have 



F=J2a ( (7). 



The Second General Law, as expressed above in equation (1), applied to the pre- 

 sent circumstances, gives immediately, 



2^ = (8) 



or, since 7 is the same for all terms of the sum, 



2y-=0 . . . . ' . . (9). 



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 Joule 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 



