DYNAMICAL THEORY OF HEAT. 135 



The general principles are most conveniently applied by restricting the num- 

 ber of metals referred to in the general equations to two ; a case which we accord- 

 ingly proceed to consider. 



115. Let the principal conductor consist of two metals, one constituting the 

 middle, and the other the two terminal portions. Let the junctions of these por- 

 tions next the terminals E, E' be denoted by A, A' respectively, and let their tem- 

 peratures be T, T'. Let also n (T), - n (T') be the quantities of heat absorbed at 

 them per second by a current of unit strength. We should have 



n(T) = n(T'), 



if the temperatures were equal, since the Peltier phenomenon consists, as we 

 have seen, of equal quantities of heat evolved or absorbed, according to the 

 direction of a current crossing the junction of two different metals ; and if these 

 quantities be not actually equal, we may consider them as particular values of a 

 function n of the temperature, which depends on the particular relative thermo- 

 electric quality of the two metals. Accordingly, the preceding notation is reduced 

 to n - 2, Tj = T, T 2 = T', u 1 =n (T), n 2 = - n (T') ; and we have 



/* T o r T i /~t 2 r r 



I a l dt+ 1 (T 2 dt+I a- 1 dt=f (a-. — cr^dt, 

 */t, Jt 2 u t Jt 



and similarly for the integral involving - . Hence the general equations become 



v 



F=j{n(T)-n(T)+/ / (^-o-.Jd*] (13) 



£(L)_5£H + ^ T £^ ( =o (U) 



If in the latter equation we substitute t for T, and differentiate with reference to 

 this variable, we have, as an equivalent equation, 



/ TT \ 



d< 



(15) 



V t ) 



+ ^ 



t 



-.0 



dt 



°"l-<3" 2 



_ n 



du 

 ~~dt 





or ^~^= T - Wt • ... Q6) 



This last equation leads to a remarkably simple expression for the electro-motive 

 force of a thermo-electric pair, solely in the terms of the Peltier evolution of 

 heat at any temperature intermediate between the temperatures of its junctions; 

 for we have only to eliminate by means of it (<r, -<r 2 ) from (13), to find 



F=jf rS! ~dt (17) 



J T t 



116. Let us first apply these equations to the case of a thermo-electric pair, 

 VOL. XXI. part i. 2 o 



