150 PROFESSOR W. THOMSON ON THE 



§§ 138-140. General Lemma, regarding relative thermo-electric properties of 

 Metals, and multiple combinations in a Linear Circuit. 



138. The general equation (11), investigated above, shows that the aggregate 

 amount of all the the7*mal effects produced by a current, or by any system of currents, 

 in any solid conductor or combination of solid conductors must be zero, if all the 

 localities in which they are produced are kept at the same temperature. 



Cor. 1. If in any circuit of solid conductors the temperature be uniform from 

 a point P through all the conducting matter to a point Q, both the aggregate 

 thermal actions, and the electro-motive force are totally independent of this inter- 

 mediate matter, whether it be homogeneous or heterogeneous, crystalline or non- 

 crystalline, linear or solid, and is the same as if P and Q were put in contact. [The 

 importance of this simple and elementary truth in thermo-electric experiments of 

 various kinds is very obvious. It appears to have been overlooked by many expe- 

 rimenters who have scrupulously avoided introducing extraneous matter (as solder) 

 in making thermo-electric junctions, and who have attempted to explain away 

 Cumming's and Becquerel's remarkable discovery of thermo-electric inversions, 

 by referring the phenomena observed to coatings of oxide formed on the metals 

 at their surfaces of contact.] 



Cor. 2. If n (A, B) , n (B, C) , n (C, D) , n (Z, A) denote the amounts of 



the Peltier absorption of heat per unit strength of current per unit of time, at the 

 successive junctions of a circuit of metals A, B, C, . . . . Z, A, we must have, 



n (A, B) + n (B, C) + + n (Z, A) = . 



Thus if the circuit consist of three metals, 



n (A, B) + n (B, C) + n (C, A) = 0; 

 from which, since n (C, A) = - n (A, C), we derive 



n (B, C) = n (A, C) - n (A, B). 



139. Now, by (19) above, the electro-motive force in an element of the two 

 metals (A, B), tending from B to A through the hot junction, for an infinitely small 



difference of temperature t, and a mean absolute temperature t, is ~ — - t , 



and so for every other pair of metals. Hence, if <p (A, B) , (B, C) , &c, denote 

 the quantities by which the infinitely small range t must be multiplied to get the 

 electro-motive forces of elements composed of successive pairs of the metals in the 

 same thermal circumstances, we have 



(f) (A, B) + $ (B, C) + + </> (Z, A) = ; 



and, for the case of three metals, 



</> (B, C) = (p (A, C)-4> (A, B). 



Since the thermo-electric force for any range of temperature is the sum of the 

 thermo-electric forces for all the infinitely small ranges into which we may divide 



