150 PROFESSOR W. THOMSON ON THE 
§§ 138-140. General Lemma, regarding relative thermo-electric properties of 
Metals, and nultiple combinations in a Linear Circuit. 
138. The general equation (11), investigated above, shows that the aggregate 
amount of all the thermal effects produced by a current, or by any system of currents, 
in any solid conductor or combination of solid conductors must be zero, tf all the 
localities in which they are produced are kept at the same temperature. 
Cor. 1. Ifin 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 
Cummine’s and BrcquEreE’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. | 
Cony 2. lr m(AlPB) 5 n0(Be ©), nr (O:mD) ha pean s 1 (Z, A) denote the amounts of 
the Pettier 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, 
M(AyB)y Lbs) eee +1 (Z, A) =0. 
Thus if the circuit consist of three metals, 
m (A, B) + 1(B, C) + m(C, A) =0; 
from which, since m (C, A) = — 11 (A, C), we derive 
mi (B, C) = m1 (A, C) — m1 (A, B). 
139. Now, by (19) above, the electro-motive force in an element of the two 
inetals (A, B), tending from B to A through the hot junction, for an infinitely small 
difference of temperature 7, and a mean absolute temperature ¢, is a T, 
and so for every other pair of metals. Hence, if p (A, B), ¢ (B, C), &e., denote 
the quantities by which the infinitely small range 7 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 
p (A, B) + $(B, C)+...... +  (Z, A)=0; 
and, for the case of three metals, 
p (B,C) = $ (A, C) — 9 (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 
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