( 586 ) 
same metal and are kept at the same temperature, to be brought in 
contact with each other. The potentials pp and pq will then become 
equal, but the stream of electrons » will no longer be 0. We shall 
have on the contrary, denoting by + the normal section, which may 
slowly ‘change from point to point, as has already been observed, 
=P 1°. . 4 Ua 
Taking this into account and using (23), we get from (21) and (80) 
dep Ih ah? md (1 m dlog A 1 
lie Male hee (=) Sn ane 
We shall integrate this along the circuit from P to Q. Since 7 
has the same value everywhere and 
Pp PQ" Vn=Vo : hp ho, 
Q 
m (ldlog A (de 
= da — î ==); 
ZO IE Che o= 
P 
we find 
Here, the first term is reduced to the form (34), if we integrate 
by parts. Hence, if we put 
the result is 
VR Gd 
R 
as was to be expected. Indeed, 6 being the coefficient of conductivity, 
R is the resistance of the circuit. 
$ 12. We shall now proceed to calculate the heat developed in a 
circuit in which there is an electric current 7, or rather, supposing 
each element of the wire to be kept at a constant temperature by 
means of an external reservoir of heat, the amount of heat that is 
transferred to such a reservoir per unit time. Let us consider to this 
effect the part of the circuit lying between the sections whose positions 
are determined by « and w+ dw and let wdt be the work done, 
during the time dt, by the forces acting on the electrons in this 
element. J)’ being the quantity of heat traversing a section per 
unit time, we may write 
d 
(WE) de 
DE 
for the difference between the quantities of heat leaving the element 
at one end and entering it at the other, and the production of heat 
is given by 
