the sums in these formulae relating to all elements of the closed 
circuit we have examined in $ 11. Now, by (42), these formulae 
become 
i da 
2e h dz 
P 
and 
Q 
al; 1 dlog A et, 
AT da laks 
P 
The first of these equations is identical with (34) and the second 
holds because 47’ has everywhere the same value. 
It must also be noticed that the formula (35) implies the existence 
of a thermo-electric series and the well known law relating to it. 
This follows at once from the fact that the value (85) may be 
written as the difference of two integrals depending, for given 
temperatures of the junctions, the one on the properties of the first 
and the other on those of the second metal. Denoting by Illa third 
metal, we may represent by /y 77, Fi, mi Fi, the electromotive 
forces existing in circuits composed of the metals indicated by the 
indices, the junctions having in all these cases the temperatures 7” 
and 7" and the positive direction being such that it leads through 
the junction at the first temperature from the metal indicated in 
the first towards that indicated in the second place. Then it is seen 
at once that 
Epa + Prime Er Or EEEN 
Strictly speaking there was no need to prove this, as it is a con- 
sequence of the thermodynamic equations and our results agree 
with these. 
§ 15. In what precedes we have assumed a single kind of free 
electrons. Indeed, many observations on other classes of phenomena 
have shown the negative electrons to have a greater mobility than 
the positive ones, so that one feels inclined to ask in the first place 
to what extent the facts may be explained by a theory working 
with only negative free electrons. 
Now, in examining this point, we have first of all to consider the 
absolute value of the electromotive force /’. If we suppose the tem- 
peratures 7" and 7” to differ by one degree and if we neglect the 
