DYNAMICAL THEORY OF HEAT. 151 
1 
the whole range (being, as proved above, equal to J ¢ dt), in the case of each ele- 
Au 
ment, the theorem expressed by these equations is true of the thermo-electric 
forces in the single elements for ail ranges of temperature, provided the absolute 
temperatures of the hot and cold junctions be the same in the different elements. 
The second equation, by successive applications of which the first may be derived, 
is the simplest expression of a theorem which was, I believe, first pointed out and 
experimentally verified by Becqueret in researches described in the second volume 
of his Traité d’Electricité. 
140. For brevity, we shall call what has been denoted by ? (B, C) the thermo- 
electric relation of the metal B to the metal C; we shall call a certain metal (per- 
haps copper or silver) the standard metal; and if A be the standard metal, we 
shall call ¢ (A, B) the thermo-electric power of the metal B. The theorem expressed 
by the last equation may now be stated‘ thus: The thermo-electric relation betiveen 
two metals is equal to the difference of their thermo-electric powers ; which is nearly 
identical with BecquEREL’s own statement of his theorem. 
§§ 141-146. Elementary Explanations in Electro-cinematics and Electro-mechanics. 
141. When we confined our attention to electric currents flowing along linear 
conductors, it was only necessary to consider in each case, the whole strength of 
the current, and the longitudinal electro-motive force in any part of the circuit, 
without taking into account any of the transverse dimensions of the conducting 
channel. In what follows, it will be frequently necessary to consider distributions 
of currents in various directions through solid conductors, and it is therefore con- 
venient at present to notice some elementary properties, and to define various 
terms, adapted for specifications of systems of electric currents and electro-motive 
forces, distributed in any manner whatever throughout a solid. 
142. It is to be remarked, in the first place, that any portion of a solid traversed 
by current electricity may be divided, by tubular surfaces coinciding with lines of 
electric motion, into an infinite number of channels or conducting arcs, each con- 
taining an independent linear current. The strength of a linear current being, as 
before, defined to denote the quantity of electricity flowing across any section in 
the unit of time, we may now define the intensity of the current, at any point of a 
conductor, as the strength of a linear current of infinitely small transverse dimen- 
sions through this point, divided by the area ofa normal section of its channel. The 
elementary proposition of the composition of motions, common to the cinematics 
of ordinary fiuids and of electricity, shows that the superposition of two systems 
of currents in a body gives a resultant system, of which the intensity and direc- 
tion at any point are represented by the diagonal of a parallelogram described 
upon lines representing the intensity and direction of the component systems 
VOL. XXI. PART I. 28 
