130 PROFESSOR W. THOMSON ON THE 



§§ 107-111. Mathematical expression of the Thermo -dynamic circumstances of 



Currents in Linear Conductors. 



107. In a heterogeneous metallic conductor the whole heat developed in a 

 given time, will consist of a quantity generated j rictionally , increased or diminished 

 by the quantities produced or absorbed in the different parts by action depending 

 on heterogeneousness of the circuit. The former, according to the law discovered 

 by Joule, may be represented by a term B 7 2 > in which B denotes a constant de- 

 pending only on the resistance of the circuit. The latter, being reversible with the 

 current, may be assumed, at least for infinitely feeble currents, to be, in a given 

 conductor, proportional simply to the strength of the current ; and hence, the 

 whole quantity of heat evolved in a given time, must be expressible by a term of 

 the form -A 7, where A, whether it varies with 7 or not, has a finite, positive, or 

 negative value, when 7 is infinitely small. Hence, the whole heat developed in 

 any portion of a heterogeneous metallic conductor in a unit of time, must be 



expressible by the formula 



- A 7 + B 7 2 ; 



where B is essentially positive, but A may be positive, negative, or zero, according 

 to the nature of the different parts of the conducting arc. It may be assumed, with 

 great probability, that the quantities A and B are absolutely constant for a given 

 conductor with its different parts at given constant temperatures, and that when 

 the temperatures of the different parts of a conductor are kept as nearly constant 

 as possible with currents of different strengths passing through it, the quantities 

 A and B can only depend on 7, inasmuch as it may be impossible to prevent the in- 

 terior parts of the conductor from varying in temperature, and so changing in their 

 resistance to conduction of electricity, or in their thermo-electric properties. In 

 the present paper, accordingly, A and B are assumed to depend solely on the nature 

 and thermal circumstances of the conductor, and to be independent of 7 ; but the 

 investigations and conclusions would be applicable to cases of action with suf- 

 ficiently feeble currents, probably to all currents due solely to the thermal electro- 

 motive force, even if A and B were in reality variable, provided the limiting values 

 of these quantities for infinitely small values of 7 be used. 



108. Let us consider a conductor of any length and form, but of comparatively 

 small transverse dimensions, composed of various metals, at different temperatures, 

 but having portions at its two extremities homogeneous, and at the same temper- 

 ature. These terminal portions will be denoted by E and E', and will be called 

 the principal electrodes, or the electrodes of 'the principal conductor ; the conductor 

 itself being called the principal conductor to distinguish it from others, either 

 joining its extremities or otherwise circumstanced, which we may have to con- 

 sider again. 



