222 Prof. Thomson on the Dynamical Theoi-y of Heat. 



reversible, gave a different, even an enormously different, and 

 either a greater or a less proportion of heat converted into work 

 to heat taken in than that law requires in all completely rever- 

 sible pi'ocesses. Still the reversible part of the agency, in the 

 thermo-electric circumstances we have supposed, is in itself so 

 perfect, that it appears in the highest degree probable it may be 

 found to fulfil independently the same conditions as the general 

 law would impose on it if it took place unaccompanied by any 

 other thermal or thermo-dynamic process. 



§§ 107-111. Mathematical expression of the lliermo-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 

 frictionally, increased or diminished by the quantities produced 

 or absorbed in the different parts by action depending on hete- 

 rogeneousness of the circuit. The formei-, according to the law 

 discovered by Joule, may be represented by a term 67^, in which 

 B denotes a constant depending 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 — Ay ; 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 deve- 

 loped in any portion of a heterogeneous metallic conductor in a 

 unit of time must be expressible by the formula 



-A7 + By^ 

 where B is essentially positive ; but A may be positive, negative, 

 or zero, according to the nature of the different parts of the con- 

 ducting arc. It may be assumed with great probability, that 

 the quantities A and B are absolutely constant for a given con- 

 ductor with its different parts at given constant temperatures ; 

 and that when the temperatm-es of the different parts of a con- 

 ductor 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 pre- 

 vent the interior parts of the conductor from varying in tempe- 

 rature, and so changing in their resistance to the 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 sufficiently feeble currents. 



