DYNAMICAL THEORY OF HEAT. 127 



This equation may be considered as the mathematical expression of the Second 

 fundamental Law of the Dynamical Theory of Heat. The corresponding expres- 

 sion of the First Law is 



W + J(H 4 + H,;+ .... +H f(n _ 1) + H t(n) ) = .... (2), 



where W denotes the aggregate amount of work spent in producing the operations, 

 and J the mechanical equivalent of the thermal unit. 



§§ 102-106. Initial examination of Thermo-dynamic circumstances regarding 

 Electric Currents in Linear Conductors. 



102. Peltier's admirable discovery that an electric current in a metallic cir- 

 cuit of antimony and bismuth produces cold where it passes from bismuth to 

 antimony, and heat where it passes from antimony to bismuth, shows how an 

 evolution of mechanical effect, by means of thermo-electric currents, involves trans- 

 ference of heat from a body at a higher temperature to a body at a lower temper- 

 ature, and how a reverse thermal effect may be produced, by thermo-electric 

 means, from the expenditure of work. For if a galvanic engine be kept in motion 

 doing work, by a thermo-electric battery of bismuth and antimony ; the current 

 by means of which this is effected passing, as it does, from bismuth to antimony 

 through the hot junctions, and from antimony to bismuth through the cold 

 junctions, must cause absorption of heat in each of the former, and evolution of 

 heat in each of the latter ; and to sustain the difference of temperature required for 

 the excitation of the electro-motive force, even were there no propagation of 

 heat by conduction through the battery, it would be necessary continually, during 

 the existence of the current, to supply heat from a source to the hot junctions, 

 and to draw off heat from the cold junctions by a refrigerator: — Or, if work be 

 spent to turn the engine faster than the rate at which its inductive reaction 

 balances the electro-motive force of the battery, there will be a reverse current 

 sent through the circuit, producing absorption of heat at the cold junctions, and 

 evolution of heat at the hot junctions, and consequently effecting the transference 

 of some heat from the refrigerator to the source. 



103. We see then, that in Peltier's phenomenon we have a reversible thermal 

 agency of exactly the kind supposed in the second Law of the Dynamical Theory 

 of Heat. Before, however, we can apply either this or the first Law, we must 

 consider other thermal actions which are involved in the circumstances of a 

 thermo-electric current ; and with reference to the second Law we shall have to 

 examine whether there are any such of an essentially irreversible kind. 



104. It is to be remarked, in the first place, that a current cannot pass through 

 a homogeneous conductor without generating heat in overcoming resistance. 

 This effect, which we shall call the frictional generation of heat, has been dis- 

 covered by Joule to be produced at a rate proportional to the square of the 



vol. xxi. part. i. 2 m 



