158 PROFESSOR W. THOMSON ON THE 



ture in an infinitely small space dx across the bar in the plane of the diagram, 

 and % an unknown element, constant or a function of the temperature, depending 

 on the nature of the substance, we may assume 



. dt_ 

 %x Tx 



as the amount of absorption, per unit of the volume of the bar, due to a current of 

 intensity i, by means of the new agency. The whole amount in a lamina of 

 thickness d x, length I, and breadth a perpendicular to the plane of the diagram, 

 is therefore 



i y •=- a I dx, 

 " dx 



or 7 £ x dt- 



As there cannot possibly be any other reversible thermal agency to be taken into 

 account, we may now assume 



2H, = 7 \ [ [n (t) - o (t')] +fj x dt } . . (22), 



XT 



The second General Law showing that 2 -=* must vanish, gives, by the second 

 of these equations, 



affl_££>+ r T Xdt = o . . . (24). 



Substituting, in place of t, t, and differentiating with reference to this variable, 

 we have, as an equivalent equation, 



, n ,n 



x_ = _Z±__L .... (25); 

 t dt dt 



and using this in (22), we have 



2H <=nX > ■ ■ w- 



This expresses the full amount of heat taken in through the agency of the cur- 

 rent 7 ; of which the mechanical equivalent is therefore the work done by the 

 current. Hence (according to principles fully explained above) the thermal cir- 

 cumstances actually cause an electro-motive force F, of which the amount is given 

 by the equation 



F = J l T f T -dt (27), 



to act along the bar from left to right of the diagram ; which will produce a cur- 

 rent, unless balanced by an equal and contrary reaction. This result both esta- 



