92 



vary in temperature^ the heat generated in a given time is expressible 

 by the formula 



Ay + B/ 



where A, which may he either positive or negative^ and B, which is 

 essentially positive, denote quantities independent of y. 



The fundamental equations of the theory are the following : — 



F 7 = J ^2 a +B 72J {a) 



2 a^=2 a fl-e"^^''^'^*) .... (6) 



where F denotes the electromotive force (considered as of the same 

 sign with y, when it acts in the direction of the current) which must 

 act to produce or to permit the current y to circulate uniformly 

 through the conductor ; J the mechanical equivalent of the thermal 

 unit ; a 7 the quantity of heat evolved in the unit of time in all 

 parts of the conductor which are at the temperature t when y is in- 

 finitely small ; ih "Carnot's function"* of the temperature ^ ; T the 

 temperature of the coldest part of the circuit ; and 2 a summation 

 including all parts of the circuit. 



The first of these equations is a mere expression of the equi- 

 valence, according to the principles established by Joule, of the work, 

 F 7,f done in a unit of time by the electromotive force, to the heat 

 developed, which, in the circumstances, is the sole effect produced. 

 The second is a consequence of the first and of the following equa- 

 tion : — 



^. 7 = ^ 2 a 7. (« — T) (c) 



where <p denotes the electromotive force when 7 is infinitely small, 

 and when the temperatures in all parts of the circuit are infinitely 

 nearly equal. This latter equation is an expression, for the pre- 

 sent circumstances, of the proposition^ (first enunciated by Carnot, 

 and first established in the dynamical theory by Clausius) that 



* The values of this function, calculated from Regnault's observations, and 

 the hypothesis that the density of saturated steam follows the " gaseous laws," 

 for every degree of temperature from 0° to 230° cent., are shewn in Table I. of 

 the author's " Account of Carnot's Theory," Transactione, vol. xvi., p. 541. 



t See Philosophical Magazine, Dec. 1851, " On Applications of the Principle 

 of Mechanical Eifect," &c. 



X " Dynamical Theory of Heat" (Transactions, vol. xx., part ii.) Prop. II., &c. 



