530 Royal Society of Edinburgh. 



electrical generation of heat in a homogeneous metallic conductor, 

 suggests the following assumption, which is the foundation of the 

 theory at present laid before the Royal Society. 



When electricity passes in a current of uniform strength y through 

 a heterogeneous linear conductor, no part of which is permitted to vary 

 in temperature, the heat generated in a given time is expressible by the 

 formula 



Ay+By 2 , 



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

 essentially positive, denote quantities independent of y. 



The fundamental equations of the theory are the following : — 



Fy=J(y2a ( + By a ) (a) 



laf*lafl — **£**). (b) 



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 ; cc t y 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 ; fi " Carnot's function" of the temperature t * ; 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 equivalence, 

 according to the principles established by Joule, of the work, F yf, 

 done in a unit of time by the electromotive force, to the heat deve- 

 loped, which, in the circumstances, is the sole effect produced. The 

 second is a consequence of the first and of the following equation : — 



<p.y=fx-Ecc y.(t-T), . ■ . -. (c) 



where <p denotes the electromotive force when y 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 present 

 circumstances, of the proposition]; (first enunciated by Carnot, and 

 first established in the dynamical theory by Clausius) that the ob- 

 taining of mechanical effect from heat, by means of a perfectly re- 

 versible arrangement, depends in a definite manner on the transmis- 

 sion of a certain quantity of heat from one body, to another at a lower 

 temperature . There is a degree of uncertainty in the present appli- 

 cation of this principle, on account of the conduction of heat that 

 must necessarily go on from the hotter to the colder parts of the 



* 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 shown in 

 Table I. of the author's " Account of Carnot's Theory," Transactions, vol. 

 xvi. p. 541. 



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

 ciple of Mechanical Effect," &c. 



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

 II. &c. 



