106 Prof. Thomson on the Dynamical Theory of Heat. 



number of perfect engines, each working within an infinitely 

 small range of temperature, and arranged in a series of which 

 the source of the first is the given source, the refrigerator of the 

 last the given refrigerator, and the refrigerator of each interme- 

 diate engine is the source of that which follows it in the series. 

 Each of these engines Avill, in any time, emit just as much less 

 heat to its refrigerator than is supplied to it from its source, as 

 is the equivalent of the mechanical work which it produces. 

 Hence if t and t + dt denote respectively the temperatures of the 

 refrigerator and source of one of the intermediate engines, and 

 if q denote the quantity of heat which this engine discharges 

 into its refrigerator in any time, and q + dq the quantity which 

 it draws from its source in the same time, the quantity of work 

 which it produces in that time will be idq according to Prop. I., 

 and it will also be q^idt according to the expression of Prop. II., 

 investigated in § 21 ; and therefore we must have 



idq = qju-dt. 



Hence, supposing that the quantity of heat supplied from the 

 first somxe, in the time considered is H, we find by integration 



But the value of q, when / = T, is the final remainder discharged 

 .into the refrigerator at the temperatm-e T ; and therefore, if this 

 be denoted by R, we have 



H 1 /^s 



from which we deduce 



R = He-T/T'''^' (6). 



Now the whole amount of work produced will be the mechanical 

 equivalent of the quantity of heat lost ; and, therefore, if this be 

 denoted by W, we have 



W = J(H-R) (7), 



and consequently, by (6), 



W = JH{l-e~T/l.V/} . . . (8). 



26. To compare this with the expression H / fidt, for the 



duty indicated by Carnot's theory*, we may expand the expo- 

 nential in the preceding equation, by the usual series. We thus 



* "Account," &c.. Equation 7, § 31, 



