567 



expansion at temperature t and pressure p, when we regard mr as 

 infinitely small, we have an amount of work equal to 



inedt 



gained from the cycle. The case of a fluid such as water below 

 39' 1 Fahr., which contracts under constant pressure, with an ele- 

 vation of temperature, is of course included by admitting negative 

 values for e, and making the corresponding changes in statement. 



Since the fluid is restored to its primitive physical condition at 

 the end of the cycle, the source from which the work thus gained 

 is drawn, must be heat, and since the operations are each perfectly 

 reversible, Carnot's principle must hold; that is to say, if denote 

 the excess of temperature of the body while taking in heat above 

 its temperature while giving out heat, and if p denote " Camot's 

 function,' ' the work gained, per unit of heat taken in at the higher 



temperature, must be equal to 



p* 



But while the fluid is giving out heat, that is to say, during operation 

 (4), its temperature is sinking from t-\-dt to t, and may be regarded 

 as being on the average t + \dt ; and while it is taking in heat, that 

 is, during operation (2), its temperature is rising from what it was 

 at the end of operation (I). to a temperature higher by dt, or on 

 the average exceeds by ^dt, the temperature at the end of operation 

 (1). The average temperature while heat is taken in consequently 

 exceeds the average temperature while heat is given out, by just as 

 much as the body rises in temperature during operation (1). If, 

 therefore, this be denoted by 0, and if K dt denote the quantity of 

 heat taken in during operation (2), the gain of work from heat in 

 the whole cycle of operations must be equal to p K dt, and hence 

 we have 



pd.Kdt=vredt. 

 From this we find e 



0= -or 

 /iK 



where, according to the notation that has been introduced, is the 

 elevation of temperature consequent on a sudden augmentation of 

 pressure from p to p -f -us ; e is the coefficient of expansion of the 

 fluid, and K its capacity for heat, under constant pressure ; and p. 

 is Carnot's function, being, according to the absolute thermodynamic 



