Alleralwns of Temperalure in Chani/es of Pressure in Fluids. 541 



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 ^t denote " Carnot's 

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

 temperature, must be equal to 



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 Avhat it was 

 at the end of operation (1) to a temperature higher by dt, or on 

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

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

 exceeds the average tem})erature while heat is given out, by just as 

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

 therefore, this be denoted by Q, and if 'Kdt denote the quantity of 

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

 the whole cycle of operations must be e([ual to /t K df, and Iience 

 ^^'chave jttfl . Krf^=CT e^<7. 



From this we find ^ 



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

 elevation of temperature consequent on a sudden augmentation of 

 pressure from p to j^ + to'; e is the coefticient of expansion of the 

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

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

 scale of temperature, simply the reciprocal of the temperature, mul- 

 tiplied by the mechanical equivalent of the thermal unit. If then 

 t denote the absolute temperature, which v\'c have shown by ex- 

 periment* agrees sensibly with tenipei'ature by the air-thermometer 

 Cent, with "17^° added, and if J denote the mechanical equivalent 

 of the thermal unit Centigrade, we have 



t e 

 JlV 



This expression agrees in reality, but is somewhat more conve- 

 nient in form, than that first given, Dynamical Theory of Heat, § 49, 

 Phil. Mag. for September 1852. 



Thus for water, the value of K, the thermal capacity of a cubic 

 foot under constant pressure, is 63'4 17, and e varies from to about 

 2 -jL^, for temperatures rising from that of maximum density to 50° 

 Cent., and the elevation of temperature produced by an augmentation 

 of pressure amounting to n times 211 7 lbs. per square foot (that is 



* Sec Part II. of our Pajjcr " On the Tlicniial Etfects of Fluids la Motion," 

 I'hilotiuphical Traneactions, 1851. 



