376 



M. CLAPEYRON ON THE MOTIVE POWER OF HEAT. 



We thus see that {p v) is given by a series'of terms, each of which 

 is obtained bj' means of the preceding one, by differentiating it in re- 



dj 



spect to V, multiplying by the ratio 3^, and integrating the result in 



dV 

 dv 



,dp 



respect of J9. Thefirstterm of this series being / 6?T, it is evident that 



dv 

 the value of ^ may be easily obtained ; substituting this value in the 

 equation (1), we have for the expression of the general integral of the 

 partial differential equation 



dO-dT _ dQdJ _ ^ 



dv dp dp dv 



the formula 



Q, = F (T) - C 



MP 



dT 

 d V 



A 

 / 



dp 



+ r^v 



- fdT 



rf_T 



dp_J:_ 

 dl d 

 dv 



dT dT 



dp d p dp 



XT 'd7/'^P d_T 

 dv dv 



J- rirt: 



dv I -r- 

 kJ d V 



+ 



This equation gives the law of the specific calorics, and of the heat 

 disengaged by the variations of the volume and of the pressure of all 

 the substances of nature, when the relation which exists between the 

 temperature, the volume, and the pressure is known. 



