126 POPULAR SCIENCE MONTHLY. 



which in the final result (i. e., upon the completion of the cycle) the transfer 

 of caloric occurs. 



When a gas passes without change of temperature from one definite volume 

 and pressure to another, the quantity of caloric absorbed or emitted is always 

 the same, irrespective of the nature of the gas chosen for the experiment. 



The difference between the specific heat under constant pressure and the 

 specific heat under constant volume is the same for all gases. 



When the volume of a gas increases in geometrical progression its specific 

 heat increases in arithmetical progress. 



Of course these last two statements are now known to be incorrect, 

 it being established that the difference between C p and C v is a con- 

 stant for any one gas, but not for all gases; and also that the specific 

 heat of permanent gases is independent of pressure and temperature. 

 These conclusions were obtained by Carnot on account of the erroneous 

 assumption of the materiality of heat. Moreover, the assumption of 

 the change of specific heat with volume led him to incorrect conclu- 

 sions in other cases. 



The deductions from Carnot 's work made by Clapeyron are correct 

 by reason of the fact that he used differential equations in the exten- 

 sion of Carnot 's ideas. For, although Carnot in considering the energy 

 changes of a body subjected to a Carnot 's cycle made the mistake of 

 equating the amount of heat-energy (H) given out by the body during 

 the isothermal change of volume and pressure at the higher tempera- 

 ture to the heat-energy (h) absorbed by the body during the isothermal 

 change at the lower temperature, Clapeyron was correct in his equa- 

 tions because they dealt only with infinitesimal changes in tempera^ 

 ture, and hence the difference H h, which is the area included between 

 the two adjacent adiabatics and the two isothermals, is an infinitesimal 

 of the second order as compared with the length of the adiabatic 

 included between the two adjacent isothermals, which was taken itself 

 as an infinitesimal of the first order. 



It is fortunate that Clapeyron was mathematician enough to use 

 differential equations in expressing these processes analytically. In- 

 deed, in contrast to Carnot he used such a method wherever he could 

 throughout all his memoirs, and always to good advantage. 



Carnot used the materialistic theory of heat; but it must not be 

 supposed that he was throughout a believer in the same. For even in 

 his memoir as published in 1824 he gives more than a suspicion of its 

 falsity, and in the extracts from his laboratory note-book,* published 

 by his brother after his death, we have direct evidence that he not only 

 foresaw the dynamical theory of heat, but even went so far as to calcu- 



* These MS. notes, and also the MS. copy of the ' Reflexions ' are on file 

 in the ' Academie des Sciences,' to which body they were presented in 1878 by 

 Carnot's brother, H. Carnot. The notes were entirely unknown to the public 

 until that late date, forty-six years after their author's death. 



