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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 Cp and Cv is a constant 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 conclusions 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 extension 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 temperature to the heat-energy (h) absorbed by the body during the isothermal change at the lower temperature, Clapeyron was correct in his equations because they dealt only with infinitesimal changes in temperature, 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. Indeed, 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