CARNOT’S THEORY OF THE MOTIVE POWER OF HEAT. 569 
Ill. On the Specijfic-Heats of Gases. 
52. The following proposition is proved by Carnot as a deduction from his 
general theorem regarding the specific heats of gases. 
The excess of the specific heat® under a constant pressure above the specific heat 
at a constant volume, is the same for all gases at the same temperatureand pressure. 
53. To prove this proposition, and to determine an expression for the “ ex- 
cess” mentioned in its enunciation, let us suppose a unit of volume of a gas to be 
elevated in temperature by a small amount, ;. The quantity of heat required to 
do this will be Az, if A denote the specific heat at a constant volume. Let us 
next allow the gas to expand without going down in temperature, until its pres- 
sure becomes reduced to its primitive value. The expansion which will take 
place will be ees if the temperature be denoted by ¢; and hence, by (8), the 
quantity of heat that must be supplied, to prevent any lowering of temperature, 
2 Ep, % Er E? p 
wil be py eee Uta Ba 
Hence, the total quantity added is equal to 

EK? p 
aaa (1+Es?" 
But, since B denotes the specific heat under constant pressure, the quantity of 
heat requisite to bring the gas into this state, from its primitive condition, is 
equal to B;; and hence we have 
FE 
= P 
a (+E? ° 
(12) 
IV. Comparison of the Relative advantages of the Air-Engine and Steam-Engine. 
54. In the use of water-wheels for motive power, the economy of the engine 
depends not only upon the excellence of its adaptation for actually transmitting 
any given quantity of water through it, and producing the equivalent of work, - 
but upon turning to account the entire available fall; so, as we are taught by 
Carnot, the object of a thermodynamic engine is to economize in the best pos- 
sible way the transference of all the heat evolved, from bodies at the temperature 
of the source, to bodies at the lowest temperature at which the heat can be dis- 
charged. With reference then to any engine of the kind, there will be two points 
to be considered. 
(1.) The extent of the fall utilised. 
(2.) The economy of the engine, with the fall which it actually uses. 
55. In the first respect, the air-engine, as Carnot himself points out, has a 
* Or the capacity of a unit of volume for heat. 
VOL. XVI. PART V. 7H 
