carnot's theory of the motive power of heat. 569 



III. On the Specific-Heats of Gases. 



52. The following proposition is proved by Caenot 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 temperatuvr and 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, r. The quantity of heat required to 

 do this will be kr, 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 



Er 



place will be Y+Wt' ^^ ^^ temperature be denoted by t ; and hence, by (8), the 

 quantity of heat that must be supplied, to prevent any lowering of temperature, 

 ^iUbe E^"3.^^, or ^' ^ ^ 



II. ■ l + E; ' " /i(l+EO^ 

 Hence, the total quantity added is equal to 



^'+^1TTE7)5' 



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 r ; and hence we have 



^=^MS7? (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 

 Caenot, 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 Caenot himself points out, has a 



* Or the capacity of a unit of volume for heat. 

 VOL. XVI. PAET v. 7 H 



