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or in the physical media employed, must derive the same equivalent 

 of mechanical effect from a given thermal agency. Carnot describes 

 a steam-engine and an air-engine, each of which satisfies the criterion 

 laid down above (the construction being however in each case, prac- 

 tically impossible) ; and ho shows how, with certain physical data, 

 with reference to steam in one case, and with reference to air or any 

 gas in the other, the equivalent of mechanical effect, derivable from 

 a given thermal agency, may be calculated. Thus, if M denote the 

 amount of mechanical effect duo to the descent of H units of heat (or 

 cal <■"■€) from a body A at the temperature S, through the medium 

 of at perfect engine of any kind, to a body B at the temperature T, 

 we find, by Carnot's method of reasoning, 

 dp 



M = H^^ (1 - .) ^dt = ^Po-oJ^ f^ l^dt dq 



In the first expression, deduced by the theory of the steam-engine, 

 p denotes the pressure, e the density, and k the latent heat of a unit 

 of volume of saturated vapour from any liquid, at the temperature 

 t. In the second, deduced by the theory of the air-engine, E denotes 

 the coefficient of expansion (-00366, if the centigrade scale of the 

 air-thermometer be adopted) of a gas ; p^ the pressure of a given 

 mass of gas when reduced to the freezing point of temperature, and 

 to the volume v^ ; p the pressure of the same gaseous mass when 

 occupying the volume v, at the temperature t ; q the quantity of 

 heat which must be added to the same mass to raise its temperature 

 from to t, when its volume is at the same time changed from v^ 

 to V ; and d q the heat absorbed by the gas when, with its tempera- 

 ture kept at t, its volume is augmented from v to v + d v. 



Hence the mechanical effect to be obtained by the letting down 

 of a unit of heat from a body A, to a body B at a lower temperature 

 t, if the interval between their temperatures be an extremely small 

 quantity t, will be, according to the first expression : 



dp 



(l-<r)-r, 



and, according to the latter, 



E ppV n /'H 1 (it; 



H 



Jo V dq 



