550 PROFESSOR WILLIAM THOMSON'S ACCOUNT OF 



for the diflference of volumes of the entu-e contents at the corresponding instants, 



Hence the expression for the area of the quadrilateral figure becomes 



7rPi H 



p^ * 



dp. 



Now, e, k, and 7?, being quantities which depend upon the temperature, may be 

 considered as functions of t ; and it will be convenient to modify the integral so 

 as to make t the independent variable. The limits will be fi'om «=T to <=rS, and, 

 if we denote by M the value of the integral, we have the expression 



dp 

 S 

 M = Rf' il-<i)^dt (1) 



for the total amount of mechanical effect gained by the operations described 

 above. 



21. If the interval of temperatures be extremely small ; so small that 

 dp 



(1-ir)— ' will not sensibly vary for values of t between T and S, the preceding ex- 



presssion becomes simply 



dp 



M = (1-.)1£.H(S-T) .... (2). 



This might, of com-se, have been obtained at once, by supposing the breadth of 

 the quadrilateral figure A Ai Aj A to be extremely small compared with its length, 

 and then taking for its area, as an approximate value, the product of the breadth 

 into the line A Ai, or the line A3 A,., or any line of intermediate magnitude. 



The expression (2) is rigorously correct for any interval S-T, if the 

 dp 



mean value of (1 - (r)-r-for that interval be employed as the coefficient of H (S- T). 



fC 



Carnot's Them'H of the Air-Engine. 



22. In the ideal air-engine imagined by Caknot fom- operations performed 

 upon a mass of air or gas enclosed in a closed vessel of variable volume, consti- 

 tute a complete cycle, at the end of which the medium is left in its primitive phy- 

 sical condition ; the construction being the same as that Avhich was described 

 above for the steam-engine, a body A, permanently retained at the temperature 

 S, and B at the temperature T ; an impermeable stand K ; and a cylinder and 

 piston, which, in this case, contains a mass of ah- at the temperature S, instead of 



