550 PROFESSOR WILLIAM THOMSON’S ACCOUNT OF 
for the difference of volumes of the entire contents at the corresponding instants, 
H 
g = (1— 6) via 
Hence the expression for the area of the quadrilateral figure becomes 
p 
af (1-0) ap. 
P 
3 
Now, », &, and p, being quantities which depend upon the temperature, may be 
considered as functions of ¢; and it will be convenient to modify the integral so 
as to make ¢ the independent variable. The limits will be from ¢=T to ¢=S, and, 
if we denote by M the value of the integral, we have the expression 
dp 
laa 
M=H/ -9pae emhin abt wena 
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 
op 
(1—o) = will not sensibly vary for values of ¢ between T and S, the preceding ex- 
presssion becomes simply 
dp 
M=(-)5*. H(S-T) .. .. @ 
This might, of course, have been obtained at once, by supposing the breadth of 
the quadrilateral figure A A, A, 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 Aj, or the line A; 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— 0) for that interval be employed as the coefficient of H (S—T). 
Carnot’s Theory of the Air-Engine. 
22. In the ideal air-engine imagined by Carnot four 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 which 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 air at the temperature S, instead of 

