CARNOT’S THEORY OF THE MOTIVE POWER OF HEAT. 567 
From this result we draw the following conclusion :— 
47. Equal volumes of all elastic fluids, when compressed to smaller equal 
volumes, disengage equal quantities of heat. 
This extremely remarkable theorem of CArRNnot’s was independently laid down 
as a probable experimental law by Dutone, in his “ Recherches sur la Chaleur 
Spécifique des Fluides Elastiques,” and it therefore affords a most powerful con- 
firmation of the theory.* 
48. In some very remarkable researches made by Mr JouLE upon the heat 
developed by the compression of air, the quantity of heat produced in different 
experiments has been ascertained with reference to the amount of work spent in 
the operation. To compare the results which he has obtained with the indi- 
cations of theory, let us determine the amount of work necessary actually to pro- 
duce the compression considered above. 
49. In the first place, to compress the gas from the volume v+d v¢ to z, the 
‘work required is pdv, or, since pv=p, v, (1+ Ed, 
dv 
Po % 1+ HA) ea 
Hence, if we denote by W the total amount of work necessary to produce the 
compression from V to V’, we obtain, by integration, 
W=p, % (1+E 2) log Y,, 
Comparing this with the expression above, we find 
go = t) (11) 
50. Hence we infer that 
(1.) The amount of work necessary to produce a unit of heat by the compres- 
sion of a gas, is the same for all gases at the same temperature. 
(2.) And that the quantity of heat evolved in all circumstances, when the 
temperature of the gas is given, is proportional to the amount of work spent in 
the compression. 
* Carnor varies the statement of his theorem, and illustrates it in a passage, pp. 52, 53, of 
which the following is a translation :— 
“ When a gas varies in volume without any change of temperature, the quantities of heat absorbed 
or evolved by this gas are in arithmetical progression, if the augmentation or diminutions of volume 
are in geometrical progression. 
« When we compress a litre of air maintained at the temperature 10°, and reduce it to half 
a litre, it disengages a certain quantity of heat. If, again, the volume be reduced from half a litre 
to a quarter of a litre, from a quarter to an eighth, and so on, the quantities of heat successively 
evolved will be the same. 
“« Tf, in place of compressing the air, we allow it to expand to two litres, four litres, eight litres, 
&c., it will be necessary to supply equal quantities of heat to maintain the temperature always at the 
same degree.” 
