
DYNAMICAL THEORY OF HEAT. 273 
and consequently, by (6), 
le hae aan eel 5 ae eT an ot Ge 
26. To compare this with the expression H i pdt, for the duty, indicated 
by Carnot’s theory,* we may expand the exponential in the preceding equation, 
by the usual series. _ We thus find 
W=(1- 35+ ppg-he.) HE fpw as | 
O=5 reas 
This shews that the work really produced, which always falls short of the duty 
indicated by Carnov’s theory. approaches more and more nearly to it as the 
range is diminished, and ultimately, when the range is infinitely small, is the 
same as if Carnot’s theory required no modification, which agrees with the 
conclusion stated above in § 22. 
27. Again, equation (8) shews that the real duty of a given quantity of heat 
supplied from the source increases with every increase of the range; but that 
where 
(9). 
instead of increasing indefinitely in proportion to ae pdt,as Carnot’s theory 
makes it do, it never reaches the value J H, but approximates to this limit, as 
i pd tis increased without limit. Hence Carnov’s remarky regarding the prac- 
tical advantage that may be anticipated from the use of the air-engine, or from 
any method by which the range of temperatures may be increased, loses only a 
part of its importance, while a much more satisfactory view than his of the 
practical problem is afforded. Thus we see that, although the full equivalent of 
mechanical effect. cannot be obtained even by means of a perfect engine, yet when 
the actual source of heat is at a high enough temperature above the surrounding 
objects, we may get more and more nearly the whole of the admitted heat con- 
verted into mechanical effect, by simply increasing the effective range of tempera- 
ture in the engine. 
28. The preceding investigation (§ 25) shews that the value of Carnor’s 
function, p, for all temperatures within the range of the engine, and the absolute 
value of JouLe’s equivalent, J, are enough of data to calculate the amount of 
mechanical effect of a perfect engine of any kind, whether a steam-engine, an air- 
engine, or even a thermo-electric engine, since, according to the axiom stated in 
§ 12, and the demonstration of Prop. II., no inanimate material agency could 
* « Account,” &c., Equation 7, § 31. 
} “ Account,” &c. Appendix, Section IV. 
VOL. XX. PART Il. 4E 
