
X.—On the Economy of Heat in Expansive Machines, forming the Fifth Section of 
a Paper On the Mechanical Action of Heat. By Witu1aAM Joun Macquorn 
Rankine, Civil Engineer, F.R.S.E., F.R.S.8.A., &c. 
(Read 21st April 1851.) 
(39.) A machine working by expansive power consists essentially of a portion 
of some substance to which heat is communicated, so as to expand it, at a higher 
temperature, being abstracted from it, so as to condense it to its original volume, 
at a lower temperature. The quantity of heat given out by the substance is less 
than the quantity received; the difference disappearing as heat to appear in the 
form of expansive power. 
The heat originally received by the working body may act in two ways: to 
raise its temperature, and to expand it. The heat given out may also act in two 
ways: to lower the temperature, and to contract the body. Now, as the conver- 
sion of heat into expansive power arises from changes of volume only, and not 
from changes of temperature, it is obvious, that the proportion of the heat re- 
ceived which is converted into expansive power will be the greatest possible, 
when the reception of heat, and its emission, each take place at a constant tem- 
perature. f 
(40.) Carnov was the first to assert the law, that the ratio of the maximum me- 
chanical effect, to the whole heat expended in an expansive machine, is a function 
solely of the two temperatures at which the heat is respectively received and emitted, 
and is independent of the nature of the working substance. But his investigations 
not being based on the principle of the dynamical convertibility of heat, involve 
the fallacy that power can be produced out of nothing. 
(41.) The merit of combining Carnét’s Lam, as it is termed, with that of the 
convertibility of heat and power, belongs to Mr Ciausius and Professor WILLIAM 
Txomson; and in the shape into which they have brought it, it may be stated thus:— 
The maximum proportion of heat converted into expansive power by any ma- 
chine, is a function solely of the temperatures at which heat is received and emitted 
by the working substance ; which function, for each pair of temperatures, is the same 
Jor all substances in nature. 
This law is laid down by Mr Cuausius, as it originally had been by Carnor, 
as an independent axiom; and I had at first doubts as to the soundness of the 
reasoning by which he maintained it. Having stated those doubts to Professor 
Tuomson, I am indebted to him for having induced me to investigate the subject 
thoroughly; for although I have not yet seen his paper, nor become acquainted with 
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