
DYNAMICAL THEORY OF HEAT. 271 
M, in § 20) M dv isthe quantity of heat absorbed in the first operation. Hence 
the value of 


d 
a T.dv aD 
Mdv M - 
must be the same for all substances, with the same values of ¢ and 7; or, since 
7 is not involved except as a factor, we must have 
== SN. + iews. old Shoe) ieote den 
where » depends only on7; from which we conclude the proposition which was 
to be proved. 
d a 
22. The very remarkable theorem that “4 — ‘ must be the same for all sub- 
stances at the same temperature, was first given (although not in precisely the 
same terms) by Carnot, and demonstrated by him, according to the principles 
he adopted. We have now seen that its truth may be satisfactorily established 
without adopting the false part of his principles. Hence all Carnor’s conclusions, 
and all conclusions derived by others from his theory, which depend merely on 
equation (3), require no modification when the dynamical theory is adopted. 
_ Thus, all the conclusions contained in Sections I., II., and III. of the Appendix to 
my Account of Carnot’s Theory, and in the paper immediately following it in the 
Transactions, entitled “Theoretical Considerations on the Effect of Pressure in 
Lowering the Freezing Point of Water,” by my elder brother, still hold. Also, 
we see that CarNot’s expression for the mechanical effect derivable from a given 
quantity of heat by means of a perfect engine in which the range of temperatures 
is infinitely small, expresses truly the greatest effect which can possibly be 
obtained in the circumstances; although it is in reality only an infinitely small 
fraction of the whole mechanical equivalent of the heat supplied; the remainder 
being irrecoverably lost to man, and therefore “wasted,” although not anni- 
_ hilated. 
23. On the other hand, the expression for the mechanical effect obtainable 
from a given quantity of heat entering an engine from a “source” at a given 
temperature, when the range down to the temperature of the cold part of the 
engine or the “refrigerator” is finite, will differ most materially from that of 
CaRNor; since, a finite quantity of mechanical effect being now obtained from a 
finite quantity of heat entering the engine, a finite fraction of this quantity must 
be converted from heat into mechanical effect. The investigation of this expres- 
sion, with numerical determinations founded on the numbers deduced from 
REGNAULT’S observations on steam, which are shewn in Tables I. and II. of 
