392 TRANSACTIONS OF SECTION A. 
theory, so far from being inconsistent with the mechanical theory of heat, is a 
direct statement of the law of conservation of energy as applied to the Carnot 
cycle. If the lower limit T, of the cycle is taken at the absolute zero of the gas- 
thermometer, we observe that the maximum quantity of work obtainable from a 
quantity of caloric Q at a temperature T is simply AQT, which represents the 
absolute value of the energy carried by the caloric taken from the source at the 
temperature T. The energy of the caloric rejected at the temperature T, is 
AQT,. The external work done is equal to the difference between the quantities 
of heat energy supplied and rejected in the cycle. 
The analogy which Carnot himself employed in the interpretation of this 
equation was the oft-quoted analogy of the waterfall. Caloric might be regarded 
as possessing motive-power or energy in virtue of elevation of temperature just 
as water may be said to possess motive-power in virtue of its head or pressure. 
The limit of motive-power obtainable by a reversible motor in either case would 
be directly proportional to the head or fall measured on a suitable scale. Caloric 
itself was not motive-power, but must be regarded simply as the vehicle or carrier 
of energy, the production of motive-power from caloric depending essentially (as 
Carnot puts it) not on the actual consumption of caloric, but on the fall of 
temperature available. The measure of a quantity of caloric is the work done 
per degree fall, which corresponds with the measure of a quantity of water by 
weight, i.c., in kilogrammetres per metre fall. 
That Carnot did not pursue the analogy further, and deduce the whole 
mechanical theory of heat from the caloric theory, is hardly to be wondered at 
if we remember that no applications of the energy principle had then been made 
in any department of physics. He appears, indeed, at a later date to have caught 
a glimpse of the general principle when he states that ‘ motive-power [his equiva- 
lent for work or energy] changes its form but is never annihilated.’ It is clear 
from the posthumous notes of his projected experimental work that he realised 
how much remained to be done on the experimental side, especially in relation to 
the generation of caloric by friction, and the waste of motive-power by conduction 
of heat, which appeared to him (in 1824) ‘ almost inexplicable in the present state 
of the theory of heat.’ 
One of the-points which troubled him most in the application of the theoretical 
result that the work obtainable from a quantity of caloric was simply propor- 
tional to the fall of temperature available, was that it required that the specific 
heat of a perfect gas should be independent of the pressure. This was incon- 
sistent with the general opinion prevalent at the time, and with one solitary 
experiment by Delaroche and Bérard, which appeared to show that the specific 
heat of a gas diminished with increase of pressure, and which had been explained 
by Laplace as a natural consequence of the caloric theory. Carnot showed that 
this result did not necessarily follow from the caloric theory; but the point was 
not finally decided in his favour until the experiments of Regnault, first pub- 
lished in 1852, established the correct values of the specific heat of gases, and 
proved that they were practically independent of the pressure. 
Another point which troubled Carnot was that, according to his calculations, 
the motive-power obtainable from a kilocalorie of heat per degree fall appeared 
to diminish with rise of temperature, instead of remaining constant. This 
might have been due to experimental errors, since the data were most uncertain. 
But if he had lived to carry out his projected experiments on the quantity of 
motive-power required to produce one unit of heat, and had obtained the result, 
424 kilogrammetres per kilocalorie, subsequently found by Joule, he could hardly 
have failed to notice that this was the same (within the limits of experimental 
error) as the maximum work AQT obtainable from the kilocalorie according to his 
equation. (This is seen to be the case when the values calculated by Carnot per 
degree fall at different temperatures were multiplied by the absolute temperature 
in each case. #.g., 1212 kilogrammetre per degree fall with steam at 79° C. or 
352° Abs. 1:212 x 352 = 426 kilogrammetres.) The origin of the apparent dis- 
crepancy between theory and experiment lay in the tacit assumption that the 
quantity of caloric in a kilocalorie was the same at different temperatures. 
There were no experiments at that time available to demonstrate that the caloric 
measure of heat as work per degree fall, implied in Carnot’s principle, or more 
explicitly stated in his equation, was not the same as the calorimetric measure 
obtained by mixing substances at different temperatures, Even when the energy 
