PRESIDENTIAL ADDRESS. 391 
Carnot himself proceeded to test his principle in every possible way by com- 
parison with experiment so far as the scanty data available in his time would 
permit. He also made several important deductions from it, which were con- 
trary to received opinion at the time, but have since been accurately verified. 
He appears to have worked out these results analytically in the first instance, as 
indicated by his footnotes, and to have translated his equations into words in 
the text for the benefit of his non-mathematical readers. In consequence of this, 
some of his most important conclusions appear to have been overlooked or attri- 
buted to others. Owing to want of exact knowledge of the properties of sub- 
stances over extended ranges of temperature, he was unable to apply his principle 
directly in the general form for any temperature limits. We still labour to a less 
extent under the same disability at the present day. He showed, however, that 
a great simplification was effected in its application by considering a cycle of 
infinitesimal range at any temperature ¢. In this simple case the principle is 
equivalent to the assertion that the work obtainable from a unit of heat per degree 
fall (or per degree range of the cycle) at a temperature ¢, is some function F’¢ of 
the temperature (generally known as Carnot’s Function), which must be the same 
for all substances at the same temperature. From the rough data then available 
for the properties of steam, alcohol, and air, he was able to calculate the 
numerical values of this function in kilogrammetres of work per kilocalorie of 
heat at various temperatures between 0° and 100° C., and to show that it was 
probably the same for different substances at the same temperature within the 
limits of experimental error. For the vapour of alcohol at its boiling-point, 
78°-7 C., he found the value F’¢=1-230 kilogrammetre per kilocalorie per 
degree fall. For steam at the same temperature he found nearly the same value, 
namely, F’¢ = 1212. Thus no advantage in point of efficiency could be gained 
by employing the vapour of alcohol in place of steam. He was also able to show 
that the work obtainable from a kilocalorie per degree fall probably diminished 
with rise of temperature, but his data were not sufticiently exact to indicate the 
law of the variation. 
The equation which Carnot employed in deducing the numerical values of 
his function from the experimental data for steam and alcohol is simply the direct 
expression of his principle as applied to a saturated vapour. It is now generally 
known as Clapeyron’s equation, because Carnot did not happen to give the 
equation itself in algebraic form, although the principle and details of the calcula- 
tion were most minutely and accurately described. In calculating the value of 
his function for air, Carnot made use of the known value of the difference of the 
specific heats at constant pressure and volume. He showed that this difference 
must be the same for equal volumes of all gases measured under the same tempera- 
ture and pressure, whereas it had always previously been assumed that the ratio 
(not the difference) of the specific heats was the same for different gases. He also 
gave a general expression for the heat absorbed by a gas in expanding at constant 
temperature, and showed that it must bear a constant ratio to the work of 
expansion. These results were verified experimentally some years later, in part 
by Dulong, and more completely by Joule, but Carnot’s theoretical prediction has 
generally been overlooked, although it was of the greatest interest and import- 
ance. The reason of this neglect is probably to be found in the fact that Carnot’s 
expressions contained the unknown function F’t of the temperature, the form of 
which could not be deduced without making some assumptions with regard to the 
nature of heat and the scale on which temperature should be measured. 
It was my privilege to discover a few years ago that Carnot himself had 
actually given the correct solution of this fundamental problem in one of his most 
important footnotes, where it had lain buried and unnoticed for more than eighty 
years. He showed by a most direct application of the caloric theory, that if 
temperature was measured on the scale of a perfect gas (which is now universally 
adopted) the value of his function F’¢ on the caloric theory would be the same 
at all temperatures, and might be represented simply by a numerical constant A 
(our ‘ mechanical equivalent’) depending on the units adopted for work and heat. 
In other words, the work W done by a quantity of caloric Q in a Carnot cycle of 
range T to T, on the gas scale would be represented by the simple equation : 
W = AQ(T —-T,). 
It is at once obvious that this solution, cbtained by Carnot from the caloric 
