390 TRANSACTIONS OF SECTION A. 
cycle, to their initial state. Nothing but heat, or its equivalent fuel, may be 
supplied to the engine. Otherwise part of the motive power obtained might be 
due, not to heat alone, but to some change in the working substance, or in the 
disposition of the mechanism. Carnot here assumes the fundamental axiom of 
the cycle, which he states as follows: ‘ When a body has undergone any changes, 
and, after a certain number of transformations, is brought back identically to its 
original state, considered relatively to density, temperature, and mode of aggre- 
gation, it must contain the same quantity of heat as it contained originally.’ 
This does not limit the practical application of the theory, because all machines 
repeat a regular series of operations, which may be reduced in theory to an 
equivalent cycle in which everything is restored to its initial state. 
The most essential feature of the working of all heat-engines, considered 
apart from details of mechanism, is the production of motive-power by alternate 
expansion or contraction, or heating and cooling of the working substance. This 
necessitates the existence of a difference of temperature, produced by com- 
bustion or otherwise, between two bodies, such as the boiler and condenser of a 
steam-engine, which may be regarded as the source and sink of heat respectively. 
Wherever a difference of temperature exists, it may be made a source of motive- 
power, and conversely, without difference of temperature, no motive-power can 
be obtained from heat by a cyclical or continuous process. From this considera- 
tion Carnot deduces the simple and sufficient rule for obtaining the maximum 
effect : ‘In order to realise the maximum effect it is necessary that, in the process 
employed, there should not be any direct interchange of heat between bodies at 
sensibly different temperatures.’ Direct transference of heat between bodies at 
sensibly different temperatures would be equivalent to wasting a difference of 
temperature which might have been utilised for the production of motive-power. 
Equality of temperature is here assumed as the limiting condition of thermal 
equilibrium, such that an infinitesimal difference of temperature will suffice to 
determine the flow of heat in either direction. An engine satisfying Carnot’s 
rule will be reversible so far as the thermal operations are concerned. Carnot 
makes use of this property of reversibility in deducing his formal proof that an 
engine of this type possesses the maximum efficiency. If in the usual or direct 
method of working such an engine takes a quantity of heat Q from the source, 
rejects heat to the condenser, and gives a balance of useful work W per cycle, 
when the engine is reversed and supplied with motive-power W per cycle it will 
in the limit take the same quantity of heat from the condenser as it previously 
rejected, and return to the source the same quantity of heat Q as it took from it 
when working direct. All such engines must have the same efficiency (measured 
by the ratio W/Q of the work done to the heat taken from the source) whatever 
the working substance, provided that they work between the same temperature 
limits. For, if this were not the case, it would be theoretically possible, by 
employing the most efficient to drive the least efficient reversible engine back- 
wards, to restore to the source all the heat taken from it, and to obtain a balance 
of useful work without the consumption of fuel; a result sufficiently improbable 
to serve as the basis of a formal proof. Carnot thus deduces his famous principle, 
which he states as follows: ‘7'he motive-power obtainable from heat is indepen- 
dent of the agents set at work to realise it. Its quantity is fixed solely by the 
temperatures between which in the limit the transfer of heat takes place.’ 
Objection is commonly taken to Carnot’s proof, on the ground that the’combina- 
tion which he imagines might produce a balance of useful work without infringing 
the principle of conservation of energy, or constituting what we now understand 
as perpetual motion of the ordinary kind in mechanics. It has become the fashion 
to introduce the conservation of energy in the course of the proof, and to make 
a final appeal to some additional axiom. Any proof of this kind must always be 
to some extent a matter of taste; but since Carnot’s principle cannot be deduced 
from the conservation of energy alone, it seems a pity to complicate the proof 
by appealing to it. For the particular object in view, the absurdity of a heat- 
engine working without fuel appears to afford the most appropriate improbability 
which could be invoked. The final appeal must be to experiment in any case. 
At the present time the experimental verification of Carnot’s principle in its widest 
application so far outweighs the validity of any deductive proof, that we might 
well rest content with the logic that satisfied Carnot instead of confusing the issue 
by disputing his reasoning. 
