In the Carnot cycle the working substance--which might be a gas or a vapour, or a liquid, or a vapour and its liquid in contact: it did not matter what for the result--was supposed to be put through a succession of changes in which the final state coincided with the initial. Thus the substance having been brought back to the same physical condition as it had when the cycle began, has the same internal energy as it had at the beginning, and in the reckoning of the work done by or against external forces, nothing requires to be set to the account of the working substance. This is the first great advantage of the method of reasoning which Carnot introduced. The ideal engine was a very simple affair: but the notion of reversibility is difficult to express in a form sufficiently definite and precise. Carnot does not attempt this; he merely contents himself with describing certain cycles of operations which obviously can be carried through in the reverse order. Nor does Thomson go further in his "Account of Carnot's Theory," though he states the criterion of a perfect engine in the words, "A perfect thermodynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency, if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced." This proposition was proved by Carnot: and the following formal statement in the essay is made: "La puissance motrice de la chaleur est independante des agents mis en œuvre pour la réaliser: sa quantité est fixée uniquement par les temperatures des corps entre lesquels se fait, en dernier résultat, le transport du calorique." The result involved in each, that the work done in a cycle by an ideal engine depends on the temperatures between which it works and not at all on the working substance, is, as we shall see, of the greatest importance. The proof of the proposition, by supposing a more efficient engine than the ideal one to exist, and to be coupled with the latter, so that the more efficient would perform the cycle forwards and the ideal engine the same cycle backwards, is well known. In Carnot's view the former would do more work by letting down a given quantity of heat from the higher to the lower temperature than was spent on the latter in transferring the same quantity of heat from the lower to the higher temperature, so that no heat would be taken from or given to source or refrigerator, while there would be a gain of work on the whole. This would be equivalent to admitting that useful work could be continually performed without any resulting thermal or other change in the agents performing the work. Even at that time this could not be admitted as possible, and hence the supposition that a more efficient engine than the reversible one could exist was untenable.