Thomson's demonstration that a reversible engine is the most efficient is well known, and need not here be repeated in detail. The reversible engine may be worked backwards, and the working substance will take in heat where in the direct action it gave it out, and _vice versa_: the substance will do work against external forces where in the direct action it had work done upon it, and _vice versa_: in short, all the physical and mechanical changes will be of the same amount, but merely reversed, at every stage of the backward process. Thus if an engine A be more efficient than a reversible one B, it will convert a larger percentage of an amount of heat H taken in at the source into work than would the reversible one working between the same temperatures. Thus if h be the heat given to the refrigerator by A, and h' that given by B when both work directly and take in H; h must be less than h'. Then couple the engines together so that B works backwards while A works directly. A will take in H and deliver h, and do work equivalent to H － h. B will take h' from the refrigerator and deliver H to the source, and have work equivalent to H － h' spent upon it. There will be no heat on the whole given to or taken from the source; but heat h' － h will be taken from the refrigerator, and work equivalent to this will be done. Thus _by a cyclical process_, which leaves the working substance as it was, work is done at the expense of heat taken from the refrigerator, which Thomson's postulate affirms to be impossible. Therefore the assumption that an engine more efficient than the reversible engine exists must be abandoned; and we have the conclusion that all reversible engines are equally efficient. Thomson acknowledged in his paper the priority of Clausius in his proof of this proposition, but stated that this demonstration had occurred to him before he was aware that Clausius had dealt with the matter. He now cited, as examples of the First Law of Thermodynamics, the results of Joule's experiments regarding the heat produced in the circuits of magneto-electric machines, and the fact that when an electric current produced by a thermal agency or by a battery drives a motor, the heat evolved in the circuit by the passage of the current is lessened by the equivalent of the work done on the motor. [Illustration: FIG. 12.]