This paper, though subsequent to that of Clausius, is very different in character. Many of the results are identical with those previously obtained by Clausius, but they are reached by a process which is preceded by a clear statement of fundamental principles. These principles have since been the subject of discussion, and are not free from difficulty even now; but a great step in advance was made by their careful formulation in Thomson's paper, as a preliminary to the erection of the theory and the deduction of its consequences. Two propositions are stated which may be taken as the First and Second Laws of Thermodynamics. One is equivalent to the First Law as stated in p. 116, the other enunciates the principle of Reversibility as a criterion of "perfection" of a heat engine. We quote these propositions. "Prop. I (Joule).--When equal quantities of mechanical effect are produced by any means whatever from purely thermal sources, or lost in purely thermal effects, equal quantities of heat are put out of existence or are generated." "Prop. II (Carnot and Clausius).--If an engine be such that when worked backwards, the physical and mechanical agencies in every part of its motions are all reversed, it produces as much mechanical effect as can be produced by any thermodynamic engine, with the same temperatures of source and refrigerator, from a given quantity of heat." Prop. I was proved by assuming that heat is a form of energy and considering always the work effected by causing a working substance to pass through a closed cycle of changes, so that there was no change of internal energy to be reckoned with. Prop. II was proved by the following "axiom": "It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects." This is rather a postulate than an axiom; for it can hardly be contended that it commands assent as soon as it is stated, even from a mind which is conversant with thermal phenomena. It sets forth clearly, however, and with sufficient guardedness of statement, a principle which, when the process by which work is done is always a cyclical one, is not found contradicted by experience, and one, moreover, which can be at once explicitly applied to demonstrate that no engine can be more efficient than a reversible one, and that therefore the efficiency of a reversible engine is independent of the nature of the working substance.