this time, it is true, other molecules emerge from the space enclosed by the circuit; but the two effects do not counterbalance each other, and the resulting current is maintained. There is elevation of temperature in the circuit in accordance with Joule's law; and this phenomenon, under such conditions, is incompatible with the principle of Carnot.

It is possible—and that, I think, is M. Lippmann's idea to draw from his very ingenious criticism an objection to the kinetic theory, if we admit the absolute value of the principle; but we may also suppose that here again we are in presence of a system where the prescribed conditions diminish the complexity and render it, consequently, less probable that the evolution is always effected in the same direction.

In whatever way you look at it, the principle of Carnot furnishes, in the immense majority of cases, a very sure guide in which physicists continue to have the most entire confidence.

§ 4. THERMODYNAMICS

To apply the two fundamental principles of thermodynamics, various methods may be employed, equivalent in the main, but presenting as the cases vary a greater or less convenience.

In recording, with the aid of the two quantities, energy and entropy, the relations which translate analytically the two principles, we obtain two

relations between the coefficients which occur in a given phenomenon; but it may be easier and also more suggestive to employ various functions of these quantities. In a memoir, of which some extracts appeared as early as 1869, a modest scholar, M. Massieu, indicated in particular a remarkable function which he termed a characteristic function, and by the employment of which calculations are simplified in certain cases.

In the same way J. W. Gibbs, in 1875 and 1878, then Helmholtz in 1882, and, in France, M. Duhem, from the year 1886 onward, have published works, at first ill understood, of which the renown was, however, considerable in the sequel, and in which they made use of analogous functions under the names of available energy, free energy, or internal thermodynamic potential. The magnitude thus designated, attaching, as a consequence of the two principles, to all states of the system, is perfectly determined when the temperature and other normal variables are known. It allows us, by calculations often very easy, to fix the conditions necessary and sufficient for the maintenance of the system in equilibrium by foreign bodies taken at the same temperature as itself.

One may hope to constitute in this way, as M. Duhem in a long and remarkable series of operations has specially endeavoured to do, a sort of general mechanics which will enable questions of

statics to be treated with accuracy, and all the conditions of equilibrium of the system, including the calorific properties, to be determined. Thus, ordinary statics teaches us that a liquid with its vapour on the top forms a system in equilibrium, if we apply to the two fluids a pressure depending on temperature alone. Thermodynamics will furnish us, in addition, with the expression of the heat of vaporization and of the specific heats of the two saturated fluids.

This new study has given us also most valuable information on compressible fluids and on the theory of elastic equilibrium. Added to certain hypotheses on electric or magnetic phenomena, it gives a coherent whole from which can be deduced the conditions of electric or magnetic equilibrium; and it illuminates with a brilliant light the calorific laws of electrolytic phenomena.

But the most indisputable triumph of this thermodynamic statics is the discovery of the laws which regulate the changes of physical state or of chemical constitution. J. W. Gibbs was the author of this immense progress. His memoir, now celebrated, on “the equilibrium of heterogeneous substances," concealed in 1876 in a review at that time of limited circulation, and rather heavy to read, seemed only to contain algebraic theorems applicable with difficulty to reality. It is known that Helmholtz independently succeeded, a few years later, in introducing thermodynamics into the

domain of chemistry by his conception of the division of energy into free and into bound energy: the first, capable of undergoing all transformations, and particularly of transforming itself into external action; the second, on the other hand, bound, and only manifesting itself by giving out heat. When we measure chemical energy, we ordinarily let it fall wholly into the calorific form; but, in reality, it itself includes both parts, and it is the variation of the free energy and not that of the total energy measured by the integral disengagement of heat, the sign of which determines the direction in which the reactions are effected.

But if the principle thus enunciated by Helmholtz as a consequence of the laws of thermodynamics is at bottom identical with that discovered by Gibbs, it is more difficult of application and is presented under a more mysterious aspect. It was not until M. Van der Waals exhumed the memoir of Gibbs, when numerous physicists or chemists, most of them Dutch-Professor Van t'Hoff, Bakhius Roozeboom, and others-utilized the rules set forth in this memoir for the discussion of the most complicated chemical reactions, that the extent of the new laws was fully understood.

The chief rule of Gibbs is the one so celebrated at the present day under the name of the Phase Law. We know that by phases are designated the homogeneous substances into which a system is

divided; thus carbonate of lime, lime, and carbonic acid gas are the three phases of a system which comprises Iceland spar partially dissociated into lime and carbonic acid gas. The number of phases added to the number of independent componentsthat is to say, bodies whose mass is left arbitrary by the chemical formulas of the substances entering into the reaction-fixes the general form of the law of equilibrium of the system; that is to say, the number of quantities which, by their variations (temperature and pressure), would be of a nature to modify its equilibrium by modifying the constitution of the phases.

Several authors, M. Raveau in particular, have indeed given very simple demonstrations of this law which are not based on thermodynamics; but thermodynamics, which led to its discovery, continues to give it its true scope. Moreover, it would not suffice merely to determine quantitatively those laws of which it makes known the general form. We must, if we wish to penetrate deeper into details, particularize the hypothesis, and admit, for instance, with Gibbs that we are dealing with perfect gases; while, thanks to thermodynamics, we can constitute a complete theory of dissociation which leads to formulas in complete accord with the numerical results of the experiment. We can thus follow closely all questions concerning the displacements of the equilibrium, and find a relation