The daily more numerous applications of the laws of corresponding states have rendered highly important the determination of the critical constants which permit these states to be defined.  In the case of homogeneous bodies the critical elements have a simple, clear, and precise sense; the critical temperature is that of the single isothermal line which presents a point of inflexion at a horizontal tangent; the critical pressure and the critical volume are the two co-ordinates of this point of inflexion.

The three critical constants may be determined, as Mr S. Young and M. Amagat have shown, by a direct method based on the consideration of the saturated states.  Results, perhaps more precise, may also be obtained if one keeps to two constants or even to a single one—­ temperature, for example—­by employing various special methods.  Many others, MM.  Cailletet and Colardeau, M. Young, M.J.  Chappuis, etc., have proceeded thus.

The case of mixtures is much more complicated.  A binary mixture has a critical space instead of a critical point.  This space is comprised between two extreme temperatures, the lower corresponding to what is called the folding point, the higher to that which we call the point of contact of the mixture.  Between these two temperatures an isothermal compression yields a quantity of liquid which increases, then reaches a maximum, diminishes, and disappears.  This is the phenomenon of retrograde condensation.  We may say that the properties of the critical point of a homogeneous substance are, in a way, divided, when it is a question of a binary mixture, between the two points mentioned.

Calculation has enabled M. Van der Waals, by the application of his kinetic theories, and M. Duhem, by means of thermodynamics, to foresee most of the results which have since been verified by experiment.  All these facts have been admirably set forth and systematically co-ordinated by M. Mathias, who, by his own researches, moreover, has made contributions of the highest value to the study of questions regarding the continuity of the liquid and gaseous states.

The further knowledge of critical elements has allowed the laws of corresponding states to be more closely examined in the case of homogeneous substances.  It has shown that, as I have already said, bodies must be arranged in groups, and this fact clearly proves that the properties of a given fluid are not determined by its critical constants alone, and that it is necessary to add to them some other specific parameters; M. Mathias and M. D. Berthelot have indicated some which seem to play a considerable part.

It results also from this that the characteristic equation of a fluid cannot yet be considered perfectly known.  Neither the equation of Van der Waals nor the more complicated formulas which have been proposed by various authors are in perfect conformity with reality.  We may think that researches of this kind will only be successful if attention is concentrated, not only on the phenomena of compressibility and dilatation, but also on the calorimetric properties of bodies.  Thermodynamics indeed establishes relations between those properties and other constants, but does not allow everything to be foreseen.