to-t, and to the surface of contact of these parts Σ; C is a specific constant. An analogous law no doubt exists for the velocity with which mechanical equilibrium is established in a homogeneous or heterogeneous system dp=C'Σ(po-p). Provided no other dr phenomena interfere*, it will be directly proportional to the difference of pressure in the different parts of the homogeneous system, and directly proportional to the difference of pressure and to the surface of contact of the different parts of the heterogeneous system, when an equalization of pressure takes place between them; C is a specific constant. Passing now to the velocity of chemical reaction, we find that this region is exceedingly complicated and, experimentally, very difficult. It is enough to mention that after a period of about 40 or 50 years of research we have been successful only in the determination of the laws concerning the velocity of reaction in homogeneous systems (as far as gases or solutions are concerned), but our attempts to obtain a knowledge of the velocity of reaction in heterogeneous systems, one may well say, remained theoretically and experimentally unsuccessful. The object of this paper is to furnish a solution of this problem, guided and helped in the first instance by carefully executed experiments. Before passing to the subject of this paper a few words should be added concerning Gibbs's rule of phases, as the present work has much to do with it, and this will introduce order in the different subjects dealt with here. Gibbs's rule of phases is given in the Transactions of the Connecticut Academy, 1874-78, pp. 152-156. The quintessence of the rule can be formulated thus :: (i.) If the system consists of n kinds of molecules, and the number of parts or phases in the heterogeneous system r=n+2 (experience shows that r can never be greater than n+2), no variation is possible in the parts of the system which are capable of coexisting, because the concentration of each kind of the molecules, as well as temperature and * Prof. FitzGerald was kind enough to draw my attention to the fact that the equation =C'( po−p) well expresses the law of equalization of pressure only when the resistances to motion due to viscous causes are so great that the inertia of the masses set in motion can be neglected owing to the small velocities generated; for example, owing to inertia a body may move against the pressure, as in the case of air in a tube expanding and rarefying itself to a lower pressure than the external atmosphere. dp pressure, can have in every part of the system in this case only one definite value. There is therefore only one point at which equilibrium is possible (van't Hoff's "point of transition"). (ii.) If the number of parts or phases in the heterogeneous system r=n+1, all variables up to one are determined; to each temperature corresponds also a definite pressure and a definite concentration of each of the substances in the parts of the system, i. e., we have not only one point of equilibrium but a continuous series of them, a curve. This kind of equilibrium is called "complete." (iii.) Lastly, if the number of phases r=n, at a given temperature or pressure the values of the concentration of each kind of molecules in the phases remain undetermined, i. e., each part or phase may contain different concentrations of each kind of the molecules, and the system may remain in equilibrium. This kind of equilibrium is called "incomplete." The object of the author's experimental investigations was in the first instance to find the law concerning the velocity of reaction in the system before "complete equilibrium " or its "point of transition" is reached. For this systems were investigated on the freezing-point and the solubility-curve, where n=2, r=n+1=3, e. g., systems consisting of pure solid solvent, solution and vapour, or systems consisting of solid salt, the saturated aqueous solution and vapour. Again, systems were investigated where n=1 or 2 and r=n+2 i.e.=3 or 4, e. g. systems consisting of ice, water, and vapour, or consisting of ice, solid salt, saturated solution, and vapour. II. Velocity of Reaction before Complete Equilibrium and the Point of Transition are reached. The velocity of reaction before complete equilibrium has formed the subject of my investigation since 1895. In the Zeitschrift für physikalische Chemie, January 1896, I published a short communication on the velocity of separation of ice from overcooled water and from aqueous solutions. I have had, however, to repeat my experiments more carefully, with a more sensitive thermometer. Herr Götze, at Leipzig, has constructed for me an exceedingly sensitive 1/100° mercury-thermometer. The mercury-bulb of the thermometer is very thin and long, and the glass wall of the bulb is as thin as the skill of Herr Götze allowed him to construct. The time was read with accuracy to second, and in this Mr. Still, of Christ Church, Oxford, assisted me. The arrangements of the experiments were those which I used in bene andisers, ze in the case of the very a curate treezing-point The velocity with which my mercury-therumed the temperature of the Baia was neverthe soup abest 20 minit. By this methoi I investigated separation of the soll Fided solvent from overprojeto ligous or solutions, the velocity of separation of salts from engenaturated solutions, and the velocity of melting of ice in water ani aqueous solutions: the results of Investigation were published in the Report of the British Aeration 16. in Sections III. IV and V. Besides I investigated the velocity of soll lineation of overcooled or solutions: 1 cases in which pure solvent separates, 2 as in which a solid solution parates. The results entale investigation were published in the same Report, in Lati II., and are here given in a little more detail. 2 1. Velocity of Solidification of Overcooled Liquids and Solutions. Phenol and Solutions of Water in Phenol.) The method ud was that of Gernez-Moore. in the glass Torbe, however, one of the arms was replaced by a tube of sery thin platinum, the diameter of the tube being about 5 mm, for reasons which are given below. The tube was £t with the liquid to be investigated, and a small, very entive mercury-thermometer was immersed in the liquid, enabling the temperature to be read to 01. The U-tube was placed in a tali glass jar containing about & litres of water, and the temperature of the water was kept constant to 0°-05. Temperture of the water was successively arranged at lower and lower temperatures below that of the freezing-point of the Huia contained in the tube. The time was read to of a con. If the freezing-temperature of the pure liquid or of the solution in the tube be T., the temperature of the bath and of the liquid in the U-tube TB (read directly on the small thermometer before the reaction of the solidification was started, and the temperature of the liquid in the U-tube during the process of solidification T (the maximum tem perature read on the small thermometer, while the solidifying mass is passing its bulb), then the rise of temperature of the liquid from its initial temperature TB-T is found to be =40 per cent. and more of the total value TB-To. The investigation was carried out as near to the melting-point as possible (i. e. as far as the abnormalities in the crystallization, which appear near the freezing-point, did not hinder the investigation), and again, so far from the melting-point, until spontaneous crystallization set in; in other words, the whole length of the curve accessible for the investigation has been studied. If the undercoolings T.-T (as read on the small thermometer) be taken as the abscissæ, and the velocities of crystallization (or the reciprocal values of time which are necessary for the solidifying mass to pass from the lower end of the platinum tube to its upper end) as ordinates, straight lines are obtained, which on continuation pass the freezingpoint. If, on the contrary, I calculate my results so that I assume the temperature of the liquid at the surface of contact with the solid to be that of the surrounding bath, the curves become less regular, and on continuation they cut the abscissæ, not at the freezing-point, but considerably below it, since a shifting of all the points of the curve by about 40 per cent. is thus caused." This leads either to the impossible conclusion that overcooled liquids cannot freeze below the freezing-point, or to the result that the otherwise more or less straight lines finish up irregularly on approaching the freezing-point. The obtained result is, therefore, = = C(T.-T), where is the time, T. the melting-point, T the temperature of the liquid in contact with the solid while the reaction is going on. dt de No capillary tubes were used for the investigation since I found in my investigation of mercury thermometers that the velocity with which the mercury thread moves in the capillary tube, when the thermometer assumes the temperature of a liquid, is about 40 per cent. greater when it rises than when it falls. Consequently there is no free movement of the liquid in a capillary tube, even when this liquid is mercury, and still more in the case of other liquids which adhere to the glass or platinum when they are crystallizing out. A tube of very thin platinum was used instead of a glass tube, as it is necessary that the heat capacity of the mass of the tube should be so very small in comparison with that of the liquid that it may be neglected, and also that the conductivity for heat of the tube should be very great. We must otherwise take into consideration the tube as well. In that case the conditions become very complicated and the necessary data cannot be experimentally determined with any degree of accuracy, since the velocity with which the tube assumes the temperature of the bath and of the liquid which it contains cannot be investigated. Under the conditions of my experiments the following relations hold good (though only approximately): -The velocity with which the liquid in the U-tube (and therefore also the liquid at its surface of contact with the solid) is cooled by the bath is (d+)=C(T, —T), where T is the temperature of the upper layer of the liquid in contact with the solid while the reaction is going on. The velocity with which the temperature of the liquid is raised during the reaction by the latent heat of melting is (d)'=C'(T.—T), i. e. is directly pro portional to the velocity of reaction (T, is the melting-point of the liquid). Therefore the total rate of variation of the temperature of the liquid at the surface of contact with the dT solid is, during the reaction, =C(T„—T)+C'(T —T). If we assume that the temperature of the liquid in contact with the solidified mass remains constant during the reaction (this ατ C' dT dr cannot, however, be strictly correct), we get =0, and = =constant, or T-TC+0=k, i. e. the remoteness of the temperature of the liquid in contact with the solidified mass T from the melting-point T. is, for all values of the undercoolings, the same fraction of T-Tв (T cannot be T, during the reaction, since at the point of equilibrium T, no more reaction, i. e. no solidification, is possible). I found, however, that this does not strictly hold true, if the maximum temperature indicated by the small thermometer in the platinum tube be taken as the true temperature of the liquid at the surface of contact with the solid. I concluded from this that the maximum temperature indicated by the small thermometer T is still removed by about 0°1 or 0.2 from the real temperature of the liquid at the surface of contact with the solid. At the end of the curve a diminution in the proportion between velocity of reaction and the value T.-T was found. This I attributed partly to the incomplete indications of the temperature by the small thermometer, partly to the variation of temperature of liquid in contact with the solid as the reaction is proceeding, partly to the expected variation of the velocity constant |