an equation from which the value of k, the absorption velocity of the nucleus, is computed at once in centim./sec., supposing decay (k') to be a vanishing quantity. Waiving the more refined methods of the kinetic theory of gases, if but of all the nuclei wander in a given direction, the term expressing absorption of the wall of the tube in the differential equation would be k(n/3) 2πrdæ, or k/3 replaces k. Hence the data in the above tables should be increased threefold to meet this point of view, as stated in the first paragraph of this paper. The value of k is given for each series in the tables, computed from three points of the observational data corresponding to 0 and the maximum and mean lengths. It will be noticed that V1='60 litre/min. is nearly the same for all the absorption-tubes, as it should be for initially saturated air, and has been so taken. From the value of k found for each tube I then computed the corresponding curves, these being given in the last columns of Table I. The computed curves are constructed in the chart to show the distribution of the observation with respect to them. The agreement is throughout surprisingly good; it would be impossible to get a better interpretation of the observations in view of the difficulty of colour experiments. If we compare the nuclear velocities k with the radii of the absorption-tubes, with which they were obtained, we find that they vary for the wide tubes (grey rubber and lead) as much as for the narrower tubes (lead, pure rubber, and glass). Hence k must be regarded as independent of r; and the variations found are observational errors. I conclude, therefore, that the proposition which considers decay (k) to be relatively negligible and the absorption effect of the tubes of velocity k, or an ionic velocity 3k, to be real, is one of great probability. The whole ionized region is under volume expanding stress, much like an osmotic pressure. 6. The case of the wide tubes of tin plate (2r=5 cms.) is different in character; for here the different lengths correspond to different initial densities no and n'o, while the radius of the tube and the velocity of the air-current are the same. One may assume that the initial densities are to each other as the litres per minute (V) of air saturated with phosphorus emanation put into the tube at distances and a from the jet. Thus and therefore no/n'。=V/V', k=(rv/2(x-x')) log (V/V'). Since ru/2(x-x') is about 093, the volume ratio, V/V', should be between 15 and 20. The observed values rarely exceed 2, often falling much below this. The coefficients k are thus too small as compared with the preceding set. It is interesting to compare the degree of dilution here with the above cases. The volume of saturated air added rarely exceeded 1 or 2 litres/min. The volume of air traversing the tube and due to the jet is 120 litres/min. The dilution is thus from 50 to over 100. The above cases of dilution with narrow tubes would be given by while in the tube. Otherwise, since noVo=nV for a given colour, the chart gives n/no at once. The range of values does not exceed 11 (grey rubber), the other maxima being 7 (lead, rubber), 4 (lead, glass), terminating with 1. Hence the orders of dilution in the two experiments are very different. The results for the tin tube are naturally much less certain, because the colours become dull after the dust has passed through great lengths (50) feet) of tube, or they flicker, and because is not easily found. Still the preservative qualities of dilution are undeniable; and my observations again lead me to disbelieve that diffusion within this wide, eddying current (tested) can be in question. Ignoring it, it seems most probable to adopt recent suggestions (J. J. Thomson, Rutherford, Townsend) that new ions are being continually reproduced from neutral molecules by collisions. From the present experiments with wide tubes, it would then follow that when the ionized air is markedly diluted, the new ions produced are in excess of the old. From this one may argue that their velocity must diminish, for a corresponding excess of energy is being potentialized. Admitting the delicate nature of this speculation*, it is certain, unless I have misunderstood the difficult observations with wide tubes, slow currents, and weak ionization; k here in all my experiments has never exceeded , and often fallen below, of the very definite values for small-bore tubes, swift air-currents, and nearly saturated ionization. Brown University, Another way out, possibly, would be the introduction of the above coefficient of decay k'. Observations of a different character, which I will communicate in the next of the present series of papers, seem to justify the position taken in the text. Phil. Mag. S. 6. Vol. 2. No. 7. July 1901. E III. On the Velocity of Reaction before Complete Equilibrium and before the Point of Transition, &c. Part I. By MEYER WILDERMAN, Ph.D., B.Sc. (Oxon.) *. [Plates I. & II.] PART I. I. J. W. Gibbs's general thermodynamic principles concerning equilibrium of heterogeneous systems. The general principles concerning velocity of reaction. Extension of the same for components with several potentials. Gibbs's rule of phases. II. Experiments on the velocity of reaction before complete equilibrium and the point of transition are reached: (a) The results achieved from earlier experiments. (b) The later research at the Davy-Faraday Laboratory. III. The method employed. IV. The results obtained. The general law concerning all velocities of reaction before complete equilibrium and before the point of transition of the system are reached. PART I. I. J. W. Gibbs's General Thermodynamic Principles concerning Equilibrium in Heterogeneous Systems. The General Principles concerning Velocity of Reaction. Extension of the same for Components with several Potentials. Gibbs's Rule of Phases. In his classical work, 1, 9 N his classical work "Graphical Methods in the Thermodynamics of Fluids" (Transactions of the Connecticut Academy of Arts and Sciences, 1873, vol. ii. p. 309) and "On the Equilibrium of Heterogeneous Substances" (18751878, vol. iii. pp. 108, 343), J. Willard Gibbs gave us a very detailed theoretical investigation of all kinds of chemical and physical equilibrium. Since my investigation concerns in the first instance complete equilibrium, the velocity of reaction before complete equilibrium, &c., I would first recapitulate some of the results of his work with which the present paper is connected. Gibbs gives us "the criterion of equilibrium and stability thus "For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system, which do not alter its entropy, the variation of its energy shall either vanish or be positive, i. e. (de), ≥0," where e denotes the energy and ʼn the entropy of the system. Equivalent to this is the theorem that (dn),0 *Communicated by the Author. (vol. iii. p. 109). On p. 110 of the same volume he distinguishes the different kinds of equilibrium in respect t› stability (having regard to the absolute values of the variations: it is sufficient and necessary η for stable equilibrium that (An).<0, i. e. (▲e), >0; for neutral equilibrium that (An).=0, i. e. (▲e),=0; while in general (▲n).≤0, i. e. (▲€)n ≥0 ; for unstable equilibriumn that (An). >0, i. e. (Ae),<0 ; while in general (An).≤0, i. e. (A€),≥0. A more detailed consideration and proof of the above theorem is given in the same chapter. On page 116 Gibbs gives us "the conditions relating to the equilibrium between the initially existing homogeneous parts of the given mass thus:"Let us first consider the energy of any homogeneous part of the given mass and its variation for any possible variation in the composition and state of this part. (By homogeneous is meant that the part in question is uniform throughout not only in chemical composition but also in physical state.) If we consider the amount and kind of matter in this homogeneous part as fixed, its energy e is a function of its entropy and its volume v, and the differentials of these quantities are subject to the relation de=tdn-pdr, t denoting the (absolute) temperature of the mass, and p its pressure. For tdn is the heat received, and pdv the work done by the mass during its change of state. But if we consider the matter in the mass mass as variable, and write m1, m2... m2 for the quantities of the various substances S1, S. . . . Sn of which the mass is composed, e will evidently be a function of ŋ, v, m1, mq.. mn, and we shall have for the complete value of the differential of € ... de=tdn―pdv +μ¿dm ̧ +μ¿dm2 . . . +μndmn, (12) H112...μ denoting the differential coefficients of e taken with respect to m1, m2 ・ ・ ・ Mn.' Gibbs then passes to heterogeneous systems (p. 118):-" We will now suppose that the whole mass is divided into parts so that each part is homogeneous, and consider such variations in the energy of the system as are due to variations in the composition and state of the several parts remaining (at least approximately) homogeneous, and together occupying the whole space within the envelope. We will at first suppose the case to be such that the component substances are the same for each of the parts, each of the substances S1, S2. . . S being an actual component of each part (i. e., each of the masses m1, mq . . . Mn in each part of the heterogeneous system may be either increased or diminished). If we distinguish the letters referring to the different parts by accents, the variation in the energy of the system may be expressed by de +de" + . . ., and the general condition of equilibrium requires that de+de" + ... &c. >0 for all variations which do not conflict with the equations of condition. These equations must express that the entropy of the whole given mass does not vary, nor its volume, nor the total quantities of any of the substances S1, S2 ... Sn. We will suppose that there are no other equations of condition. It will then be necessary for equilibrium that t'dŋ' —p'de' +μ{'dmy' +μ2'dm2' tri âm + t"dn"-p"do" +μ,"dm," +μ2"dm2" . . . +μn"dm"n + &c. >0 for any values of the variations for which dn'+dn'+dn'" + &c. =0, dmn'+dmn" +dm„"" + &c. =0. For this it is evidently necessary that Equations (19) and (20) express the conditions of thermal and mechanical equilibrium, viz. that the temperature and the pressure must be constant throughout the whole mass. In equations (21) we have the conditions characteristic of chemical equilibrium. If we call a quantity μ, as defined by such an equation as (12), the potential for the substance m in the homogeneous mass considered, these conditions may be expressed as |