and the link AB turning about a fixed vertical pin at A; the same pin for all the corpuscles, which move in horizontal planes that are indefinitely close to each other. Each nucleus repels any other within a minute circle of influence with a force depending on the horizontal distance only, according to some unknown law. Let the angles made by AB and BC with the horizontal be and respectively. The present velocities of any molecule p' and ' are considered as each an undetermined function of previous p's and 's; the two velocities of any molecule among the progenitors being not now independent of each other. Identifying the mean energy of a molecule as deduced (by expansion of the functions) from that consideration with what it is as given by dynamical theory, we find by parity of reasoning that p', the present velocity of a molecule, is in effect the (weighted) sum of innumerable prior p's and 's, and likewise ', of 's and o's. Accordingly, the distribution of o' and is normal; but as among the constituents, each pair, e. g., p, and are now correlated, the exponent of the resulting error-function may be expected to involve the product, as well as the squares of velocities*. As before, we may pass from the distribution of the "universe" to that of the genus †. Parity of reasoning is applicable to molecular motion of the most general character, admitting movements of translation and other degrees of freedom. II. The result is reached more readily by the second argument with regard to the free molecules moving (with any number of degrees of freedom) in the space outside the spheres of influence. We have now to determine the law of distribution f, so that flogf should be a minimum, subject to the conditions that Ef=const., EƒT= [T], where Σ is used to denote integration with respect to all the velocities (or components of momentum), but not the co-ordinates. Tis the quadratic expression in terms of the velocities (or components of momentum) for any assigned values of the co-ordinates, say =A11Q12+2A12Q1Q2 + A 22Q22 + ... + α 1912+2α127192... 11 As to the formation of correlated compounds from correlated elements, and other propositions implied in this paragraph, see Camb. Phil. Trans. loc. cit. p. 116 et seq. Cp. above, p. 263. where the A's are, in general, functions of the co-ordinates Q1, Q2...; and the a's are likewise functions of the g's. Whence at once there is obtained for the ƒ form J exp-hT. As before, we may pass from this expression for the law of distribution of the genus to that of the universe *. III. The third argument in the simple form so far adopted is not applicable to cases in which the co-ordinates are changed by an encounter. Recourse must be had to that theorem of Liouville which leading writers have called in at an earlier stage. E. It would be possible to advance further in other directions-in particular, where a field of force occurs-on the lines of the first and second arguments, without the aid of Hamiltonian Dynamics, by mere Probabilities. XXX. On the Permeability of Thin Fabrics and Films to Hydrogen and Helium. By Prof. J. C. MCLENNAN, F.R.S., and W. W. SHAVER, B.A., University of Toronto †. IN I. Introduction. Na recent paper by R. T. Elworthy‡ and V. F. Murray the diffusion of hydrogen and helium through thin rubber fabrics was discussed, and the results of measurements made by them on several samples of balloon fabrics were given. In these experiments the amount of gas diffusing through the fabrics was measured by a Shakespear Katharometer and by a Jamin Interferometer. As the method was one capable of wide application it was decided to use it in determining the permeability of liquid films to various gases, and the following paper describes some experiments made upon the passage of hydrogen and helium through soap films. The study of gas transfusion through membranous tissues is an important physiological problem, and it was thought on this account that it would be useful and might. prove interesting to measure the rate of gas diffusion through films of various materials, with a view to formulating a more exact theory of the process of gas transfusion than exists at present. * Above, p. 264. + Communicated by the Authors. II. Preliminary Experiments. In order to test the apparatus and to acquire a working familiarity with the instruments, a preliminary study of the diffusion of hydrogen through the fabrics used by Elworthy and Murray was made. The apparatus used and the method of assembling it was the same as described in their paper. The fabrics used by them were inserted as a separating diaphragm in an air-tight drum-like vessel. Two gases were brought into this drum, one on either side of the fabric, and their transfusion was determined by tests on the gases by means of the instruments mentioned above. For a full description of the Shakespear apparatus the reader is referred to the paper by Elworthy and Murray. It will suffice here to say that this apparatus was made by the Cambridge Scientific Instrument Co., and that its principle is based on the variation in resistance of a heated platinum coil, constituting one branch of a Wheatstone Bridge circuit, when the gas mixture surrounding the cell has its thermal conductivity varied by changes in its component parts. The two methods adopted were (1) to pass a continuous stream of pure air and one of pure hydrogen on opposite sides of the fabric as a dividing diaphragm, and (2) to enclose a known quantity of pure air on one side and to pass a continuous stream of pure hydrogen past the other side of the fabric. In the present experiments both methods were followed, but gas tests were made with the katharometer only. It was found that 20° C. was a more suitable temperature for working at than 15°5 C. as previously used by Elworthy and Murray. The measurements obtained were made by keeping the permeameter and connexions in a thermostat at 20°0 C., the variation in temperature being not more than 0.2 C. III. Calibration. The katharometer used to detect small percentages of hydrogen or of helium in air had already been calibrated for both gases; but this calibration was checked by noting the galvanometer deflexions for a given sample of gas, deducing the percentage of helium or hydrogen present from the calibration curve and then checking the result by actually weighing a known volume of the sample studied. It was found that the values obtained by the latter method fitted in very closely with the calibration curve of Elworthy and Murray. It may be stated here that in their work it had been well established that the curve obtained by plotting galvanometer deflexions against percentages of hydrogen or helium present in air was a straight line through the origin. The calibration showed that (1) 259 mm. deflexion on the scale 1 metre from the galvanometer represented 1 per cent. hydrogen in air, and (2) 163 mm. deflexion on the scale 1 metre from the galvanometer represented 1 per cent. of helium in air. The following table gives a comparison of the results obtained in the present experiments with those obtained by Elworthy and Murray when using the same fabrics. In each case the permeability is given as being the number of litres of gas permeating 1 square metre of a fabric in 24 hours :: After the preliminary experiments had been made, an attempt was made to employ the same method in making a determination of the transfusion of hydrogen and of helium through a soap film. Sir James Dewar † in a paper presented at a meeting of the Royal Institution of Great Britain in Jan. 1917, described many interesting experiments with long-lived soap bubbles and films, among them being a determination of what he calls "gas transference" through The fabric numbers refer to samples of balloon fubrics described in the papers by Elworthy and Murray. + Dewar, Paper, "Soap Bubbles of Long Duration," presented at weekly meeting of the Royal Institution of Great Britain, Jan. 19, 1917. a soap bubble, by blowing a hydrogen bubble in hydrogen and noting the decrease in diameter as time went on, due to the slight excess pressure inside the bubble. What he measured was the excess of the rate of gas diffusion outward over the rate inward through the film, and he found that as the soap bubble became thinner the gas transference became greater. In the present experiment the endeavour was to determine the actual rate of gas flow per square centimetre through the film, keeping the film as nearly constant in composition and thickness as possible. V. Description of Apparatus. A small cylindrical brass chamber (see fig. 1) was made for the film in two sections with a ground brass joint, which, when covered with soft wax and pressed together, made the vessel air-tight. Each section was 41 cm. in diameter and 70 cm. in height, having inlet and outlet tubes as shown in the diagram. The top section A, fig. 1, was closed by a window of plate glass, G, put on with hard wax, so that when a source of light was held directly over the chamber, its image in the film could be distinctly seen and in this way the character of the surface of the film-whether concave, convex, or plane-was known at once by the character of the image produced. Knowing the curvature of the film one could adjust the pressures of hydrogen and air on either side very accurately and so as to keep the film plane and therefore eliminate the diffusion due to excess pressure on either side. The brass ring C, fig. 1, supporting the film was 4·95 cm. in diameter, and ground down to a sharp edge. An annular channel, D, was made in the outer part of the supporting ring, and the whole soldered in the lower section of the film chamber, leaving about 0.6 cm. of the brass ring projecting above the wax surface. In this way the soft wax used in making the joint air-tight was prevented from contaminating the film and destroying its surface tension. To overcome the difficulty of evaporation and drainage from the film, that is to keep its composition and thickness constant, the air and hydrogen used were both saturated with water vapour before entering the chamber, and, in addition, a means of adding solution to the film was provided in the following way. A bent tube, T, was inserted in the upper chamber as indicated in the diagram, having a thistle tube connected to the outer end by rubber tubing. A small amount of the same soap solution used in making the film was poured into the thistle tube and a drop of this was |