where R is the gas-constant, T the absolute temperature of the medium, and N the number of molecules per gram-molecule. In the deduction of this formula, Einstein makes two important assumptions. The first is that Stokes' law holds concerning forces of diffusion. Stokes' law states that a force F will carry particle of radius r through a fluid of viscosity n with velocity F÷6r. Einstein's second assumption is that the displacement of a particle in some interval of time small in comparison with those which we can observe is independent, to all intents and purposes, of its entire antecedent history. It is the purpose of this paper to show that even without this assumption, under some very natural further hypotheses, the departure of di/t from constancy will be far too small to observe. In this connection, it is well to take note of just what the Brownian movement is, and of the precise sense in which Stokes' law holds of particles undergoing a Brownian movement. In the study of the Brownian movement, our attention is first attracted by the enormous discrepancy between the apparent velocity of the particles and that which must animate them if, as seems probable, the mean kinetic energy of each particle is the same as that of a molecule of the gas. This discrepancy is of course due to the fact that the actual path of each particle is of the most extreme sinuosity, so that the observed velocity is almost in no relation to the true velocity. Now, Stokes' law is always applied with reference to movements at least as slow as the microscopically observable motions of a particle. It hence turns out that Stokes' law must be treated as a sort of average effect, or in the words of Perrin,4 "When a force, constant in magnitude and direction, acts in a fluid on a granule agitated by the Brownian movement, the displacement of the granule, which is perfectly irregular at right angles to the force, takes in the direction of the force a component progressively increasing with the time and in the mean equal to Ft÷6πçα, F indicating the force, t the time, the viscosity of the fluid, and a the radius of the granule." It is a not unnatural interpretation of this statement to suppose that we may assume the validity of Stokes' law for the slower motions which are all that we see directly of the Brownian motion, so that we may regard the Brownian movement as made up (1) of a large number of very brief, independent impulses acting on each particle and (2) of a continual damping action on the resulting velocity in accordance with Stokes' law. It is to be noted that the processes which we treat as impulsive forces need not be the simple results of the collision of individual molecules with the particle, but may be highly complicated processes, involving intricate interactions between the particle and the surrounding molecules. It may readily be shown by a numerical computation that this is the case. It follows from considerations discussed at the beginning of my paper on The Average of an Analytic Functional that after a time the probabil ity that the total momentum acquired by a particle from the impacts of molecules will lie between x。 and x, is of the form T Superimposed on this momentum is that due to the viscosity acting in accordance with Stokes' law, namely 6πrηV, where V is the velocity of the particle. We shall write Q for 6π÷M, where M is the mass of a particle. Let us write 7 for ct. Let the total impulse received by a given particle in time t, neglecting the action of viscosity, be ƒ (7). actual momentum of the particle, as a function of t. m(t + dt) = m(t) +f(ct + cdt) − f(ct) — Qm(ct + cvdt)dt (o≤ v ≤ 1). We cannot treat this as a differential equation, as we have no reason to suppose that ƒ has a derivative. We can make it into an integral equation, however, which will read m(t) — m(o) = f(ct) — QS''m(t)dt. Clearly one solution of this integral equation is Consider m (t), the m(t) = m(o)e ̄o1 +f(ct) — Qe ̄o1ƒ'ƒ(ct)eodt, and there is no difficulty in showing that an integral equation of this sort can have only one continuous solution. Another integration gives for the distance traversed by the particle in Applying the methods of my previous paper, we get for the mean value of d, in accordance with (1); Therefore d; /t—c/(2M2Q2)| ≤1/t[m(o) (1—e ̃o)/(MQ) +c(3-e ̃o1)(1-e ̃o1)/(4M2Q3t) 2 ≤[m(o)/(MQ)]*+3c/(4M2Q3) This represents the absolute departure of d/t from constancy. Writing v for m(o)/M, the initial velocity of the particle, we get ບ This is a measure of the relative departure of d/t from constancy. cannot exceed, on the average, the velocity given on the average to the particle on the basis of the equipartition of energy; actually it is much smaller. c/(2M2Q2) can be found directly, as it is nearly the observed value of d/t. Q can be readily computed from the constants of the particles. Taking as a typical case one of Perrin's experiments on gamboge, Q turns out to be of the order of magnitude of 108, c/2M2Q2 of the order of magnitude of 10-8, and the kinetic energy velocity of the order of magnitude of 10-1. Hence the proportionate error is of the order of magnitude of 10-8. A proportionate error thus small is quite beyond the reach of our methods of measurement, so that we are compelled to conclude that ď2/t, under the hypotheses we have here formulated, is sensibly constant. There are cases, however, which seem to give a slightly different value of d/t for small values of the time than for larger values. The explanation has been suggested that over small periods the Einstein independence of an interval on previous intervals does not hold. The result of the present paper would be to suggest strongly, if not to demonstrate, that the source of the discrepancy, if, as appears, it is genuine, and not due to experimental error, is in the fact that Stokes' law itself is only a rough approximation, and that the resistance does not vary strictly as the velocity. 1 Paris, Bull. Soc. Math. France, 1919, pp. 47-70. 2 "The Average of an Analytic Functional," in the last number of these PROCEEDINGS. 3 Leipzig, Ann. Physik, 17, 1905 (549). Ann. Chim. Phys., Sept., 1909; tr. by F. Soddy. 5 Cf. Kleeman, A Kinetic Theory of Gases and Liquids, §§ 56, 60. MEASUREMENTS OF THE DEVIATION FROM OHM'S LAW IN METALS AT HIGH CURRENT DENSITIES BY P. W. BRIDGMAN JEFFERSON PHYSICAL LABORATORY, HARVARD UNIVERSITY Communicated August 10, 1921 The chief Any picture of the mechanism of current conduction in metals which takes account of the part played by the electrons would lead to the expectation of departures from Ohm's law at high current densities. On the classical free electron basis J. J. Thomson has shown that at currents of the order of 109 amp./cm2. the current would be expected to increase as the square root of the applied E. M. F., and hence that the resistance will increase indefinitely. Many attempts have been made to detect the existence of this effect experimentally, but without success. source of difficulty has been the necessity for separating the change of resistance due to the great temperature rise under the heavy current from the change due to a departure from Ohm's law. The best known attempt in this direction is perhaps that of Maxwell.1 Assuming that the departure from Ohm's law must be proportional to the square of the current, which is plausible on grounds of symmetry, he showed that at a density of 1 amp./cm2. the resistance of platinum, iron, and German silver does not differ by more than 1 part in 1012 from the resistance at infinitely small currents. His maximum density was about 5 X 10 amp./cm2. By the application of a new method I have been able to eliminate the source of error due to temperature rise, to detect the existence of the effect, and to measure it with a fair degree of accuracy. The specimen is made one of the arms of a bridge, and is traversed simultaneously by a heavy direct current and a small superposed alternating current of acoustical frequency. The resistance of the specimen to the direct current is measured with an ordinary galvanometer, and the resistance to the alternating current is measured at the same time with a telephone. If there is a departure from Ohm's law under the heavy current, that is if the relation between current and E. M. F. is not linear, the two resistances will not be equal, and from their difference the departure from Ohm's law may be calculated. The reason for F. Ө CURRENT this will be evident from an inspection of figure 1. It is to be remembered that a bridge is an instrument for balancing potentials in different parts of a net-work. At D. C. balance the potential drop corresponding to tane in figure 1 is measured and at A. C. balance the potential drop corresponding to tanë'. It is evident that the direct effect of the unknown rise of temperature is eliminated by this method, because the temperature of the conductor is the same to both the direct and alternating current. There is, however, an indirect effect which is important, and which must be eliminated. Under the joint action of the direct and alternating current the wire receives a comparatively large supply of heat steadily, with a small heating and cooling effect superposed. Under this superposed heating and cooling the wire experiences alternations of temperature which give rise to fluctuations of resistance. The large direct current flowing through the fluctuating resistance gives rise to an alternating difference of potential in one of the bridge arms, which affects the A. C. balance. This action is like that of a microphone. It becomes vanishingly small at high frequencies of the alternating current, because the fluctuations of temperature become vanishingly small under these conditions. The "microphone" effect was eliminated by making readings at a number of frequencies, plotting against the reciprocal of frequency and extrapolating to zero (that is, infinite frequency). The range of frequencies employed was from 320 to 3750 cycles per second. The extrapolation is therefore over a range only one-tenth of the observed readings. As further adding to the certainty of the extrapolation, it may be shown by a dimensional argument that the curve extrapolates to zero as a straight line. The extrapolated difference between D. C. and A. C. resistance gives the sought for departure from Ohm's law. Measurements were made on gold and silver. These metals were in the form of thin leaf, cemented to a glass backing, cut into the shape of a narrow isthmus at the part intended to carry the high current density, and cooled by a stream of distilled water flowing over the glass. Two thicknesses of gold were used, 8×10-6 and 1.67 × 10-5, and one thickness of silver, 2X 10-5 cm. Greater thicknesses of gold were tried, but good results could not be obtained. It was possible to reach current densities up to about 5X10-6 amp./cm2. Further details of the experiment and of the electrical arrangements, which were sufficiently simple and obvious, will be described elsewhere, probably in the Proc. Amer. Acad. Arts. and Sci. All of the experimental results obtained on eleven different samples of 8X10-6 gold (three of these were films formed by cathode deposit) are collected in figure 2. The ordinates are the extrapolated difference between D. C. and A C. resistance in terms of the initial resistance, and |