I. On the velocity distribution function, and on the stresses in a non-uniform rarefied monatomic gas

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Abstract
The application of the kinetic theory to the investigation of viscosity and other physical phenomena of a gas involves, in general, the determination of the distribution of the molecular velocities in the non-uniform state. This is unnecessary in the case of one special molecular model, successfully dealt with by Maxwell, but for more general models his treatment is inapplicable. It was shown by Boltzmann that the function which expresses the law of distribution of the molecular velocities must satisfy a certain integral equation, but no solution of the equation was discovered nor any further progress made until Enskog, in 1911, applied to it the method of solution by series, and thereby obtained the form of the function. Later the same writer, by a more elaborate treatment of the problem, succeeded in obtaining a solution applicable to the most general molecular model of a monatomic gas, and used it to calculate the numerical values of the coefficient of viscosity, heat conduction and of diffusion. About two years before the appearance of Enskog’s later work, a paper was published by Chapman, in which he obtained the form of the velocity distribution function, partly by using the fact that it must be an invariant. The complete solution of the function was found by applying Maxwell’s equation of transfer to certain odd and even functions of the molecular velocity. The application of the solution to the evaluation of the coefficients of viscosity, heat conduction, and so on, yielded the same results as those afterwards obtained by Enskog.

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