116 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
a simple gas.* * * § ** For the same reason the results are not carried to a higher degree 
of approximation than that attained, in regard to the same phenomena, in an earlier 
memoir, t The present method, however, enables the approximation to be carried 
to any degree of accuracy, which was not formerly possible. Also a certain mistake 
in the previous investigation of the conductivity of a composite gas is indicated and 
corrected (cf. § 18). 
By its very nature, the problem of diffusion requires the consideration of molecules 
of two kinds, a complication which is unnecessary in the construction of a theory of 
viscosity and conduction. Perhaps this circumstance largely explains the greater pro¬ 
gress which had hitherto been made in the latter theory, as compared with the theory 
of diffusion. Until recently the only accurate expression which had been obtained 
for the coefficient of diffusion D 12 was that deduced by Maxwell in his second great 
memoir| on the dynamical theory ; it referred exclusively to a gas whose molecules 
inter-act according to the inverse fifth-power law of force. The best available value 
of I) 12 for molecules of other kinds was due to Langevin,§ but the formula, unlike 
Maxwell’s, was only approximate. It was determined on the assumption that the 
distribution of velocities in each group of molecules, relative to the mean velocity 
of the group, was according to Maxwell’s well-known law appropriate to a gas 
in the uniform steady state. The amount of error (if any) introduced by this 
assumption was unknown. In the present paper the true law of distribution is 
determined, and an exact expression is obtained for D I2 which is applicable to the 
most general case of a composite monatomic gas. It is found on comparison that 
the error of the above approximate formula is as great as 13 per cent, in extreme 
cases (§13 (e )). 
A particular case of L angevin’s formula, relating to rigid elastic spherical 
molecules, had previously been deduced by Stefan in 1871. The theories of 
Maxwell,|| Stefan, 1 ! Boltzmann, ## and Langevin, and my own earlier theory, 
all agreed in predicting no change in I) 12 with the relative proportion of the two sets 
of molecules. Another theory, originated by Meyer! f, asserted that there would 
be a large variation in D 12 as the proportion of either component varied from 0 to 1. 
* This has been dealt with in detail in my recent memoir, ‘ Phil. Trans.,’ A, vol. 216, pp. 279-348, 1915. 
t ‘Phil. Trans.,’ A, vol. 211, pp. 433-483, 1911. 
1 Maxwell, ‘Collected Works,’ ii., p. 27. His formula for D 12 is a special case of the general 
result (13'03) of this paper. 
§ Langevin, ‘Ann. de Chimie et de Physique,’ (8), v., 245 (1905); cf. also Enskog, ‘Phys. Zeit.,’ xii., 
533 (1911). The same result was independently discovered by the present writer, ‘Phil. Trans.,’ A 
vol. 211, p. 499 (1911). 
[| Maxwell, ‘ Collected Works,’ i., p. 392 ; ii., p. 57, p. 345. 
Stefan, ‘Wien. Sitzb.,’ 63, (2), p. 63, 1871 ; 65, p. 323, 1872. 
** Boltzmann, ‘ Wien. Sitzb.,’ 66, p. 324, 1872 ; 78, p. 733, 1878; 86, p. 63, 1882 ; 88, p. 835, 1883 ; 
also ‘ Vorlesungen,’ i., p. 96. 
ft Meyer, ‘ Kinetic Theory of Gases,’ p. 255 (English ed.); also Gross, ‘ Wied. Ann.,’ 40, p. 424, 1890. 
