280 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
In one special case, as Maxwell found, the actual determination of this function 
proves to be unnecessary for the purpose mentioned ; this is the case of a gas composed 
of molecules which are point centres of force varying inversely as the fifth power of 
the distance. The reasons for the peculiarity in this instance are analytical and not 
physical, and unfortunately for the simplicity of the mathematical theory of gases, 
Maxwell’s results* for such a gas do not accord with the observed data of actual 
gases. This particular molecular model is therefore interesting chiefly on theoretical 
grounds, and it is important to develop the theory for molecules of other types, 
which may better represent the behaviour of real molecules. . 
Until recently no progress had been made towards the determination of the velocity- 
distribution function for a non-uniform gas, beyond a theorem by Boltzmann,!' who 
proved that the function must satisfy a certain integral equation. In 1911, EnskogJ 
applied the method of solution by series to this equation ; he determined the form of 
the function, but without evaluating its coefficients, and his numerical approximations 
proved far from satisfactory. In 1912, Hilbert § showed that if the molecules of the 
gas are rigid elastic spheres, Boltzmann’s equation may be transformed into a linear 
orthogonal integral equation of the second kind with a symmetrical kernel, and 
deduced the existence of a unique solution. Lunn|| and Pidduck’F have since removed 
Hilbert’s' restriction to a special type of molecule, and by means of the transformed 
equation Pidduck has worked out a numerical solution of a special problem on 
diffusion. These researches are of much importance and interest, especially from the 
logical standpoint of the pure mathematician. The use of Boltzmann’s equation, 
howevever, does not appear to be the best method of actually determining the formal 
solution; thus Pidduck states that the symmetrical kernel of the transformed 
equation shows no special properties in the case of Maxwellian molecules, and in the 
numerical solution it appears to be necessary to repeat all the calculations, which are 
very laborious, in every special case which is worked out. 
In 1911, by the assumption of a simple form for the velocity-distribution function, 
I endeavoured to extend Maxwell’s accurate theory of a gas to molecules of the most 
general kind compatible with spherical symmetry.** Subsequent acquaintance with 
Enskog’s work convinced me of the approximate nature of my results, and during the 
last few years I have given much thought to the determination of the general velocity- 
distribution function. By a method which is quite distinct from that based on 
* Maxwell, ‘ Scientific Papers,’ II., p. 23. Molecules which are point centres of force varying 
inversely as the fifth power of the distance will, for the sake of brevity, be referred to as Maxwellian 
molecules. 
t Boltzmann, ‘ Vorlesungen iiber Gastheorie,’ I., p. 114. 
J Enskog, ‘ Physikalische Zeitschrift,’ XII., 58, 1911. 
§ Hilbert, ‘Math. Annalen,’ 1912, or ‘ Linearen Integralgleichungen ’ (Teubner), 1912. 
|| Lunn, ‘Bull. Amer. Math. Soc.,’ 19, p. 455, 1913. 
11 Pidduck, ‘ Proc. Lond. Math. Soc.,’ (2), 15, p. 89, 1915 ; cf. p. 95 for the statement quoted. 
** Chapman, ‘Phil. Trans.,’ A, vol. 211, p, 433, 1911, 
