118 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
as to suggest a doubt whether some compensating influence has not been overlooked 
in the theory, and it is desirable that the matter should be put to the test of 
experiment (see Note A, p. 196). In §§ 12, 19, it is shown that diffusion is neces¬ 
sarily accompanied by a transfer of thermal energy, and a new physical constant, 
the “ specific energy of diffusion,” is introduced and discussed. 
The method used to determine the velocity-distribution function is similar to that 
published in my recent ‘ memoir on a simple gas ; the details of the work are, of 
course, more complicated in the present case. The formulation of the equations 
of diffusion and energy for a composite gas, executed in §§10 and 12, embodies 
certain features which seem to be novel. 
We may here remark also upon some by-products of the analysis which suggest 
interesting developments in the field of pure mathematics. The comparison of 
Lorentz’s solution of the problem of electronic diffusion with my own has led to 
expressions for it and sin itx of an altogether new form. Lorentz used Boltzmann’s 
integral equation for the velocity-distribution function, and obtained a solution in 
finite terms involving it ; the solution arrived at in this paper is determined by the 
use of the aggregate of the equations of transfer (§2), which is really equivalent 
to Boltzmann’s equation. The result is expressed, however, in terms of the quotient 
of a symmetrical infinite determinant by its principal minor, and formulae of this 
kind are hence found for it (and also for sin i tx). The elements of the determinant 
are expressible simply, in terms of gamma functions. A further study of the subject 
from the analytical point of view would probably be fruitful in results of interest and 
importance (see Note B, p. 196). 
I hope later to apply the present methods to the examination of the problems 
offered by rarefied gases. So far, however, as concerns the mean-free-path phenomena 
in monatomic gases under normal conditions, the investigation imperfectly attempted 
in my memoir of 1911 is completed by this and the second paper already referred to 
(‘Phil. Trans.,’ A, vol. 216). 
It is a pleasure here to make grateful acknowledgment of my indebtedness 
to Sir Joseph Larmor throughout these investigations, which were started under 
his influence, and would hardly have been carried to this stage but for the inspiration 
afforded by his continued encouragement and interest. 
§ 1. Analysis of the Dynamical State of a Composite Gas. 
(a) Notation * 
The gas considered in this paper is one composed of molecules of two kinds, each 
having the property of spherical symmetry (or, in brief, each being “monatomic”). 
* In numbering the equations I have adopted the decimal method introduced by Peano. The 
number to the left of the decimal point is the number of the section, and within any section the numbers 
to the right, if read as decimals, are in numerical order. With this method it is possible, by the 
introduction of a third or even fourth figure, to number equations inserted between others already 
numbered, without having to alter the references to all succeeding equations. 
