166 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
in view of the convention in § 13 (a) as to the value of a 00 . 
have 
(13'05) 
3 (m 1 +m 2 )RT 
2xj' 0 ?n 1 m 2 K / 12 (0) 
Hence, by (13'03), we 
as the exact value of D 12 when the molecules are Maxwellian. The same result 
follows readily also from (10‘05), (6'03), and (6*04). It is the same, except as regards 
the notation, as the formula deduced by Maxwell in his second great memoir 
on the dynamical theory of gases.* 
(c) A First Approximation to the Coefficient of Diffusion. 
Only in the case just considered does our general formula for D 22 reduce to a 
simple finite form : usually we must, for practical purposes, be content to make 
approximations to the exact result. As in § 9, this may be effected by taking 
successive finite convergents to our infinite determinants, which is equivalent to 
neglecting all terms in the expansion of f { U, V, W) after the first one, two or more 
at the beginning. Thus for a first approximation, taking only the central element of 
V (S mn a mn ), it is clear that V'/V is equal to 1 /a 00 or unity simply ( cf § 13a). Hence 
we have 
(13-06) 
3 (m 1 + m 2 ) RT 
27n' 0 m 1 m 2 K / 12 (0) 
(1st approximation), 
a result which also follows from (10"05) and (9"03). This, it will be noticed, is the 
same as (13‘05), showing that what is in general only a first approximation to D ]2 is 
in the case of Maxwellian molecules a strictly accurate result. 
The formula (13"06) is not new ; it was first given by L angevin,! and subsequently 
by myself | independently. In all these cases, and also in Maxwell’s investigation 
the method used was an approximate one which involves the assumption that 
the peculiar velocities of the molecules of the two constituent gases are distributed 
about the separate mean velocities c 1} c 2 according to Maxwell’s law for the steady 
state of a gas. The method of the present paper is based on an actual determination 
of the law of distribution. The assumed law just mentioned, if expressed in the 
manner adopted in § 3, would involve no a-coefficient beyond a 0 (neglecting squares 
or higher powers of the velocity of diffusion c' 0 ). This coincides with the true law 
only in the case of Maxwellian molecules (§ 6), so that only in that case is (13"06) 
exact : Maxwell himself did not prove this rigorously, though he obtained the 
* Maxwell, ‘ Phil. Trans.,’ vol. 157 (1866); or ‘ Scientific Papers,’ ii., p. 27. 
f Langevin, ‘Ann. de Chimie et de Physique’ (8), v., 245, 1905: cf. also Enskog, ‘ Phys. Zeit.,’xii. 
533, 1911. 
t ‘ Phil. Trans.,’ A, vol. 211, p. 499 (35), 1911. 
