172 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
to calculate the exact values of these functions, but fortunately the present calculation 
does not require them to be known with any great accuracy. When n = 5 we have 
I 2 (5)/I 1 (5) = 0'501, and when n — oo the value is 0'333 (cf. Jeans’ treatise, 2nd ed., 
§33), so that the assumed values when n = 9 and n — 7 cannot be materially 
in error. 
The value of V' 0 /V 0 is in all the cases considered (n = 5 to n = o°) very nearly 
equal to unity, the correction introduced by a second approximation being so small 
that further approximations are not likely to lead to any but a negligible increase in 
accuracy. Thus the exact value of V' 0 /V in the case of rigid elastic spherical 
molecules, for instance, is not likely to differ from 1*017 (slightly greater than 1*015) 
by more than one part in a thousand. 
The exact expression for D n corresponding to molecules of the type just mentioned 
is consequently given by the following equation :— 
(13*22) 
(13*23) 
(13*24) 
D u = 1*017 
3 
32n 0 o- 2 (2/t7rm) 
0*1520 
v (2<t) 2 (hm)' 12 
= 1*200 -, 
P 
where in the last line we have made use of the formula for the coefficient of viscosity 
k for a simple gas, which has already been given by the author (‘ Phil. Trans.,’ A, 
vol. 216, § 11 (D), p. 337), viz., 
(13*25) 
= 1*016 
64 (27r) 1,2 per 2 (Il'IJl) ^ 
In my first paper on the kinetic theory (‘ Phil. Trans.,’ A, vol. 211, p. 477, 1912) 
the formulae (13'22)-(13*25) were given as above except for the omission of the 
factors 1*017 and 1*016 in the first and last, which resulted in 0*150 taking the place 
of 0*1520 in (13*23). 
The expression (13*23) for D n agrees almost exactly also with a result obtained by 
Pidduck* for the same quantity, by an entirely different method. Mr. Pidduck’s 
work is based on Hilbert’s transformation of Boltzmann’s integral equation for the 
velocity-distribution function.! His formula for D n (loc. cit., p. 101, 4l) is the same 
as (13*23) except that the numerical constant, there given only to three places of 
decimals, is 0*151. 
* Pidduck, ‘Proc. Lond. Math. Soc.,’ (2), 15, p. 89, 1915. 
t Hilbert, ‘ Math. Ann.,’ 72, p. 562, 1912; Boltzmann, ‘Yorlesungen liber Gastheorie,’ I. 
