146 
DR. S. CHAPMAN ON THE KINETIC THEORY OF A COMPOSITE 
Consequently, in the case of Maxwellian molecules, F (c 2 ) and H (c 2 ) each reduce to 
their first terms, while G (c 2 ) reduces to two terms only, the solution being quite 
finite. 
§7. The General Solution when m l : m 2 is Very Large. 
When the mass m, of the heavier molecules is very large compared with that of 
the lighter molecules, so that mo/m, may be supposed zero, an exact general solution 
may be obtained in simple terms. This was first proved by Lorentz,* in connection 
with the theory of electrons. His method is much simpler than that of this paper, 
from which, however, his results may, with little difficulty, be deduced as a very 
special case. The deduction will be described in some detail, since the knowledge 
of the exact solution throws valuable light on the convergence of our successive 
approximations ; it also leads to an expression for x which is of interest, as being in a 
form which, so far as my knowledge goes, is new. 
We suppose that the effect of collisions between the molecules m 2 is negligible, so 
that p 22 and p' 22 may be omitted from our calculations ; if the molecules are rigid 
elastic spheres of radii ay: <r 2 , this amounts to the neglect of o-^/oy 2 ( cf ’. § 9 (f )) or, in 
general, to <f> 22 /<p n . 
It may readily be seen from (4T3), (4T4) that if m 2 = 0 the value of B k (m, n), 
with whatever suffix, is zero except when k = 0, and that 
(7-01) 
f B° 12 i 2 (m, n) = (2x 2 ) m+n , 
iBV {m, n) = (2 y 2 ) m+n , 
B° 2112 (m, n) = 2 m+n y 2m x 2n , 
B u 122 i (m, n) — 2 m+n x 2m y 2n . 
We also require B 1 (m, n) to the first order in m 2 , as follows :—• 
(7'02) 
B , 2 , 2 [yxi) ?&) 
B^m (m, n) 
^mnp. 2 2 m+n x 2( ' m+n ~ 1) y 2 , 
jjmnpo 2 m+ ”?/ 2(m+ n ~ 1) x 2 , 
B l 2112 ( m , n ) — 0 mn p.2^ m + n y 3m ^ 2n , 
Bh 221 (m, n) = ^mnp..^2 mJrn x 2m y 2n . 
From these, by means of (4'07)-(4'12) we deduce the following expressions for 
p (r, s ), p (r, s), retaining only the terms of highest order :— 
(/ 03) P 12 ( r i s i) — (2rs + r + $+■§-) {r + s+-|-) r+s Ei / 12 (0), 
(7 04) Pi 2 { r 2 s i) = ~~ ^ 7r) T I '2/ w i2 r+s (r + f) r+ i ( s + ‘f)sGK- / i 2 (o), 
(7 05) p2i( r i s -i) — ~ ‘977j/ 1 i' 2 /* 2 2 r+i ( , r+f-) r+ i (s+f-^^K^ (0), 
* Lorentz, ‘Archives N^erlandaises,’ 10, p. 336, 1905 ; ‘Theory of Electrons,’ p. 268. A more 
general form of Lorentz’s theory is given in Jeans’ ‘ Dynamical Theory of Gases,’ 2nd ed., §§ 314, et seq . 
I am indebted to Mr. Jeans for pointing out the interest of a detailed comparison of Lorentz’s theory 
with this special case of my own, and in consequence I have rewritten § 7 with greater fullness than at 
first. 
