296 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
The Expression for A ]2 Qx. 
§ 5 (B) The number of molecules m 1 having velocity components lying between the 
limits (Ui, V,, Wj) and (if + dUi, Vx + dV 1} Wx + dW,) is, by our definition of f (Ui, Vi, Wj), 
equal to 
vifi( Ux, Vi, W,) dUxdVx dWi 
per unit volume. The number of encounters in time dt of any one of these, with a 
molecule m 2 having velocity components lying between the limits (U 2 , V 2 , W 2 ) and 
(U 2 + dU 2 , V 2 + dV 2 , W 2 + dW 2 ), the variables p, e of the encounter lying between p and 
p + dp, e and e + de , is equal to the number of such molecules m 2 contained within a 
small cylinder of length (mi/^) - '^^ dt and of sectional area p dp de , i.e., to 
v2 ( M 1 M 2 ) ^ f 2 ( U 2 , V 2 , W 2 ) Or p dp de d / ij2 d\/2 dW 2 dt . 
Thus the total number of encounters of the above type, per unit volume per unit 
time, is 
(61) (U x , V x , W X )/ 2 (U 2 , V 2 , W 2 ) C R pdpdedU , dV : dW x d U2 dV 2 dW 2 .' 
At each such encounter the change in the value of Qi(Lh, V l5 W x ) is clearly 
(62) Q x (U' x , V' x , W'j)-Qj (Ux, Vx, Wx), 
or Q 7 ! — Qi, as we shall write it for brevity. 
We shall include the effect of all possible encounters per unit volume per unit time 
if we integrate the product of (6l) and (62) over all values of e (0 to r), p (0 to 00 ) 
and (U 1} Vx, Wj), (U 2 , V 2 , W 2 ) (each from — co to + co). Such an integration will 
include encounters which are not binary, but our postulate that the gas is nearly 
perfect (§ 2) implies that our integral would be altered only inappreciably if the 
upper limit of integration for p were not infinity but equal to the very small distance 
at which two molecules cease to exercise any appreciable inter-action. Hence, 
throughout this paper, where no limits of integration are specified, it is to be under¬ 
stood that they have the above values. Thus we have 
(63) A 12 Qx = viv a (mim 2 )~ 1/2 j |j 111 | |(Q / i-Qi)/i/ 2 CrP dp de dUi dVi d\Ni dU 2 dV 2 dW 2 . 
The term/x/ 2 in the integrand may be written 
(64) v „ w,) + F 2 (U„ V 2 , W 2 )} 
= Mfff (1 + F, + F a ), 
where, in the first line, we have neglected FxF 2 , which is a second-order quantity, 
while in the second line we have made use of (45). 
