282 DR. S. CHAPMAN ON THE LAW OF DISTRIBUTION OF MOLECULAR VELOCITIES, 
of force to the whole gas, the mean velocity and acceleration at the particular point 
and time under consideration are reduced to zero, the velocity and acceleration at 
other points throughout the gas being small, though not actually zero. 
Notation. 
§ 2 (B) We shall denote the mass of a molecule by m, the number of molecules per 
unit volume at the point (x, y, z ) by v, the components of external force acting on a 
molecule at (x, y, z ) by (X, Y, Z), the components of the velocity of a typical molecule 
by (u, v, w), and the components of the mean velocity of the gas at the point ( x , y, z) 
by (u 0 , v 0 , w 0 ). The vector difference between the velocity of a typical molecule and 
the mean velocity (u 0 , v 0 , w 0 ) will be called the peculiar velocity of the molecule; we 
shall denote its components by (U, V, W), so that 
(l) U = u—u 0 , V = v—v 0 , W = w—w 0 . 
The Distribution of Velocities. 
§ 2 (C) The distribution of the molecular velocities may be specified by ( u 0 , v 0 , iv 0 ) 
together with a function f( U, V, W), called the velocity-distribution function, 
which is defined by the following property: the number of molecules contained 
within a volume-element dx cly dz about the point ( x , y, z) which possess peculiar 
velocities whose three components lie respectively between (U, V, W) and (U+dU, 
V + dV, W + dW) is 
(2) vf( U, V, W) dl) d\l d\N dx dy dz. 
Besides being a function of U, V, W, f will depend on the mass m, the absolute 
temperature T and its space derivatives at the point (x, y, z), and on the space 
derivatives of (u 0 , v 0 , w 0 ), but not on the absolute magnitudes of the latter: for we 
may evidently impart an arbitrary additional velocity (u', v', w'), to the whole mass 
of gas without affecting the distribution of the peculiar velocities of the molecules 
at any point. It is therefore legitimate, and it will prove convenient, to suppose 
that, at the actual point under consideration, u 0 — v 0 = w 0 — 0 ; where u 0 , v 0 , w 0 
occur in any expression which has to be differentiated, however, they must not be 
made equal to zero till after the differentiation lias been performed. 
In consequence of the definition of f and of U, V, W, f must satisfy the following 
equations :— 
( 3 ) 
/(U, V, \N)dl}dMd\N = 1, 
(I) 
U/(U. V, \N)d\JdWd\N = Jjj V/(U, V, W) dl) dW d\N 
W/ (U, V, W) d\J dy dW = 0. 
