THEORY OF VISCOSITY AND THERMAL CONDUCTION, IN A MONATOMIC GAS. 283 
If Q denotes any function of the velocity components (u, v, w) of a typical molecule, 
while Q denotes its mean value at the point (x, y, z), we have 
(5) Q = jjj Q/(U, V, \N)d\JdVd\N, 
in which, for purposes of integration, Q would be expressed in terms of w„+U, % + V, 
w 0 + W by (2). In the integrals (3) to (5), and elsewhere throughout the paper, 
integrations with respect to the velocity components are understood to be taken 
oyer all values of the variables, from — go to + oo. 
The equations (4) may, in the notation just introduced, be expressed as follows :— 
(6) D = V = W = 0. 
The Velocity-distribution Function for a Uniform Gas. 
§ 2 (D) When the gas is uniform, all the derivatives of T and of (u 0 , v 0 , w 0 ) are zero, 
and f must depend only on m, T, and (U, V, W). It has, in fact, been shown by 
Maxwell and others* that 
(7) 
where 
f = 
hmf' 1 
IT 
—lim (u 2 +V 2 +W 2 ) 
5 
( 8 ) 
2 h 
= RT, 
and R is the universal gas constant in the characteristic equation of a gas: 
(9) p = Ri/T. 
The Distribution Function for a Non-uniform Gas. 
§ 2 (E) When the gas is slightly non-uniform, f will differ slightly from the value 
given by (7), which we shall denote by f : we may therefore write 
(10) /(U, V, W) =/ 0 (U, V, W){1 + F(U, V, W)} = ^y'V Am(u2+v2+w2) {l + F(U, V, W)}. 
The function F will be of the same order of magnitude as the variations of 
temperature and velocity in the gas ; these space derivatives we shall regard as being 
of the first order, and as we shall neglect second order quantities throughout our 
work, no products of derivatives will occur in F. Hence, since F vanishes when the 
variations in the gas are zero, it must be a linear function of the space derivatives 
of T and (wo, v 0 , w 0 ), with no term independent of these derivatives. The coefficients 
will be functions of m, T, and U, V, W. 
Clearly the form of F cannot depend upon any special choice of axes of reference 
(these are throughout taken to be mutually perpendicular), so that F is an invariant 
* Cf. Jeans’ ‘Dynamical Theory of Gases.’ 
