MONATOMIC GAS: DIFFUSION, VISCOSITY, AND THERMAL CONDUCTION. 121 
As regards A' 0 , this is useful for the purpose of imparting a symmetrical form to 
some of our equations ; if D denotes a differential operator of any kind with respect 
to (x, y, 2 , t), since A, + A 2 = 1 we have, by (l‘02), 
(ri2) Da, = — Da 2 = £D(a,-a 2 ) = Da' 0 . 
(c) The “ Peculiar ” Motions of the Molecules. 
So far we have been concerned with the mean velocities of the constituents of the 
gas, without considering the actual motions of individual molecules. We shall denote 
the velocities of typical molecules of the two kinds by (c),, (c) 2 or {(u) 1} (v) 1} (w)jj, 
{(u) 2 , (v) 2 , (w) 2 j, when referred to the co-ordinate axes, or, when referred to axes 
moving with the velocity c 0 appropriate to the point and time in question, by C,, C 2 
or (U„ Vi, W,), (U 2 , V 2 , W 2 ). Thus 
(l 13) C, = (c), c 0 , C 2 = (c) 2 c 0 . 
The velocities C„ C 2 will be called the “ peculiar ” velocities of the molecules. 
We have no means of determining the individual values of C,, C 2 for the molecules 
near ( x , y, 2 , t), but for a give.n state of the gas, as specified by its composition, mean 
motions, pressures, and temperature (these are expressible in terms of mean values of 
functions of C), there will be a certain frequency law, or function representing the 
distribution of various values of the velocity among the molecules. The determination 
of this velocity-distribution function is fundamental in the method of this paper. It 
will clearly involve C or its components as independent variables, together with 
certain parameters ( e.g ., pressures or mean velocities) which are dependent on 
{ x , y, 2 , t). 
The mean value of any function of the molecular velocities will be denoted by 
placing a bar over the expression representing the function. Thus, for instance 
(of. 1-07), 
(l 14) (c), = c,, (c) 2 - c 2 , G, — (c), = c, c 0 = c q/a,, G 2 (c) 2 — c 2 c 0 = c (I /A 2 . 
It is convenient at this stage to modify the meaning of our symbols C„ C 2 , which 
have so far represented vector quantities ; henceforward they will denote not the 
vectors themselves, but their amplitudes. These, of course, are essentially positive, 
scalar quantities. Thus 
(i t5) C, 2 - ud+vd + wd, C 2 2 = u 2 2 +v 2 2 +w 2 2 . 
The mean energy of peculiar motion per molecule is jmC 2 , and we shall write 
m ,Ci a = i = 3RT -> 
3ET„ 
(1-16) 
