68 
ME. CLERK MAXWELL ON THE DYNAMICAL TLIEOEY OF GASES. 
(y) As the expressions for the variation of functions of three dimensions in mixed 
media are complicated, and as we shall not have occasion to use them, I shall give the 
case of a single medium, 
|(S;+4,»;+l,K)=-3*,f 1 A s ®+f^+«,?;)+X(3|;+,;+f9 + 2Y|, !!l +2Z| 1 ?,. (54) 
Theory of a Medium composed of Moving Molecules. 
We shall suppose the position of every moving molecule referred to three rectangular 
axes, and that the component velocities of any one of them, resolved in the directions of 
x, y , z, are 
*«+!> v+'C, 
where u, v, w are the components of the mean velocity of all the molecules which are 
at a given instant in a given element of volume, and £, f], £ are the components of the 
relative velocity of one of these molecules with respect to the mean velocity. 
The quantities u, v, w may be treated as functions of x, y, z , and t, in which case differ- 
entiation will be expressed by the symbol d. The quantities £, pj, £, being different for 
every molecule, must be regarded as functions of t for each molecule. Their variation 
with respect to t will be indicated by the symbol }>. 
The mean values of £ 2 and other functions of £, ??, £ for all the molecules in the ele- 
ment of volume may, however, be treated as functions of x, y, z, and t. 
If we consider an element of volume which always moves with the velocities u, v, w, 
we shall find that it does not always consist of the same molecules, because molecules 
are continually passing through its boundary. We cannot therefore treat it as a mass 
moving with the velocity u, v, w, as is done in hydrodynamics, but we must consider 
separately the motion of each molecule. When we have occasion to consider the vari- 
ation of the properties of this element during its motion as a function of the time we 
shall use the symbol b. 
We shall call the velocities u, v, w the velocities of translation of the medium, and 
£, v, £ the velocities of agitation of the molecules. 
Let the number of molecules in the element dx dy dz be N dx dy dz, then we may call 
N the number of molecules in unit of volume. If M is the mass of each molecule, and 
g the density of the element, then 
MN=g> (55) 
Transference of Quantities across a Plane Area. 
We must next consider the molecules which pass through a given plane of unit area in 
unit of time, and determine the quantity of matter, of momentum, of heat, See. which 
is transferred from the negative to the positive side of this plane in unit of time. 
We shall first divide the N molecules in unit of volume into classes according to the 
value of I, tj, and £ for each, and we shall suppose that the number of molecules in unit 
of volume whose velocity in the direction of x lies between | and f n and r\-\-dq , 
£ and %-\-d% is dN, <?N will then be a function of the component velocities, the sum of 
