ME. CLEEK MAXWELL ON THE DYNAMICAL THEOEY OF GASES. 
69 
which being taken for all the molecules will give N the total number of molecules. 
The most probable form of this function for a medium in its state of equilibrium is 
IV 
dM=~e — dnd% (56) 
In the present investigation we do not require to know the form of this function. 
Now let us consider a plane of unit area perpendicular to x moving with a velocity 
of which the part resolved parallel to x is u!. The velocity of the plane relative to the 
molecules we have been considering is u ! — (% + |), and since there are dN of these mole- 
cules in unit of volume it will overtake 
(vl— (m+|))<ZN 
such molecules in unit of time, and the number of such molecules passing from the 
negative to the positive side of the plane, will be 
(u-\-%—u')dN. 
Now let Qbe any property belonging to the molecule, such as its mass, momentum, vis 
vivd, &c., which it carries with it across the plane, Q being supposed a function of g or of 
§, tj, and £, or to vary in any way from one molecule to another, provided it be the same 
for the selected molecules whose number is <ZN, then the quantity of Q transferred 
across the plane in the positive direction in unit of time is 
J(w— .w'+DQdN, 
or 
(u- u!)§Q,dN +j£QdN (57) 
If we put QN for JQ^N, and £QN for JIQ^N, then we may call Q the mean value of 
Q, and |Q the mean value of |Q, for all the particles in the element of volume, and we 
may write the expression for the quantity of Q which crosses the plane in unit of time 
(tt-tt')QN+|QN (58) 
(a) Transference of Matter across a Plane — Velocity of the Fluid. 
To determine the quantity of matter which crosses the plane, make Q equal to M 
the mass of each molecule; then, since M is the same for all molecules of the same kind, 
M=M ; and since the mean value of \ is zero, the expression is reduced to 
(u— «')MN=(tt- u')o (59) 
If u=u', or if the plane moves with velocity u , the whole excess of matter transferred 
across the plane is zero ; the velocity of the fluid may therefore be defined as the velo- 
city whose components are u, v, w. 
((3) Transference of Momentum across a Plane — System of Pressures at any point 
of the Fluid. 
The momentum of any one molecule in the direction of x is M(w+I)- Substituting 
this for Q, we get for the quantity of momentum transferred across the plane in the 
