MR. CLERK MAXWELL ON THE DYNAMICAL THEORY OF GASES. 
63 
we shall class as encounters of the given kind. The number of such encounters in unit 
of time will be 
n^n^Fde, (17) 
where n, and n. 2 are the numbers of molecules of each kind under consideration, and F 
is a function of the relative velocity and of the angle 0, and de depends on the limits of 
variation within which we class encounters as of the same kind. 
Now let A describe the boundary of an element of volume dV while AB and A'B' 
move parallel to themselves, then B, A', and B' will also describe equal and similar 
elements of volume. 
The number of molecules of the first kind, the lines representing the velocities of 
which terminate in the element dV at A, will be 
n i=fi(a)dV (18) 
The number of molecules of the second kind which have velocities corresponding to OB 
will be 
n,=f,(i)dV; (19) 
and the number of encounters of the given kind between these two sets of molecules 
will be 
f>(a)f,(b)dN^de. (20) 
The lines representing the velocities of these molecules after encounters of the given 
kind will terminate within elements of volume at A' and B', each equal to dV. 
In like manner we should find for the number of encounters between molecules 
whose original velocities corresponded to elements equal to dV described about A' and 
B', and whose subsequent velocities correspond to elements equal to dV described about 
A and B, 
ftaY^yV’F'de, (21) 
where F' is the same function of BA.' and A'GA that F is of BA and AGA'. F is there- 
fore equal to F'. 
When the number of pairs of molecules which change their velocities from OA, OB 
to OA' OB' is equal to the number which change from OA', OB' to OA, OB, then the 
final distribution of velocity will be obtained, which mil not be altered by subsequent 
exchanges. This will be the case when 
mm=mm w 
Now the only relation between a , b and «', V is 
(23) 
whence we obtain 
/(«)= cyS, f 2 (b)= c 2 <r?, (24) 
where 
M 1 oj 2 =M 2 |3 2 (25) 
-d • • CCC 
By integrating 1 1 iC,e d% dr, d%, and equating the result to N n we obtain the 
value of Cj. If, therefore, the distribution of velocities among N, molecules is such that 
