64 
ME. CLEEK MAXWELL ON THE DYNAMICAL THEOEY OF GASES. 
the number of molecules whose component velocities are between f and % + d%, n and 
and £ and %-\-d% is 
TV P+*+P 
—z-d*didi;, (26) 
then this distribution of velocities will not be altered by the exchange of velocities among 
the molecules by their mutual action. 
This is therefore a possible form of the final distribution of velocities. It is also the 
only form ; for if there were any other, the exchange between velocities represented by 
OA and OA' would not be equal. Suppose that the number of molecules having velo- 
city OA' increases at the expense of OA. Then since the total number of molecules 
corresponding to OA' remains constant, OA' must communicate as many to OA", and so 
on till they return to OA, 
Hence if OA, OA', OA", &c. be a series of velocities, there will be a tendency of each 
molecule to assume the velocities OA, OA', OA", &c. in order, returning to OA. Now 
it is impossible to assign a reason why the successive velocities of a molecule should be 
arranged in this cycle, rather than in the reverse order. If, therefore, the direct exchange 
between OA and OA' is not equal, the equality cannot be preserved by exchange in a 
cycle. Hence the direct exchange between OA and OA' is equal, and the distribution 
we have determined is the only one possible. 
This final distribution of velocity is attained only when the molecules have had a great 
number of encounters, but the great rapidity with which the encounters succeed each 
other is such that in all motions and changes of the gaseous system except the most 
violent, the form of the distribution of velocity is only slightly changed. 
When the gas moves in mass, the velocities now determined are compounded with the 
motion of translation of the gas. 
When the differential elements of the gas are changing their figure, being compressed 
or extended along certain axes, the values of the mean square of the velocity will be 
different in different directions. It is probable that the form of the function will then be 
(27) 
where a, (3, y are slightly different. I have not, however, attempted to investigate the 
exact distribution of velocities in this case, as the theory of motion of gases does not 
require it. 
When one gas is diffusing through another, or when heat is being conducted through 
a gas, the distribution of velocities will be different in the positive and negative directions, 
instead of being symmetrical, as in the case we have considered. The want of symmetry, 
however, may be treated as very small in most actual cases. 
The principal conclusions which we may draw from this investigation are as follows. 
Calling a the modulus of velocity, 
1st. The mean velocity is 
(28) 
