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Proceedings of the Poyal Society 
fraction of the whole group. After a very long period the final state 
will be one of irregular fluctuation about this “ special ” distribution, 
a fluctuation confined within limits which are (relatively) narrower 
as the whole number of particles is greater. This special distribu- 
tion, of course, is that of uniform number of particles, zero of average 
velocity in every direction, and “ error-law ” distribution of equal 
amounts of kinetic energy, in every region of given volume large 
enough to contain a very great number of particles. Of this pro- 
position satisfactory proof has been given by Maxwell ; but we may 
obtain it very simply by the following considerations. 
4. The tendency is to levelling all round. The only things to be 
levelled are the distribution of the whole momentum in each direc- 
tion, and the distribution of energy among the various velocities. 
The first depends on direction cosines, the second on their squares. 
From a point, lay olf lines representing the velocities of the various 
particles, The ends of these lines of lengths r to r + dr must be 
uniformly spread in the volume 47 rr 2 dr. This secures that the 
momentum is equally arid similarly distributed in all directions. 
The energy condition requires that, if there be a final state at all$ 
the number of ends in unit of that volume shall be subject to the 
“ error-law,” i.e ., expressed by Ae _r2 / a2 . This law is the only one 
which (when the momentum condition is secured) does not make 
the calculated number of “ends” in a given volume dependent 
upon our choice of rectangular axes. We have now to show how 
the collisions tend to produce this result, and also to prove that 
they tend to maintain it. 
Impacts on the containing vessel do not alter r, and thus 
can shift only the position of an “ end ” on the spherical surface 
of which r is the radius. And the impact of two equal particles 
(as we saw above), does not alter the distribution of velocity along 
the line of centres, nor in any direction perpendicular to it. 
Hence impacts, in all of which the line of centres is parallel to 
one common line, produce no change in the arrangement of velocity- 
components along that line, nor along any line at right angles to it. 
But there will be, in general, changes along every other line. It is 
these which lead gradually to the final result, in which the distribu- 
tion of velocity -components is the same for all directions. 
When this is arrived at, collisions will not, in the long run, tend 
