MR. S. H. BUR BURY OX THE COLLISION OF ELASTIC BODIES. 
415 
or, shortly, 
f. dp, +l . . . d/p n d/jp r+y ■ ■ . dp„ 
The number of pairs of systems, each consisting of one system from each set, whose 
coordinates and velocities at any instant lie within the above limits, is 
d'pi . . . d'p r dp r+ i . . . dp n Yf. 
Now let \p be a function of the coordinates which cannot become positive, and 
such that when \fj — 0 the velocities change discontinuously, and a collision occurs. 
We may use \p for one of our coordinates, expressing p n in terms of p x . . . p n -\ 
and and in like manner p n in terms of p x . . . p„-i and \p. All those pairs for which 
at the given instant i p lies between zero and — \pbt, \p being positive, will undergo 
collision within the time at after that instant. And so the number of such collisions 
in unit time is, writing E for \p, 
dpi . . . dp n _i dpi . . . d/p, l _ l dR . F/E. 
Here F is a function ofyq . . . p r only, but f by virtue of the elimination of p n and 
p > n , is now a function^ . . . p n _\ andy> x . . . p n _i E. 
Each of these collisions changes p> l into P\, &c., that is, changes the system from 
the first into the second state. The number in unit time of collisions, which with the 
same coordinates change the system from the second state with reversed velocities 
into the first state with reversed velocities, is 
dpi . . . dp H _i dp\ . . . dp' n _i dR F'/'E, 
in which Y',f' are the same functions of p\, &c., as F, f are of pj v &c. 
By a known theorem 
dp\ . . . dp n _ j = d/pi . . . dp, t _ u 
and in Maxwell’s distribution Yf—Y'f. And so the number of reverse collisions 
is in that distribution equal to the number of direct collisions. And this insures the 
permanence of the distribution. It is assumed that there are always as many systems 
with any given set of velocities as with those velocities reversed. And so we speak 
of the second state with reversed velocities as equivalent to the second state. 
12. If Yf = Yf the number of reverse collisions is not equal to the number of 
direct collisions. And, therefore, more (or fewer) pairs of systems pass out of the 
first state into the second than vice versa. In that case the number of systems M. 
whose coordinates and velocities are within the limits A. A', Is increased by collisions 
