MR. S. H. BURBURY ON THE COLLISION OF ELASTIC BODIES. 
417 
and. therefore, 
= i f j| ■ • • (L/'' - F/) R log S. d Pi .,. d p K _, dB . 
Now let 
H = HI. . . F (log F — 1 )dp l . . . dp r + HI. . . /(log/ — 1) dp r+l . . . dp„ 
and, therefore, 
^ = ill • • • f lo § F d P> ■ ■ ■ l! P' + ![[••• % l °sf d P'+‘ ■ ■ ■ 4. 
= i fj| • • • (W - rn R log d Pl ... dp„_, c m, 
as we have seen. 
Now, this expression is necessarily negative, unless F f = F/ whenever the pair of 
systems having coordinates and velocities p x . . . can pass by collision, and, there¬ 
fore, with unchanged kinetic energy, into the state in which they are ■p l . . . p',„ that 
is, unless the Maxwell-Boltzmann distribution exist, and is then zero. H therefore 
tends to a minimum which it reaches when F/’= F f. 
We will now make 
H = H, + K 
where 1/ is the minimum value assumed by H when F f — F/i Then dll’dt = 0, 
and dK/dt = dH/dt, and is always negative. We may define the function K to be the 
disturbance, and (l/K) (dK/dt) to be the rate of subsidence of the disturbance by 
collision. In certain cases, we can calculate the rate. 
13. We have assumed that / varies only as the result of collisions. That is, if 
bf/dt denote the time variation of / due to causes other than collisions, and 8H jdt be 
formed from df/dt as dK/dt from df jdt, then 8H /dt = 0, on average. It is worth 
while to consider on what condition this nra}^ be safely assumed. 
Let 
As we are dealing with rigid elastic bodies under the action of no forces, we may 
treat / as a function of three translation velocities, and three angular velocities, 
w v iv 2 , iv z , about three principal axes of the solids. Let A, B, C, be the principal 
moments of inertia. Evidently, there being no forces, the translation velocities cannot 
vary except as the result of collisions. But for each solid, w v w. z , may vary, the 
law of their variations being Euler’s equations. We may, therefore, in calculating 
dH/bt treat/as a function of u\, iv 2 , only. Then 
MDCCCXCII. —A. 3 H 
