410 
MR. S. H. BURBURY ON THE COLLISION OF ELASTIC BODIES. 
that when \Jj = 0 a collision— i.e., a discontinuity of the velocities—occurs, we may 
treat the pair of systems M, m in all respects as a single system within the preceding 
investigation. 
7. All those systems, or pairs of systems, for which at any instant xjf lies between 
zero and — ( chjt/dt) St, dxfj/dt being positive, will undergo collision within the time St 
after that instant. We may, therefore, take dxfj/dt or It as measuring the frequency 
of collisions for given values ofyq, &c. 
8. From the linear equations (B) above given we can find any of the velocities, for 
instance, p }) as a linear function of Sj . . . S„_j It, and p\ will be the same function 
with — It written for It. Hence p 1 3 — pp = 4R2pS where 
YpS = pi&j + poS^ ~b &c., 
and the p’s are functions of the coordinates and constants. 
Also (pp — p\ 2 ) It = 4R 2 2pS. 
Now, without altering E or It, or the coordinates, let us make S 2 . . . S, t _! pass 
through the whole range of values consistent with 
2E = \ 3 R 3 +/(S 1 ...S„_ 1 ) .(E). 
Also let (j) (Sj . . . dS 1 . . . dS /l _ l be the number of systems for which these 
variables lie between the limits 
S x and Sj -f dS, 
S«_i and S„_! + 
E and R and the coordinates being constant. 
Then 
jj • •. (k - pV) r • t (s.... s,_,) ds l ... ds._, 
= 4R J ||... (f> (Sj. . . S„_]) S/xS ciS,. . . c/S,,-!, 
the integrations being over all values consistent with (E). 
Now, in the Maxwell-Boltzmann distribution </> (S x . . . S«_ x ) is a function of 
the kinetic energy only, and is therefore constant throughout this integration. There¬ 
fore the right-hand member of the last equation is zero, because for any given set of 
values of Sj . . . S»_j satisfying (E), there is a corresponding set with the opposite 
signs also satisfying it. Therefore 
jj" . . . R<£ (Sj. . . S„_j) (_pj 2 — p j 3 ) dSj . . . dS il _ l = 0, 
and, therefore, the average value of^pj 3 — p\ 2 for all collisions given E and R is zero. 
The same is true for p. 2 2 — p'p, &c. 
