4J8 
MR. S. H. BUBBURY ON THE COLLISION OF ELASTIC BODIES. 
and 
df 
dt 
df dvj j df dw 2 df dw s 
dw x dt dw 2 dt dio % dt 
B - C df 
C - A df 
= - ,— w. 2 Wo H- -—~ ivy Wo -f 
A du\ " 3 B dv:>„ 1 J C 
A - B tf/ 
dVJn 
W, 10. 
1 
0H B - C 
df 
df 
A JJJ du\ 
log/, ivo Wo . dw x dwo dw z , 
with two other corresponding terms, the limits being in each dz co . Now, with these 
limits, 
j]( log/- W-2 u h • dw i dw 2 d *h 
= || div 2 dw z w\ 2 io 3 {/ log ./ 1 = 00 -/ log /*=_»}. 
Now, we may assume /= 0 and/log/= 0, when any one of the three variables is 
infinite, whether positive or negative. And this assumption is sufficient to justify 
the statement 0H /dt = 0. 
14. It is possible to calculate, in a simple case, the rate at which a disturbance subsides 
by collisions. For example, two sets of elastic spheres, N of mass M, and n of mass 
m, in unit of volume. In the normal state, the number in unit of volume, whose 
velocities are represented by lines drawn from an origin to points within an element 
of volume U 3 sin 9 dO df dXJ is for the M spheres 
N t^~J e -U 3 sin 9 d9 df dXJ, 
where U, 9, f are usual spherical coordinates ; or, let us say, 
F(U) = N^Je- /iMu b 
Similarly, for the m spheres, 
f(u) = e~ hm '* 
expresses the law of distribution of velocities in the undisturbed state. We will 
write F and / for these expressions. 
We will now suppose there is a small disturbance consisting in h having different 
values for the two sets. ~Leth be written h (l + F) for the M spheres, and h (1 + d) 
for the m spheres. We shall neglect third and higher powers of D, d. Then F 
becomes in the disturbed state 
N ( ~J ( 1 + D) l e _,l(1+D)MU2 , 
