2 
MR. S. H. BURBURY ON THE APPLICATION 
by the constant finite force Mg' cos 6/2dt acting on each sphere in the line of centres 
during the small time 2 dt. 
Let us define this short but finite time 2 dt as the time during which the two 
spheres are in collision, or the duration of a collision. Then, during collision, if dt 
be small enough, the virial of the supposed force is sensibly constant, and we calculate 
its value as follows. Let X, /a, v, be the direction cosines of the line of centres referred 
to any axes. The coordinates of the point of contact shall be x, y, z. Then those of 
the centres of the two spheres are, c being the diameter of either sphere, 
x -f gXc, y + bye, z -f- b. vc f°r one sphere, 
and x — tj;Xc, y — \yc, z — bvc for the other. 
'Fhe component forces acting at the centre of the first sphere are 
XMg cos 6j2dt, /aM q cos 6/2dt, vM.q cos 6j2dt. 
Those acting at the centre of the second sphere are the same with reversed signs. 
For two spheres colliding with relative velocity q we find that the virial is 
cM.qcosO/2dt at each instant during collision. We have to multiply this by the 
chance that, given two spheres A and B with relative velocity q, they shall be in 
collision at any given instant. 
It is assumed that we are dealing with a space throughout which T,. is constant, 
and therefore the fact that the relative velocity is q, affords no presumption with 
regard to the relative position of the two spheres. About the centre of sphere A 
suppose a spherical surface described with radius c. An element of that surface is 
2ttc^ cos 0 sin 0 dO. Upon that element of surface form the element of volume 
27rc 2 cos 6 sin 9 dO qdt. And form a similar element of volume on the other side of 
the sphere A, that is, using tt — 6 for 6. Then the two spheres are at this instant in 
collision if the centre of the sphere B is within either of those elements. 
Let V be the volume in which n spheres are moving with T, constant. Then by 
our assumption B is as likely to be in any part of V as in any other. Therefore the 
chance that A and B, having relative velocity q, are in collision, is 
Xird cos 6 sin 0 d6 qdt 
V 
The average virial for two spheres with relative velocity q is then at each instant 
,r ' 2 47 nr cos 6 sin 6 qdt My cos 0 
= M ~ Trey = M 4 - «y + (t> - v'f + (w - w'f] 
2dt 
2 
3V 
dO 
s> 
