of Perfectly Elastic Spheres. 21 
exactly reversed, while those perpendicular to that line are un- 
changed. Compounding these velocities again, we find that the 
velocity of each ball is the same before and after impact, and 
that the directions before and after impact lie in the same plane 
with the line of centres, and make equal angles with it. 
Prop. IT. To find the probability of the direction of the velo- 
city after impact lying between given limits. 
In order that a collision may take place, the line of motion of 
one of the balls must pass the centre of the other at a distance 
less than the sum of their radii; that is, it must pass through 
a circle whose centre is that of the other ball, and radius (s) the 
sum of the radii of the balls. Within this circle every position 
is equally probable, and therefore the probability of the distance 
from the centre being between r and r+ dr is 
2rdr 
wee 
Now let # be the angle APa between the original direction and 
the direction after impact, then APN =}¢, and r=ssin 3¢, and 
the probability becomes 
isin ¢ dd. 
The area of a spherical zone between the angles of polar distance 
and ¢+d¢ is 
27 sin ¢ do ; 
therefore if w be any small area on the surface of a sphere, radius 
unity, the probability of the direction of rebound passing 
through this area is 
2 . 
Acar’ 
so that the probability is independent of ¢, that is, all directions 
of rebound are equally likely. 
Prop. ILL. Given the direction and magnitude of the veloci- 
ties of two spheres before impact, and the line of centres at im- 
pact ; to find the velocities after impact. 
Let O A, O Bre- 
present the veloci- 
ties before impact, 
so that if there had 
been no action be- 
tween the bodies 
they would have 
been at A and Battheendofasecond. Join A B, and let G be their 
centre of gravity, the position of which is not affected by their 
mutual action. Draw GN parallel to the line of centres at im- 
pact (not necessarily in the plane AO B). Draw aGé in the 
