I mentioned that in my misspent youth, long before I studied physics, I
hung around bowling alleys shooting pool. (I also played quite a bit at
the Hollywood YMCA after the war.) I was interested in trick shots.
Many trick shots depend upon the "natural", two balls in contact on the
table. If one strokes the cue ball into one of the balls in a natural,
the other ball comes out of the collision moving* along the initial
line of the centers of the balls in contact. Thus a very precisely
aimed ball could be produced as an intermediate in what might look like
a very casual initial stroke.
I discovered (or was shown, perhaps) that if one lined up several balls
periodically with spaces separating them and then drove the cue ball
into the first ball at almost any angle that would result in the first
ball striking the second ball in the line, that subsequent collisions
would result in the last ball coming off in the direction of the line.
This was more magical; it didn't look like a natural at all, and it
could be set up with as few as three or four balls. (This trick can be
extended to spaced arcs of balls, too.)
In the summer of 1958 I attended a seminar on the phenomenon of
"channeling" in crystals. This is the odd phenomenon observed when a
fast moving particle is incident on one face of crystal. Sometimes an
atom is seen to emerge from the opposite face of the crystal, along a
crystal axis in line with the incident collision, The talk was given at
General Atomic (Hi, Larry!) by Conyers Herring. Herring introduced his
talk by deriving the condition for focussing collisions, exactly the
phenomenon my trick shots employed.
It is a nice problem to derive the critical condition on interball
spacing in a regularly spaced line of billiard balls that will produce
the result described above.
*We assume that the natural is not hit in such a manner as to leave the
second ball stationary.