Answer :
If you mark a point
A on the circumference of a
wheel that runs on the surface of a level road, like an ordinary
cart-wheel, the curve described by that point will be a common cycloid,
as in Fig. 1.

But if you mark a point B on the circumference of the
flange of a locomotive-wheel, the curve will be a curtate cycloid, as
in Fig. 2, terminating in nodes.

Now, if we consider one of these nodes or loops, we shall see that "at
any given moment" certain points at the bottom of the loop must be
moving in the opposite direction to the train.

As there is an infinite number of such points on the flange's
circumference, there must be an infinite number of these loops being
described while the train is in motion.

In fact, at any given moment
certain points on the flanges are always moving in a direction opposite
to that in which the train is going.

In the case of the
two wheels, the wheel that runs
round the stationary one makes two revolutions round its own
centre.

As
both wheels are of the same size, it is obvious that if at the start we
mark a point on the circumference of the upper wheel, at the very top,
this point will be in contact with the lower wheel at its lowest part
when half the journey has been made.

Therefore this point is again at
the top of the moving wheel, and one revolution has been made.

Consequently there are two such revolutions in the complete journey.