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.
[Illustration: 1]
[Illustration: 2]
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.