The construction of the toothed wheels used in machinery gives rise to some very interesting investigations in the geometry of motion. The general problem is so to shape the contours as that they shall remain in contact while the wheels turn on their centres with uniform angular velocities.

The inquiry becomes more extensive when the velocities of the wheels are to be variable; as, for example, when we seek to imitate the revolutions of the planets round the sun, and for that purpose introduce the equation of the centre.

In these cases the wheels are supposed to turn on fixed centres; but we may still farther extend the scope of our researches by removing the centres and subjecting the discs to the single condition that they roll upon each other.

If two discs A and B touch at the point S, and if they so move as that the point of contact shifts equally along the two boundaries, they are said to roll on each other; that is to say, if we measure equal distances ST, SV along the two boundaries, the points T and V will come together in the course of the movements.