**Epicycloidal** Teeth.The most convenient rolling curve is the circle.

In fig~ I04 let BB be part of the pitch-circle of a wheel with **epicycloidal** teeth; dC the line of centres; I the pitch-point; EIE.

Nearly **Epicycloidal** Teeth: Williss Method.To facilitate the drawing of **epicycloidal** teeth in practice, Willis showed how to approximate to their figure by means of two circular arcsone concave, for the flank, and the other convex, for the faceand each having for its radius the mean radius of curvature of the **epicycloidal** arc. \Villiss formulae are founded on the following properties of epicycloids Let R be the radius of the pitch-circle; r that of the describing circle; 8 the angle made by the normal TI to the epicycloid at a given point T, with a tangent-to the circle at Ithat is, the obliquity of the action at T.

The smallest number of teeth in a pinion for **epicycloidal** teeth ought to be twelve (see 49)but it is better, for smoothness of motion, not to go below fifteen; and for involute teeth the smallest number is about twenty-four.

The epicycloid was so named by Ole Romer in 1674, who also demonstrated that cog-wheels having **epicycloidal** teeth revolved with minimum friction (see Mechanics: Applied); this was also proved by Girard Desargues, Philippe de la Hire and Charles Stephen Louis Camus.