X The **involute** of the catenary is called the tractory, tractrix or antifriction curve; it has a cusp at the vertex of the catenary, and is asymptotic to the directrix.

- Transverse section of an **involute** leaf.

**Involute** TeethThe simplest form of tooth which fulfils the conditions of 45 is obtained in the following manner (see fig.

~s Now, suppose a tracing point T Pa to be fixed to the cord, so as to be carried along the path of con- Dz a tact P11P2, that point will trace on a plane rotating along with the wheel I part of the **involute** of the base-circle DfD1, and on a plane rotating along with the wheel 2

Part of the **involute** of the base- C2

Consequently, one of the forms suitable for the teeth of wheels is the **involute** of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.

All **involute** teeth of the same pitch work smoothly together.

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.

U=cv~-+-a To apply this to **involute** teeth, let ci be the length of the approach, c2 that of the recess, u1, the mean volocity of sliding during the approach, u2 that during the recess; then civil i\ c,v/1 I

Which, substituted in equation (63), gives the work lost in a unit of time by th~ friction of **involute** teeth.

This result, which is exact for **involute** teeth, is approximately true for teeth of any figure.