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.

39), as in banana and apricot; or its edges are rolled inwards, **involute** (fig.

- 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 teeth of a rack, to work correctly with wheels having **involute** teeth, should have plane surfaces perpendicular to the line of connection, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action.

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.