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.
~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.
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.