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