The direction of motion of a point in a turning piece being perpendicular to a plane passing through its axis, the condition that each pair of points in contact with each other must move in the same direction leads to the following consequences:—
I. That, when both pieces rotate, their axes, and all their points of contact, lie in the same plane.
II. That, when one piece rotates, and the other shifts, the axis of the rotating piece, and all the points of contact, lie in a plane perpendicular to the direction of motion of the shifting piece.
The condition that the velocity of each pair of points of contact must be equal leads to the following consequences:—
III. That the angular velocities of a pair of turning pieces in rolling contact must be inversely as the perpendicular distances of any pair of points of contact from the respective axes.
IV. That the linear velocity of a shifting piece in rolling contact with a turning piece is equal to the product of the angular velocity of the turning piece by the perpendicular distance from its axis to a pair of points of contact.
The line of contact is that line in which the points of contact are all situated. Respecting this line, the above Principles III. and IV. lead to the following conclusions:—
V. That for a pair of turning pieces with parallel axes, and for a turning piece and a shifting piece, the line of contact is straight, and parallel to the axes or axis; and hence that the rolling surfaces are either plane or cylindrical (the term “cylindrical” including all surfaces generated by the motion of a straight line parallel to itself).
VI. That for a pair of turning pieces with intersecting axes the line of contact is also straight, and traverses the point of intersection of the axes; and hence that the rolling surfaces are conical, with a common apex (the term “conical” including all surfaces generated by the motion of a straight line which traverses a fixed point).
Turning pieces in rolling contact are called smooth or toothless wheels. Shifting pieces in rolling contact with turning pieces may be called smooth or toothless racks.