vx/vr = p/2πr = tan θ.
(14)
where θ denotes the angle made by the helical path of the point with a plane perpendicular to the axis.
Division 4. Elementary Combinations in Mechanism
§ 33. Definitions.—An elementary combination in mechanism consists of two pieces whose kinds of motion are determined by their connexion with the frame, and their comparative motion by their connexion with each other—that connexion being effected either by direct contact of the pieces, or by a connecting piece, which is not connected with the frame, and whose motion depends entirely on the motions of the pieces which it connects.
The piece whose motion is the cause is called the driver; the piece whose motion is the effect, the follower.
The connexion of each of those two pieces with the frame is in general such as to determine the path of every point in it. In the investigation, therefore, of the comparative motion of the driver and follower, in an elementary combination, it is unnecessary to consider relations of angular direction, which are already fixed by the connexion of each piece with the frame; so that the inquiry is confined to the determination of the velocity ratio, and of the directional relation, so far only as it expresses the connexion between forward and backward movements of the driver and follower. When a continuous motion of the driver produces a continuous motion of the follower, forward or backward, and a reciprocating motion a motion reciprocating at the same instant, the directional relation is said to be constant. When a continuous motion produces a reciprocating motion, or vice versa, or when a reciprocating motion produces a motion not reciprocating at the same instant, the directional relation is said to be variable.
The line of action or of connexion of the driver and follower is a line traversing a pair of points in the driver and follower respectively, which are so connected that the component of their velocity relatively to each other, resolved along the line of connexion, is null. There may be several or an indefinite number of lines of connexion, or there may be but one; and a line of connexion may connect either the same pair of points or a succession of different pairs.
§ 34. General Principle.—From the definition of a line of connexion it follows that the components of the velocities of a pair of connected points along their line of connexion are equal. And from this, and from the property of a rigid body, already stated in § 29, it follows, that the components along a line of connexion of all the points traversed by that line, whether in the driver or in the follower, are equal; and consequently, that the velocities of any pair of points traversed by a line of connexion are to each other inversely as the cosines, or directly as the secants, of the angles made by the paths of those points with the line of connexion.
The general principle stated above in different forms serves to solve every problem in which—the mode of connexion of a pair of pieces being given—it is required to find their comparative motion at a given instant, or vice versa.