Its value at intermediate instants is given by the following equations: let φ1, φ2 be the angles respectively made by the central planes of the forks and shafts with the plane OC1C2 at a given instant; then
cos θ = tan φ1 tan φ2,
| α2 | = − | dφ2 | = | tan φ1 + cot φ1 | . |
| α1 | dφ1 | tan φ2 + cot φ2 |
(40)
§ 67. Intermittent Linkwork—Click and Ratchet.—A click acting upon a ratchet-wheel or rack, which it pushes or pulls through a certain arc at each forward stroke and leaves at rest at each backward stroke, is an example of intermittent linkwork. During the forward stroke the action of the click is governed by the principles of linkwork; during the backward stroke that action ceases. A catch or pall, turning on a fixed axis, prevents the ratchet-wheel or rack from reversing its motion.
Division 5.—Trains of Mechanism.
§ 68. General Principles..—A train of mechanism consists of a series of pieces each of which is follower to that which drives it and driver to that which follows it.
The comparative motion of the first driver and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the directional relations of the several elementary combinations of which the train consists.
§ 69. Trains of Wheelwork.—Let A1, A2, A3, &c., Am−1, Am denote a series of axes, and α1, α2, α3, &c., αm−1, αm their angular velocities. Let the axis A1 carry a wheel of N1 teeth, driving a wheel of n2 teeth on the axis A2, which carries also a wheel of N2 teeth, driving a wheel of n3 teeth on the axis A3, and so on; the numbers of teeth in drivers being denoted by N′s, and in followers by n’s, and the axes to which the wheels are fixed being denoted by numbers. Then the resulting velocity ratio is denoted by
| αm | = | α2 | · | α3 | · &c. ... | αm | = | N1 · N2 ... &c. ... Nm−1 | ; |
| α1 | α1 | α2 | αm−1 | n2 · n3 ... &c. ... nm |