§ 74. Differential Screw.—On the same axis let there be two screws of the respective pitches p1 and p2, made in one piece, and rotating with the angular velocity α. Let this piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut A. The velocity of advance of B relatively to C is (according to § 32) αp1, and of A relatively to B (according to § 57) −αp2; hence the velocity of A relatively to C is

α (p1 − p2),

(46)

being the same with the velocity of advance of a screw of the pitch p1 − p2. This combination, called Hunter’s or the differential screw, combines the strength of a large thread with the slowness of motion due to a small one.

§ 75. Epicyclic Trains.—The term epicyclic train is used by Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels or assist in driving them. The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in §§ 30, 31.

§ 76. Link Motion.—A slide valve operated by a link motion receives an aggregate motion from the mechanism driving it. (See [Steam-engine] for a description of this and other types of mechanism of this class.)

Fig. 113.

§ 77. Parallel Motions.—A parallel motion is a combination of turning pieces in mechanism designed to guide the motion of a reciprocating piece either exactly or approximately in a straight line, so as to avoid the friction which arises from the use of straight guides for that purpose.

Fig. 113 represents an exact parallel motion, first proposed, it is believed, by Scott Russell. The arm CD turns on the axis C, and is jointed at D to the middle of the bar ADB, whose length is double of that of CD, and one of whose ends B is jointed to a slider, sliding in straight guides along the line CB. Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis of the bar ADB; and the direction of motion of A is at every instant perpendicular to EA—that is, along the straight line ACa. While the stroke of A is ACa, extending to equal distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by the sliding of B through a comparatively short stroke, and by rotatory motions at the joints C, D, B.

Fig. 114. Fig. 115.