whose axis of motion is fixed; the mangle-rack Ce is formed upon a separate plate, and in this example has the teeth upon the inside of the projecting ridge which borders it, and the guide-groove formed within the ring of teeth, similar to fig. 160.

This rack is connected with the piece Bb in such a manner as to allow of a short transverse motion with respect to that piece, by which the pinion, when it arrives at either end of the course, is enabled by shifting the rack to follow the course of the guide-groove, and thus to reverse the motion by acting upon the opposite row of teeth.

The best mode of connecting the rack and its sliding piece is that represented in the figure, and is the same which is adopted in the well-known cylinder printing-engines of Mr. Cowper. Two guide-rods KC, kc are jointed at one end K,k to the reciprocating piece B b, and at the other end C, c to the shifting-rack; these rods are moreover connected by a rod Mm which is jointed to each mid-way between their extremities, so that the angular motion of these guide-rods round their centers K, k will be the same; and as the angular motion is small, and the rods nearly parallel to the path of the slide, their extremities C, c, may be supposed to move perpendicularly to that path, and consequently the rack which is jointed to those extremities will also move upon Bb in a direction perpendicular to its path, which is the thing required, and admits of no other motion with respect to Bb.

The earliest shifting rack of this kind is to be found in the work of De Caus*, in which the rack is moved from one side to the other at each end of its trip by a pair of camplates, turned by the same pinion which drives the rack.

De Caus, Les Raisons des forces mouvantes, 1615. L. 1. probs. xvi. and XVII. Copied in Bockler's Theatrum Machinarum, 1662, pl. 94.

322. In the works of the early mechanists a variety of contrivances for reversing motion are to be found, in which the teeth of a driving wheel or pinion are made to quit one set of teeth and engage themselves abruptly with another set, and so on alternately; the two sets being so disposed upon the reciprocating follower as to produce motion respectively in the opposite directions in it.

For example, Aa, fig. 164, is an axis which revolves continually in the same direction, Bb an axis to which is to be communicated a few rotations to right and left alternately.

a

m

164

B

This axis carries two pinions, B and b, and the first axis has a crown-wheel at its extremity, of which the teeth extend only through half its circumference, as from m to n.

In the figure the crown-wheel is supposed to revolve in the direction from n towards m, and its teeth will accordingly act upon those of b, and cause the shaft Bb to revolve. When the last tooth n has quitted b this rotation will cease, but at that moment the first tooth m of the series will begin to act upon the lower pinion B, and turn it in the opposite direction. This contrivance is so manifestly faulty for the two reasons already discussed, of the shock at each change of motion, (Art. 314), and the danger of the first teeth that come together becoming entangled (Art. 271), that I should hardly have thought it worth describing, were it not for the numerous similar forms that present themselves in the early history of machinery, more especially in the work of Ramelli, in which this principle is exhibited in a great variety of forms, and applied not only to wheels but also to racks*.

• Vide Ramelli, I. II. III. IV. et passim. De Caus, pr. 111. and Iv. Bockler, 109, 110, 111, copied from Ramelli. Bessoni, Theatrum Instrumentorum, 1569. pl. 34.

323. Fig 165 is an application of the same principle to a double rack*, which deserves attention on account of the provision which is made to diminish the shock, and ensure the first engagement of each set of teeth.

Aa is the frame to which the reciprocating motion is to be given, A B the driving pinion; this is made in the form of a lantern, and the teeth confined to about a quarter of its circumference.

165

B

Oa

These teeth act alternately upon racks fixed to the opposite sides of the frame, and thus the frame receives a back and forward motion from the continued rotation of the pinion. In the figure the pinion revolving in the direction of the arrow is shewn at the moment of quitting the lower rack to begin its action upon the upper the tooth of each rack which receives the first action of the pinion is made longer than the others, and straight sided, and is so arranged that the action of the first stave upon it shall be oblique, by which the shock is diminished, while at the same time the stave sliding down the long side is safely conducted into the first space, and thus the proper action of the teeth and staves secured.

324. If the driver be a wheel A, fig. 166, and the follower

ЗА

B

166

an arm BC revolving round a center B, and having a wheel of an irregular form D turning round a pin at its extremity C; its teeth being kept in constant action with those of 4 by means of guide-plates, grooves, or any of the contrivances already described, then the rotation of A will produce a reciprocating motion in the arm BC, the law of which will vary according to the figure of the wheel

From Bockler, Theatrum Machinarum, No. 71.

D. For the distance of C from A continually increases or diminishes as A revolves, and therefore C will oscillate to and fro in its path.

CLASS C. DIVISION D. COMMUNICATION BY LINK-WORK.

325. I have thought it necessary to place Link-work in this class, immediately after Rolling Contact, because in some of the combinations by sliding contact I shall have occasion to refer to those which are included under this head. As the order in which these different divisions is taken is otherwise arbitrary, no inconvenience can arise from this change of plan.

The velocity ratio of a pair of arms connected by a link has been already determined (Art. 32); but it is often more convenient to investigate their motion by determining the relative positions of the parts of the system, as follows:

326. Let A, B (fig. 167), be two centers of motion; AP, BQ the arms, PQ the link;

let APR, BQ=r, AB = d,

167 P

or = (R sin – r sin p)2 + (d + r cos & − R cos 0)*

=

R2 + r2 + d2 − 2 Rd cos 0 - 2 Rr sin . sin

+2r (d - R cos 0) cos p.

For convenience assume m = R2 + r2 + d2 − 2 Rd cos 0 – 12,

n = 2Rr sin 0,

p = 2r (d - R cos 0);

whence, squaring and arranging the terms, we have

in which equation m, n, and p, being functions of 0, it appears that for every value of 0, sin has two values, or in other words, every given position AP of one rod has two possible corresponding positions of the other rod BQ, which is indeed evident; for with center B and radius BQ describe the circle Qq, and take Pq = PQ, then will Bq be also a position of this rod corresponding to AP.

If R = r, and l = d, we have the system of fig. 109 (Art. 196), and when these suppositions are introduced into the equation (1), we obtain

The first value corresponds to the system when in the position of a parallelogram, and the second to that in which the link lies across the line of centers.

If, on the other hand, we make R = l, and r = d, we have 2d2 Rd cos 0 = p ;

m =

and in fact it will be seen that if these proportions are given to the rods, Q will always coincide with A in the second position Bq, since AP = PQ and BQ = BA. Consequently, in that position AP will revolve without producing any change