point in the link, from its approximate circular path, is proportional very nearly to the product of the lengths of the segments into which it divides the link. For example, in fig. 209 we have greatest deviation of ƒ Lf fF = nearly; 256. Roberts's Parallel Motion. This might also be called the W parallel motion, from its shape. venient form for horizontal engines. are shown in the figures: a simpler plex case in fig. 212. It is considered to be a conTwo cases of this combination case in fig. 211; a more com I. In Fig. 211, let D E be the straight line of stroke which the guided point is to follow approximately, and A the middle of that line. Draw two p length of each of these legs should not be less than D A × 0.843; but it may be as much greater as the available space will permit; and the greater it is the more accurate will be the motion. Join B C, which of course Ꭺ Fig. 211. is = DA = AE. Then A B C represents a rigid triangular frame, of which the apex A is the guided point; while the angles of the base, B and C, are jointed to two levers or bridles, BD, CE, which turn about axes at D and E respectively. The two extreme positions of the triangle A B C are marked respectively D B'C' and E B C, the points B' and C' coinciding. The reason for prescribing that the length of each leg of the triangles shall not be less than the base × 0.843 is, that this proportion between these lengths makes the points B', C', and E lie in one straight line, and the points C", B", and D in another straight line, in the two extreme positions of the combination respectively. II. The second arrangement, fig. 312, is to be used in those cases in which it may be inconvenient to have the axes of motion D and E of the bridle-levers traversing the line of stroke. Let A' A A' be the line of stroke, and A its middle point. Draw an isosceles triangle, A B C, of a convenient size and figure, with its apex, A, at the middle point of the stroke, and its base, BC, parallel to the line of stroke. Draw the same triangle in its two extreme positions, A'B'C' and A" B' C', leaning over equally in opposite B OE Fig. 212. directions; the sides A'B' and A"C" may conveniently be made vertical; but this is not essential. Find, by plane geometry, the centre D of a circular are traversing the three points B', B, B'; also the centre E of a circular arc traversing the three points C', C, C". DB and EC will be the two bridle-levers, and D and E the bases of their axes of motion. ADDENDUM TO ARTICLE 142, PAGE 141. Intermittent Gearing--Counter-Wheels-Geneva Stop.-In the intermittent toothed wheelwork described at pages 139 to 141, the wheels and their teeth are so designed that, during the transmission of motion, the velocityratio has a constant value. In some cases, however, it is not necessary that the velocity-ratio should be constant, provided only that the follower performs a certain part of a revolution for each revolution of the driver, as in mechanism for counting revolutions. The simplest mechanism of that sort consists of a toothed wheel of the ratchet form (Article 194, page 207, fig. 145), driven by a wiper or single tooth (Article 164, page 175), which projects from a rotating cylinder, and has its length adjusted so that the arc of contact is equal to the pitch of the ratchet wheel. This requires no special explanation. But there are cases in which the abruptness of the action of the wiper would be disadvantageous, and in which it is desirable, in order to prevent shocks, that the follower should be set in motion and stopped by insensible degrees. The following is the most precise method of designing a pair of wheels turning about parallel axes, in which the follower is to count the revolutions of the driver by turning through a certain aliquot part of a revolution for each revolution of the driver; the action being absolutely without shock, and capable of taking place in either direction. In fig. 100 A, let A B be the line of centres, A the trace of the axis of the driver, and B that of the follower. In the example shown in the figure, the follower is to make one-fifth of a revolution for each revolution of the driver; but the same rules are applicable to any given number of aliquot parts. Draw straight lines radiating from B (marked by dots in the figure), so as to divide the angular space round B into twice the given number of equal aliquot parts; the line of centres, B A, being one of these radiating lines. Then from A let fall perpendiculars A C and A D on the two radiating lines which lie nearest to the line of centres, and complete the regular polygon of which B is the centre, and A C and AD are two of the half-sides. (In the = figure this is a regular pentagon.) On the line of centres lay off A E A C A D; and about B, with the radius B E, describe a circle; this circle will cut the alternate radiating lines in a set of points, such as F, which will be the centres of the semicircular bottoms of a set of notches that will gear in succession with a pin carried by the driver. Having assumed a convenient radius for that pin, draw a circle to represent it about the point C as a centre. The pin will be carried by a plate or arm projecting from the axis A, in a different plane from that of the driven wheel. For the bottom of the notch C F draw about F, as a centre, a semicircle of a radius equal to that of the pin C, increased by an allowance sufficient for clearance; and for the two sides of that notch draw two straight lines touching that semicircle, and parallel and equal to F C. Draw all the other notches of the driven wheel of the same figure and dimensions. In the intervals between the notches the rim of the driven wheel is to consist of a series of equal hollow circular arcs, described respectively about the angles of the polygon with a radius such as to leave a thickness of material sufficient for strength and durability at each side of each of the notches; for example, the arc G H K is described about the centre A, so as to leave a sufficient thickness of material at G and K. The periphery of the driver is to consist of a cylindrical surface extending round the dead arc, H K M, and fitting smoothly, but not tightly, into each of the hollows, such as G H K, in the rim of the follower; and of a hollow, HL M, of a depth and figure such as to clear the horns, such as G and K, of the notches in the follower. The angular extent of that hollow, HA M, is to be equal to the angle C A D, that is, to the supplement of CBD, and is to lie so that the radius A C shall bisect it. The effect of this construction is as follows:-While the pin moves through the arc C E D, the follower is driven through the angle CBD; and as the pitch-point evidently moves from A to E, and then back to A again, the angular velocity of the follower gradually increases from nothing to a maximum, and then gradually diminishes to nothing again. At the point D the pin leaves the notch, and while it moves through the arc D IC, the follower remains at rest, and is kept steady by the dead arc, M K H, fitting into one of the hollows in its periphery. When the pin arrives again at C, it enters and drives a second notch, and so on. The combination evidently works with rotation in either direction. The Geneva Stop is the name given to the form of this combination that is employed when the object is that the follower shall stop the driver after it has turned through a certain number of revolutions and fractions of a revolution. For that purpose one of the notches is to be filled up, as shown by the dotted semicircle at N, so as to leave only a recess fitting the pin in the position C or D. The extent of rotation to which the driver will then be limited is expressed by as many revolutions as there are intervals in the circumference of the follower, less the angle CAD; and as the angle CAD is the supplement of C B D, this is expressed in algebraical symbols as follows: Let n be the number of intervals in the circumference of the follower; then the driver is limited to the following number of turns: 5; therefore, if one of the notches For example, in the figure, we have n is stopped, the rotation of the driver will be limited to This contrivance is used in watches, to prevent their being overwound. Very often a hammer-headed tooth is used instead of the cylindrical pin C. This enables the horns of the notches to be shortened; but it gives more backlash, and less precision and smoothness of action. ADDENDUM TO ARTICLE 154, PAGE 161. Racks in Screw Gearing.—When a straight rack gears with a screw, the pitch-surface of the rack is a plane, touching the pitch-cylinder of the screw. The traces of the teeth of the rack on its pitch-plane are parallel straight lines, and are the development on that plane of the traces of the threads of the screw upon its pitch-cylinder. The principal rule to be used in designing a rack and screw is a modification of Rule II. of Article 154, and is as follows:: P In fig. 110 A, let the plane of projection be the pitch-plane of the rack; and let A IP be the projection of the axis of the screw: being also the straight line in which the pitch-surfaces touch each other. Let I c represent the direction in which the rack is to slide; and let the length I c represent the surface-velocity of the rack. Having assumed a proper transverse obliquity for the teeth of the rack, draw the straight line c P, to represent the trace of a tooth.* Draw I C perpendicular to I A, cutting Pc in C; then IC will represent the surfacevelocity of the screw; and c C will represent the velocity of transverse sliding of the threads of the screw on the teeth of the rack. On CCP let fall the perpendicular I N; this will represent the common component of the surface velocities. Also, the proportions borne to each other by the divided pitches are as follows:-as I clongitudinal pitch of rack C N Fig. 110 A. ::IN: divided normal pitch of rack and screw ::IP: divided axial pitch of screw. Having assumed a convenient absolute value for the longitudinal pitch of the rack, find, by the help of the diagram, the circular pitch of the screw; multiply that circular pitch by a convenient number of threads, for the circumference of the pitch-cylinder, and find its radius by construction or by calculation. Fig. 110B is a projection of a rack, CC, and screw, A A', showing the traces of the teeth and threads on the pitch-surfaces. During the action of a tooth and a thread, the C' Fig. 110 B. The proper obliquity for the rack-teeth depends on friction, a subject belonging to the Dynamics of Machinery; but it may here be stated, that when the screw is to drive the rack U |