Thus the link PQ is extended from P to W, and when Pits center, is rotating, the respective extremities travel in the crossed diameters IL, wx, like the pencil bar of a trammel.

Each revolution will cause the point or joint pin Q to travel from L to 1, and the point W from x to w. And thus the radial arm AP will move the bar through a course of twice the length due to its radius.

Now as AQ=2Ad in all positions, it follows that the law of the motion of Q in the line AL is the same as the motion of d, with twice its velocity, and thus the point Q and the bar to which it is attached move with velocities symmetrically equal on opposite sides of the center point A. The left side of the figure shows that the radius Ap makes an acute angle Apq with the vertical diameter which compels the link pq to push the slide point 9 at an obtuse angle pqA, which would generate jamming friction of a magnitude that would prevent the motion of the bar from taking place (vide Chapter on Friction). To overcome this difficulty short grooves w S', xs, are fixed to the frame of the machine to receive a pin fixed to the extremity W, of the prolonged link. Thus, as W is carried upwards by the rotation of P and its lower end Q guided horizontally by the sliding piece, so, when the angle PQA has nearly reached a degree of obliquity that generates injurious friction, the upper end W of the link enters the guide groove. Its pin acts as a fulcrum against the side of the groove as at w, and the joint pin p of the radius acts transversely on the link so as to press the sliding piece in the direction of the longitudinal motion required.

PROB. To determine the motion of a slide when the path of the end of the link travels in a line that does not meet the

Let A be the center of motion of a revolving driving arm AP (r), PQ a link (1) jointed to AP at P. Its extremity Q is compelled to move in a right line LK, which for comparison with the previous formulæ may be considered as a circle of infinite radius,

AC, perpendicular from A upon II, will therefore be a portion of the line of centers. The link may either be directed to the right as at PQ, or to the left as gl. Let an arc with center A and radius AK-1-r intersect the rectilinear path at k and K. This is the shortest possible distance of the extremity of the link from A, and gives inward dead points. Similarly an arc struck from A with radius AL=r+l gives two outward dead points L and . The motion of the outer end of the link is limited to either of the right lines KL or Al, in which it travels back and forward when r revolves.

The position of Q corresponding to any given angular position of AP can be found as follows: let CAP=0, and AC=e, Pp being drawn parallel to AC, we have the distance of Q from c=cp+pQ.

=r. sin 0+ √⁄12 — Pp2.

=r. sin 0+ √12−(e±r cos 0)2, for Pp=AC±r cos 0.

Let 0, 0, be the values of that belong to the dead points.

2

Also Cl± √ Al2 — AC2, and CK= √ AK2 — AC2.

If AK=AC, the radius Af and link fh will coincide with the line of centers, and the extremity k with the point C. As the link is now perpendicular to the path of the sliding point, and the infinite radius lies beyond the link, we have an outward dead point at C simultaneous with an inward dead point at ƒ, which is a point of helplessness. But if this be overcome by any extraneous contrivance, the sliding point will move to and fro between L and l.

The preceding pages have shown that when an excentric pin, crank, or other equivalent contrivance is employed to produce back and forward motion in a sliding bar or plane surface, the length of the link, or connecting rod as it is usually termed, compared with the radius is a very important element, and therefore its influence on the motion of the reciprocating piece must be developed by formulæ and construction.

Generally speaking, the radius being supposed to revolve uniformly, the sliding piece, beginning from one of the extremities of its course, will move slowly, but its velocity will increase as it approaches the middle part of the course, and then decrease to the end, where with a slight pause it will begin to return, and

so on continually. The position of the maximum velocity is not necessarily in the middle of the course, and there are other

To illustrate these varieties of motion practically, I will explain the construction of a piece of lecture apparatus devised by me in 1857, and employed ever since.

These figures represent the side and end aspects of the apparatus in question. Fig. 206 is the front and fig. 207 the end view of the machine. On a base board Dd a standard piece Cc is fixed, in the middle of which, at A, a socket is implanted, which receives an axis rotateable by a handle aB (fig. 207) at the back and carries in front the radial arm AP, whose revolutions communicate the reciprocations, which the machine is intended to exhibit, to a sliding rod Rr. This rod is best supplied by a straight piece of brass tube three-fourths of an inch in diameter, which is sustained by two iron standards S, S (fig. 206), that allow it to slide endlong. The form of these appears in fig. 207 at SH. An iron bar of sufficient length is bent at right angles at ƒ and g, the lower end is thus provided with a foot f, by which it can be screwed to the base board, the upper end g is furnished with an angular notch, in which the tube lies and slides, and is kept in its place by a rectangular strip or cap of metal Hh, attached to the front vertical face of the standard by screws in slits, which allow the pressure on the tube to be regulated so that it may slide freely without looseness.

The radial arm AP carries a joint pin P at its extremity, which is inserted into a hole at the end of the link, and secured by a spring cotter of wire placed in the eye of the joint pin, as shown in fig. 207.

Link rods PL of wood are provided of several lengths, distinguished in the figures by accents, P.L., P.L, PP.P.L., selected to show the variations of motion. Each link has at one end one or more holes P to receive the joint pin of the radial arm as explained above. At the other end a piece of brass wire is fixed normally into the vertical face of each link at L, and connected with the sliding brass rod by simply inserting it into one of the holes drilled through the rod, which bears the same accent as the link.

In fig. 207 the link PL is seen with the joint pin P and spring cotter at its upper end, which connects it with the radial arm, and at the lower end of the link L the wire projects from its face and is passed through the brass tube.

To withdraw the link the arm AP must be set pointing upwards, as in fig. 206, the spring cotter must then be removed, and the upper end of the link drawn outwards to release it from

the pin. The cylindrical form of the tube allows of this motion by rotating upon its own axis.

On the face of the standard CD, a circle or dial is described with center A, and is divided into four quarters, indicated by the cross diameters, and each of the quadrants bisected by a short line distinguished by a circular spot, which for distinctness is in the actual machine coloured red. The end of the radial arm being pointed serves as an index by which the radius can be placed at eight equidistant points of the circle, which are sufficient to show the general nature of the inequalities of motion produced in the sliding points by varying the lengths of the link.

The motion of the sliding rod is exhibited by means of a graduated scale on the face of a vertical board EF, fixed to the base immediately below the sliding rod. An index 7 fixed to the rod slides along its edge and shows the distances through which the rod travels.

The scale is simply divided into two equal parts by a line, and each of these parts is again divided unequally by a line marked with a red spot. These spots being placed so that when the longest link PL is employed, the index of the radial arm and that of the sliding rod will coincide simultaneously on the respective scales with the rectilinear graduating line and with the red spot lines. When shorter links are substituted this coincidence fails, for in describing fig. 202 it has been shown that when the link is very long the selected point or index 7 of the sliding rod (fig. 206) will arrive at the middle of its course when the radial end of the link is on the vertical diameter of the dial, and that the positions of the sliding index corresponding to the octant points of the dial are much nearer to the extremities of the slide scale than to the center of the scale, to which, however, they are placed symmetrically.

If short links are used, this symmetry is destroyed. The whole length of the course remains unaltered, but the intermediate graduations of the slide scale corresponding to the eight points of the radial dial are all drawn towards that dial.

By putting in turn into their respective places the three links fig. 208 (beginning with the longest) P.Lˇ, P.LTM, PL, and exhibiting for each the positions of the sliding index when the radial index is placed opposite the eight points of the dial in succession, the increasing deviations from symmetry will be made apparent very strikingly.

Three links are provided, but the shortest has three holes, P1, P1, Р1, by which it is enabled to perform the functions of