с cating point to that of the revolving point. The general principle upon which that greatest ratio depends is shown in fig. 139. in which T represents the reciprocating point, and T the revolving point; T T, the line of connection; and C T, the crank-armı. Let CA be perpendicular to the direction of motion of the reciprocating point T', and let A be the point where the line of connection cuts CA; then, as has been K Fig. 139. already shown in Article 184, page, Velocity of T' T 196, and at the instant when that ratio is greatest, A is at its greatest distance from C; therefore, at that instant the direction of motion of the point A in the line of connection is along that line itself. Draw T K parallel to CA, produce C T till it cuts T' K in K, the instantaneous centre of motion of the link, and join K A; then the direction of motion of the point A in the line of connection at any instant is perpendicular to A K; and therefore, at the instant when CA is greatest, A K is perpendicular to A T. Upon this proposition depends the determination of the greatest value of the CA ratio ; but that determination cannot be completed by geometry alone; for it requires the solution of a cubic equation, as stated in the footnote.* = CT * In fig. 139, let the crank-arm CT = a; let the line of connection TT b; these two quantities being given; and when the ratio of the velocity of T to that of T is a maximum, let the angle C T T = 0, and the angle A C T = 4. so as to determine the value of sin 20, which is the only root of that equation that is positive and less than 1. Next, calculate the value of the angle 4, or those of its trigonometrical functions, by the help of one or more of the following equations (each of which implies the others): An approximate solution of this question may, however, be obtained by plane geometry, when the line of connection, TT, is not less than about twice the crank-arm, C T. It consists in treating the angle at T as if it were a right angle (from which it differs by the angle A K T); and thus we obtain When T T is great as compared with C T, the error of this solution is inappreciable, or nearly so; when T T = 2 CT, the approximate solution is too small by about one per cent., and is therefore near enough for practical purposes; when T T becomes less than 2 CT, the error rapidly increases, so as to make the approximate solution inapplicable; but cases of this last kind are very uncommon in practice. 189. Doubling of Oscillations by Linkwork.—When two reciprocating pieces are connected by means of a link, the follower may be made to perform two oscillations or strokes for one of the driver, in the following manner:-In fig. 140, let the driver be an arm or lever, A B; A its axis of motion, and B its connected point. ს Fig. 140. Let C be the connected point of the follower, and B C the link. Then the parts of the combination are to be so arranged that the straight line C c, which traverses the two ends of the stroke of the point C, shall traverse also the axis A, and shall bisect the arc of and finally, calculate the required greatest velocity-ratio by the following formula:: In the two extreme cases the values of that ratio are as follows:-When b is immeasurably longer than a, CA÷CT sensibly = 1; when b = a, CA÷CT 2. motion, b B b', of the connected point B. The result will be, that while the point B performs a single stroke, from 6 to b', the point C will perform a double stroke, from c to C and back again. If C is a point in a second lever, that second lever may, by means of a similar arrangement, be made to drive a third lever, so as again to double the frequency of the strokes; and thus, by a train of linkwork, the last follower may have the frequency of its strokes increased, as compared with those of the first driver, in a ratio expressed by any required power of 2. 190. Slew Motion by Linkwork.-As has been already explained in Article 180, page 193, when the connected point in the driver of an elementary combination by linkwork is at a dead point, the velocity of the connected point of the follower is nothing; and when the connected point of the driver is near a dead point, the motion of the connected point of the follower is comparatively very slow, and gradually increases as the connected point of the driver moves away from the dead point. When, therefore, it is Fig. 141. desired that the motion of a follower shall, at and near a particular position of the combination, be very slow as compared with that of the driver, or as compared with that of the follower itself when in other positions, arrangements may be used of the class which is exemplified in fig. 141 and fig. 141 A. In fig. 141 the lever A B, turning about an axis at A, drives, by means of the link B D, the lever C D, which turns about an axis at C. When the driving lever is in the position marked A B, it is in one straight line with the link B D; so that B is a dead point, and the velocity of the follower is null. As the connected point of the driver advances from B towards b, the connected point of the follower advances from D towards d, with a comparative velocity which is at first very small, and goes on increasing by degrees. When the motion is reversed, the comparative velocity of the latter point gradually diminishes as it returns from d towards D, and finally vanishes at the last-named point. Motions of this kind are useful in the opening and closing of steam-valves, in order to prevent shocks. Fig. 141 A shows a train of two elementary combinations of the same kind with that just described; the effect being to make the motion of a third connected point, E, quite insensible during a certain part of the motion of the first connected point, B. B Fig. 141 A 191. (A. M., 491.) Hooke's Coupling, or Universal Joint (fig. 142), is a contrivance for coupling shafts whose axes intersect each other in a point. Fig. 142. F/ C2 Let O be the point of intersection of the axes O C1, O C2, and i their angle of inclination to each other. The pair of shafts C1, C2, terminate in a pair of forks, F1, F in bearings at the extremities of which turn the pivots at the ends of the arms of a rectangular cross having its centre at O. This cross is the link; the connected points are the centres of the bearings F1, F2 At each instant each of those points moves at right angles to the central plane of its shaft and fork; therefore the line of intersection of the central planes of the two forks, at any instant, is the instantaneous axis of the cross; and the velocity-ratio of the points F1, F2 (which, as the forks are equal, is also the angular velocity ratio of the shafts), is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity-ratio is that of equality; for each successive quarter turn is made by both shafts in the same time; but its instantaneous value fluctuates between the limits, The following is the geometrical construction for finding the position of one of the shafts which corresponds to any given position of the other; also the velocity-ratio corresponding to that position:Let the shaft whose position is given be called the first shaft, and the other the second shaft; and let the corresponding arms of the cross be called the first and second arms respectively. In fig. 143, let O be the point of intersection of the axes of the two shafts, and let the plane of projection be a plane traversing O, and normal to the axis of the second shaft. Let A O a be the trace of the plane of the two axes, and C O C, perpendicular to A O a, a normal to that plane. With any convenient radius, O A, describe a circle about O. Lay off the angle A O D equal to the angle i, which the axes of the shafts make with each other. Through D, parallel to C C, draw D B, cutting O A in B; then the velocity-ratio of the second to the first axis, coincides with O C and the second with O A; and sec i is the velocity-ratio, when the first arm and the second with O C. ов = cos i is ОА when the first arm ОА 1 = = ов COS 2 coincides with O A, |