segment, F G, into which the diameter, E G, of the greater of the two circles described by the connected points is divided by the other circle. This principle holds also when those circles are equal, and is then applicable to the diameter of either of them. In other words, CD is to be made Greater than A B+ A C - B D, - A B. The comparative motions may be found by either of the following rules : : III. To find the angular velocity-ratio in a given position of the cranks: on the line of connection, C D, let fall from the axes the perpendiculars, A L, BM; then Or otherwise: produce the line of connection, C D, till it cuts the line of centres in I; then When C D is parallel to A B the angular velocities are equal. IV. To find the linear velocity-ratio of the connected points: in a given position of the cranks produce the crank-arms until they intersect; their point of intersection, K, will be the trace of the instantaneous axis of the link; then The limits between which that velocity-ratio fluctuates are BD-AB when B D traverses A, and Ᏼ Ꭰ when A C The two shafts, in their rotation, may be regarded as alternately overtaking and falling behind each other by an angle which we may call the angular displacement. The complete angular displacement is attained in two opposite directions alternately, at the two instants when the angular velocities of the shafts are equal : that is, when the line of connection is parallel to the line of centres. The following is a rule for designing a drag-link motion with equal cranks, which shall produce a given angular displacement; and although not the only rule by which that problem might be solved, it appears to be the simplest in its application. V. In fig. 134 draw two straight lines, CO c, DO d, cutting == = each other at right angles in the point O; lay off along those lines the equal lengths OC O D. From C and D draw the straight lines C A, D B, making the angles O CA ODB half the given angular displacement, and cutting Od and O c respectively in A and B. Join A B and CD. Then A B will represent the line of centres; AC and BD the two crankarms; and C D the line of connection. Fig. 134. 183. Link for Contrary Rotations. The only other elementary combination by linkwork which belongs to Willis's Class B is that in which two equal cranks, rotating about parallel axes in contrary directions, are connected by means of a link equal in length to the line of centres. This has been already described in Article 108, page 97, and represented in fig. 72, page 96, as a contrivance to aid the action of elliptic wheels. There are two dead points in each revolution which the pins pass at the instant when the line of connection coincides with the line of centres; consequently the link is not well adapted to act alone, and requires a pair of elliptic wheels, or of elliptic pulleys (Article 175, page 189), to ensure the accurate transmission of the motion. 184. Linkwork with Reciprocating Motion-Crank and Beam— Crank and Piston-Rod. (A. M., 488.)-The following are examples of the most frequent cases in practice of linkwork belonging to Willis's Class C, in which the directional relation is reciprocating; and in determining the comparative motion, they are treated by the method of instantaneous axes, already referred to in Article 179, page 193: Example I. Two Turning Pieces with Parallel Axes, such as a beam and crank (fig. 135).-Let C1, C2, be the parallel axes of the pieces; T1, T2, their connected points; C, T1, C2 T2, their crank arms; T T2, the link. At a given instant let v be the velocity of T; v that of T2. To find the ratio of those velocities, produce C, T1, C2 T2, till they intersect in K; K is the instantaneous axis of the link or connecting-rod, and the velocity-ratio is Should K be inconveniently far off, draw any triangle with its sides respectively parallel to C1 T1, C2 T2, and T, T2; the ratio of the two sides first mentioned will be the velocity-ratio required. For example, draw C2 A parallel to C1 T1, cutting T1 T2 in A; then Example II. Rotating Piece and Sliding Piece, such as a pistonrod and crank (fig. 136).-Let C2 be the axis of a rotating piece, and T, R the straight line along which a sliding piece moves. Let T1, T2, be the connected points; C, T2, the crank arm of the rotating piece; and TT, the link or connecting rod. The points T1, T2, and the line T, R, are supposed to be in one plane, perpendicular to the axis C Draw T K perpendicular to T, R, intersecting C. T2 in K; K is the instantaneous axis of the link; and the rest of the solution is the same as in Example I. 2 185. (A. M., 489.) An Eccentric (fig. 137) being a circular disc keyed on a shaft, with whose axis its centre does not coincide, and used to give a reciprocating motion to a rod, is equivalent to a crank whose connected point is T, the centre of the eccentric disc, and whose crank arm is C T, the distance of that point from the axis of the shaft, called the eccentricity. Fig. 137. An eccentric may be made capable of having its eccentricity altered by means of an adjusting screw, so as to vary the extent of the reciprocating motion which it communicates, and which is called the throw, or travel, or length of stroke. 186. (A. M., 490.) The Length of Stroke of a point in a reciprocating piece is the distance between the two ends of the path in which that point moves. When it is connected by a link with a point in a continuously rotating piece, the ends of the stroke of the reciprocating point correspond with the dead points of the continuously rotating piece (Article 180, page 193). From I. When the crank-arm and the path of the connected point in the reciprocating piece are given, to find the stroke and the dead points. If the connected point in the reciprocating piece moves in a straight line traversing and perpendicular to the axis of the turning piece, the length of stroke is obviously twice the crank-arm. If that connected point moves in any other path, let F F, in fig. 138, represent that path, A the trace of the crankaxis, and A D = A E the crank-arm. the point A to the path F F lay off the distances A B the line of connection - the crank-arm, and A C = the line of connection G+ the crank-arm; then B C will be the stroke of the connected point in the reciprocating piece. Draw the straight lines CE A and BAD, cutting the circular path of the crankpin in the points E and D: these will be the dead points. D Fig. 138. B = = II. When the crank-arm, AD A E, the length of the line of connection, and the dead points, D and E, are given, to find the two ends of the stroke of the connected point in the reciprocating piece. In D A and A E produced, make D B and E C each equal to the length of the line of connection; B and C will be the required ends of the stroke. When the path of the connected point in the reciprocating piece is a straight line, the preceding principles may be thus expressed in algebraical symbols: Let S be the length of stroke, L the length of the line of connection, and R the crankarm. Then, if the two ends of the stroke are in one straight line with the axis of the crank, and if their ends are not in one straight line with that axis, then S, LR, and L+ R, are the three sides of a triangle, having the angle opposite S at that axis; so that if be the supplement of the arc between the dead points, S22 (L2 + R2)- 2 (L2 187. Mean Velocity Ratio.-In dynamical questions respecting machines, especially when the mode of connection is by link work, it is often requisite to determine the mean ratio of the linear velocities of a pair of connected points during some definite period; which mean ratio is simply the ratio of the distances moved through by those points in that period. Three cases may be distinguished, according as the combination of linkwork belongs to Willis's Class A, Class B, or Class C. In Class A the mean velocity-ratio is identical with the velocityratio at each instant. For examples, see Article 181, page 194, and Article 182, page 194. In Class B the mean velocity-ratio of the connected points during each complete revolution is that of the circumferences of the circles in which they move. For examples, see Article 182, page 194, and Article 183, page 196. In Class C the mean velocity-ratio of the connected points may be taken either for a whole revolution of the revolving point and double stroke of the reciprocating point, or it may be taken separately for the forward stroke and return stroke of the reciprocating point, where it has different values for these two parts of the motion. In the former case it is expressed by the ratio of twice the length of stroke of the reciprocating point to the circumference of the circle described by the revolving point; that is to say, for example, in fig. 138, page 198, by the ratio 2 B C Circumference D GEH' In the latter case, the two mean velocity-ratios are expressed by the proportions borne by the length of stroke of the reciprocating point, to the two arcs into which the dead points divide the path of the revolving point. For example, in fig. 138, those two ratios are respectively The most frequent case in practice is that represented in fig. 136, page 197, where the reciprocating point moves in a straight line traversing the axis about which the revolving point moves; and in that case the mean velocity-ratio for each single stroke and for a whole revolution is 188. Extreme Velocity-Ratios.-In those cases in which one of the points connected by a link revolves continuously, while the other has a reciprocating motion, it is often desirable to determine the greatest value of the ratio borne by the velocity of the recipro |