Page:The Kinematics of Machinery.djvu/82



Prop. IV. The motion of a plane figure P Q relatively to one point A in its plane of motion is expressed by that of the points P and Q relatively to A, and; this, as we have seen (Prop. I), takes place along the lines joining P and Q with A. But the position of these lines in the plane remains indeterminate, so that complex motions in the plane may occur without any alteration in the lines defining the motion relatively to the given point. [Thus in Fig 16 the triangle PQA is similar and equal to P' Q' A, so that the position of PQ relatively to A is the same in both cases; its position in the plane is, however, very different. The same is true of P 2 Q 2 and P 3 3 .]

FIG 16.

Example. A kinematic chain adapted for motion in a plane, but having one point only fixed in that plane, gives no determinate absolute motion, although the motion of every point in it relatively to the chain may be absolutely fixed, the chain being closed. The apparent contradiction that certain recent parallel motions require only one fixed point arises from a misapprehension of their nature. If only one point, or more strictly one axis, of such mechanisms be fixed, no " constrained " motion of the other parts occurs.

6. Temporary Centre; the Central Polygon.

The foregoing four propositions constitute the groundwork of the Phoronomics of point-systems, and are in certain respects exhaustive. They give, however, no distinct idea of the way in which the relative positions assumed by the moving point or