Page:The Kinematics of Machinery.djvu/137

 §21. The Higher Pairs of Elements.

From the foregoing examination into the restraint of plane figures, we see that pairs of figures may be constructed in which the sliding of the one figure relatively to the other is prevented, while their relative rotation remains possible, and further, that if the normals of restraint, their number not being less than three, intersect in one point, the rotation which remains possible will be about this point only. Such a rotation is a definite motion, excluding the possibility of all others; and this is just what we have recognised as the distinguishing character- istic of a pair of elements. If, therefore, a pair of figures be so conditioned that after the completion of any indefinitely small turning about a centre 0, they have again three points of restraint with their normals cutting in a new point, and that this occurs continually for every new mutual position of the figures such figures may evidently be used as the foundation of a pair of elements. To construct the elements we require, e.g., only to erect cylinders upon them, and provide these with end surfaces so as to prevent axial sliding.

. 92.

If there be two figures of such form that in all their relative positions the sliding of the one relatively to the other be impos- sible, then their only relative motion at each instant must be turning. If the normals of restraint have always a common point of intersection, then this turning can take place at each instant about this point only, but if this be not the case, then turning (if it remain possible) must occur about a point outside at least one of the normals. But such a motion would cause a separa- tion of the figures at the corresponding point of restraint, and is therefore inconsistent with the assumption of continued restraint against sliding. Where, therefore, such continued restraint is required, the normals must always have a common point of