intersection, and the single motion possible at each instant will be turning about this point. We therefore have the important proposition: If it can be shown for any two figures that in all consecutive mutual positions relative slid- ing is impossible, it follows that the normals to their points of restraint intersect always in one point.
The series of consecutive centres of rotation or points of in- tersection of the normals for the two figures form the centroids, and the cylinders erected upon these the axoids of the two paired bodies.
Pairs of elements formed in this way are not closed, like the pairs we have before examined, but possess the more general and higher characteristic of mutual envelopment ( 3). We shall therefore distinguish them as higher pairs of elements from the closed pairs, which, on account of the smaller variety of their characteristics, we shall term the lower pairs. In order to under- stand the higher pairs we shall examine in detail one example.
§22.
Higher Pairs. Duangle and Triangle.
If from the ends of any straight line', P Q, with a radius equal to the length of the line, we describe intersecting arcs of a pair of circles, these will enclose a plane figure PR Q S (Fig. 93), which we may call a du angle. This will be touched in three points, Q, R, and $, by an equilateral triangle, A B C, of a height equal to 2 P Q, if Q be placed in the centre of one side of the triangle. For AB is J. to QR, (LPEA being = L B A Q = 30, and L Q R P = 60), and also Q, R, A and S are all points in a circle described about P with a radius PQ. The normals to the points of restraint Q, R, and $ cut each other in ft the angle between each pair being 120. Sliding, therefore, is entirely restrained, and rotation can take place about one point only. The same holds good also for any other position of the duangle, such, for instance, as that dotted, as will be seen from what follows.
If we consider in the first instance continuous contact between the duangle and two sides only, A B and B C, of the triangle, it
