coincide with that of the hollow cylinder which forms its seat. In the first case (Fig. 49) three small segments of the cylinder are cut away; in the second (Fig. 50) four screw-formed webs only are left, and in the third (Fig. 51) four thin strips of the cylinder parallel to its axis, and connected below by a ring forming part of its cross section. There must evidently be some general principle underlying the arrangement of these small strips or other portions of the cylindrical surface which have to be retained in order that the bodies may keep their required mutual positions, or, as we may say, in order that they may mutually restrain each other. A definite number of such points is necessary, but is at the same time sufficient, in order to ensure this mutual restraint. This minimum of points of restraint we shall now endeavour to find. It is not an investigation to which hitherto any special value has been attached, but it is unques- tionably one which should be kept in view, not only because no property of machine-elements can be unimportant in a scientific examination of the nature of machines, but also because of the important results which are directly connected with it.
Restraint against Sliding.
We shall first consider the case of a plane figure moving in a plane, or, if it be preferred, of a plane section of a cylinder prevented in any way from leaving the plane in which it lies. By the expression point of restraint of the figure we shall mean a point in its circumference towards which the figure is prevented from sliding along or parallel to a normal to the tangent at that point. Sliding of the figure implies here an equal and similar motion of all points in it.
Single Point of Eestraint. Let the given figure A be pre- vented from moving freely in its own plane by contact in one point with a second and con-plane figure B; we shall examine to what extent its motion is limited. The definition of a point of restraint just given renders it unnecessary that we should