prevent all sliding; a fourth point d must still be added. For the directions c and OR, parallel to the tangents, in which sliding can take place, are themselves 180 apart. The addition there- fore of two points of restraint, one between Fand W, and another between T and U, is required in order that the angle between every pair of consecutive normals may be less than 180. The
Fig. 58.
Fig. 59.
directions of restraint at c and d may in these circumstances again be 180 apart, their tangents thus becoming parallel.
The minimum number of points of restraint which can com- pletely prevent the sliding of a plane figure is therefore three, or if two of the directions of restraint enclose between them 180, four. In the illustrations given in 17 the sections of the
Fig. 60.
Fig. 61.
valve show that an arrangement is used which is in one case equivalent to three and in the others to four points of support. Every plane figure which has an outline returning upon itself can be completely restrained from sliding. It is entirely indif- ferent for this purpose whether the figure be restrained by external or internal contact, as Fig. 61, in which all the conditions of
