another in one point, at least five points of restraint are required to render turning of the figure impossible.
It will be seen that it is a much more difficult problem so to restrain a figure that it shall not turn than to prevent its sliding. As a rule, it requires at least four points of restraint (three suffice in an exceptional case, Fig. 76, only), while very fre- quently five are necessary. The form of the figure, also, cannot vary within such wide limits as when sliding only is to be prevented, its profile must have a curvature varying in such a way that it may be pos- sible to find normals occupying the required relative positions. Hence the rotation of a circle, as of course can be recognised a priori, cannot be prevented by any number of points of restraint. When this form occurs
in machine construction, (as on account of the ease with which it can be made it so often does), as the cross section of a body which it is desired to restrain from turning, it is necessary in some way to convert its form into one which can be so restrained. W r e have already ( 15) looked at this fact from another point of view, and can now examine it in the light of the foregoing investigation.
The fastening of a wheel or pulley upon a cylindrical shaft furnishes us with a very familiar example (Fig. 81). Here, if the original form of the shaft were retained, all the normals would cut at the centre 0. A rectangular groove is therefore made in it, against the sides of which the key which holds
shaft and wheel together can press at e or /. The one normal ee covers with its field of left-, and the other//' with its field of right-handed turning, exactly as we found above (Fig. 80) to be necessary. In similar cases the turning is often restrained simply by flattening a portion of the shaft. The key then exerts pressure at such points as e and / (Fig. 82), so that the normals
Fig. 81.
