shown in Fig. 43. The same process of converting a cylinder into a prism is often used in cases where two bodies have to be so connected that they may resist all forces tending to move one upon the other; here keys, cutters, and so on are employed. In short, the machine-maker is accustomed to fulfil the above de- scribed condition in most numerous ways in his practical work.
§16.
Motion in Closed Pairs.
We have found in the last section that there are three pairs of elements which fulfil the conditions necessary for the complete and continuous enclosure of the bodies of which they consist. It is specially notable that there are only three, in itself a remarkable result of the investigation, for judging from the immense variety of cases which occur in machinery we might have been inclined beforehand to assume the existence of a very much larger number. These three single cases are, however, still further characteristic, on account of the nature of the constrained motions which can be carried out by their means.
In the screw-pair all points in the nut describe helices, and equal helices if the describing points lie at equal distances from the axis. These motions are compounded of a rotation about an axis and a sliding along it, and this axis is always that of the screw-spindle itself. The axoid of the screw-spindle (see 13) is hence a straight line coinciding with the axis of the screw. We can find the axoid of the nut at once, by supposing it fixed and causing the spindle to move: all points of the spindle then describe helices relatively to the nut, and equal helices if the points are equally distant from the axis, exactly the same motion, that is, as that of the screw. The axoid of the nut is therefore likewise a straight line coincid- ing with its geometrical axis. This axoid slides endlong upon the first and at the same time revolves about it, the angular motion bearing always a constant relation to the sliding. We have then before us in this pair of elements, the screw and nut, the most
