Page:The Kinematics of Machinery.djvu/109

 3. must be so arranged as to prevent every motion of the second element except the one which is required.

The stationary element then holds the moving one as it were imprisoned, preventing every motion except a single one, forcing every point in it, when it has begun to move, to travel in a determinate path, on which account such a pair of bodies may be called constrained. We referred in the last article to the immense number of forms which the relative motion of two bodies might take; remembering this, it is easily seen that pairs of constrained bodies may have very many geometrical forms. All pairs of such forms, however, which conform to the two last of the above conditions, have this in common, that they are envelopes, and indeed reciprocal envelopes, for the given motion, which can be represented through their axoids. Hence they may, like their axoids, be more or less simple. We can imagine a case fulfilling both the conditions, and in which at the same time the one element not merely forms an envelope for the other, but encloses it, in which, that is to say, the forms of the elements are geometrically identical, the one being solid or full and the other hollow or open. Such a pair of bodies may be called a closed pair.

It is evident that, in their simplicity, closed pairs differ notably from pairs in which the elements are not identical in form. We shall on this account consider them separately, and in the first instance.

§15. The Determination of Closed Pairs.

The geometrical properties of the bodies from which closed pairs can be constructed are so well defined that we do not require to look first for these pairs in existing machinery, but may attempt to discover them by a priori reasoning.

Two bodies forming a closed pair cover each other with their surfaces; on these we may imagine any number of pairs of coincident curves, and among these some may be supposed to be such that the single motion possible for the time being occurs along them, such in other words, as slide on one another. If