(3) Only by four suitably placed points of restraint, and with certain profiles only by five, can turning and sliding be prevented simultaneously.
We shall apply these propositions to the closed pairs.
Restraint in the Sliding-pair. Let it be required with a minimum number of points so to restrain a solid prism that it shall be prevented from having any other relative motion than that which it would have if it formed one element, along with an open prism, of a closed pair. This can be done if, taking any two plane sections of the prism, perpendicular to its axis, we prevent either the sliding or the turning of their profiles. This requires four suitably placed points of restraint on each section (Fig. 87), so that no turning, and sliding parallel to the axis only, can take place, which is the characteristic of the sliding-pair. This gives eight points of restraint a, I, c, d, e, /, g,
Fig. 87.
Fig. 88.
and h, the four in each section being placed as in Fig. 87. If, however, a third plane section, parallel to and between the two first be taken, two pairs of similarly-situated points, as c and g, d and h, may be placed together upon it. Two out of the eight points are thus dispensed with, and the required restraint is obtained with six a, &, c, d, e, and/ (Fig. 88).
Restraint in the Turning-pair. To examine what minimum of points is necessary in order that a solid circular cylinder with flat ends (Fig. 89) may be restrained as completely as if it formed one element of a closed turning-pair, we may in the same way take two plane sections of it normal to its axis, and restrain these from cross sliding, which requires three points in each, a, I, c, and d, e, f, and at the same time restrain a longitudinal section from sliding in the direction of the axis, which can be done by taking one point of restraint upon each end, as g and h. Here, therefore, eight points of restraint are again required. If, however, the first
