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KINEMATICS OF MACHINERY.

for which we have used the contracted expression (P^r). The keyed joints occurring in machines have indeed always this form, neglect- ing the occasional force- or pair-closure of incomplete pairs.

The familiar case of the keying of a wheel upon a shaft, Fig. 295, shows all three links, a, I and c. The prism pairs 1 and

FIG. 294.

FIG. 295.

2 can be at once recognised, each one incomplete in itself but closed by the other. The pair 3 is omitted, but the wheel, which is to be moved by the key only in a direction perpendicular to the axis of c, is prevented by force-closure from moving in any other direction.

FIG. 296.

FIG. 297.

In the case of a round bar keyed into a socket, Fig. 296, we find all three links and all three pairs. The pairs 1 and 2 are at the under and upper surfaces of the key, and the pair 3 appears in the cylindric surfaces of 5 and c as well as the sides of the openings in c through which the key passes. It is these which make the cylinder into a prism pair. In a "gib and cutter" joint such as is