PARALLEL CEANKS. 287

belong to different classes. The three different kinds of motion obtained are, as we know, simply those relative motions in the chain which we have made absolute, or more strictly speaking absolute " for us," by fixing one or other of the links (see 3.) The most frequently used of the four mechanisms is ((7 4 ) d ((7' 4 ') b ; or putting the two formula} together, (" 4 ') d=b .

66. Parallel Cranks.

It is obvious that by altering the relative lengths of the links in the chain (C") we alter the mechanisms to be obtained from it, and therefore the resulting motions, for by extending the angle of oscillation we can convert relative swinging into rotation and vice versd. We shall consider the most important special cases which arise here. In the original mechanism we had a <^c, if the difference between them be reduced until a = c, and if at the same time b be made = d the crank chain becomes a parallelogram, as Fig. 205. The lever c becomes a crank equal to a, and (d being fixed) it moves always through the same angle.

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FIG. 205.

The contracted symbol for the chain, the opposite links being always parallel/is (C'i\\C'). It is unnecessary to use the sign #, for the || is by itself sufficient to exclude the crossing which, as far as the construction of the chain itself is concerned, is possible ( 47). The sign of equality, on the other hand, would not be sufficient by itself, for the equality of pairs might be a = b and c = d, which would allow a <C or >> c, and would therefore be inconsistent with our conditions. The sign jfr may be reserved