Page:The Kinematics of Machinery.djvu/311

 COMBINED CHAINS. 239

We have here chain-closure. It may therefore be indicated, as mentioned in 57, by placing the sign k as a divisor below the original formula, so that both chains could be written

(C" II C"}

>_ 2 " 2 But the addition to the Jc of the sign of equality, and Ic

the inclosure of both in brackets will allow us to make distinct that the closing chain is equal to the one closed. The formula

(C" \\ C"}

would then be TTTZA or ^ u words : a pair of parallel cranks (K ;

closed by another pair of parallel cranks. We may, however, choose a still more convenient way of indicating the combination of two chains which are both equal and reciprocally closing, namely, by adding the factor 2 to the formula for the single chain: 2 (C" a ' || C" 2 ')-

There is, lastly, another doubtful point to make clear, the difference between the two arrangements of Figs. 206 and 207. In the first case the cranks of the closing chain are rigidly con- nected into links with those of the other ; in the second, one of the cranks of the closing chain appears to be identical with one of those of the primary chain, the other being separately constructed but connected by a coupler also with the second primary crank. If however we compare the two chains more carefully, and in their most abstract forms, so as to see distinctly what is actually before us, we find that the two chains (not the mechanisms) are identical. The ternary links a a' andcc' of the chain Fig. 206 correspond to the ternary links a a' and c c of Fig. 207, and the binary links d, b and I' of the first to those similarly lettered in the second. If then, as the figures indicate, the chains be made into mechanisms by placing them upon dd' and a a! respectively, the second is nothing more than an inversion of the first, so that the difference between the two will be indicated in the general formulae, 2 (C" 2 || C 2 ) d for Fig. 206, and 2 (C'l \\ C 2 ) a for Fig. 207. They are both formed from the same five-linked chain, and they are examples of the only two classes of mechanisms into which this chain can be formed.

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