THE SIMPLE CHAINS. 545

enough to show us the notable fact that the number of possible pairs of elements is limited, and that the whole can be determined collectively by a synthetic treatment. We ma}' now assume this to have been done, and proceed to the synthetic determination of kinematic chains.

�150.
The Simple Chains.

We cannot adopt so direct and definite a treatment in the case of kinematic chains as was possible with pairs of elements. In treating the latter we could build directly upon the definite and limited series of axoidal forms discussed in Chapter II ; here, how- ever, we can make but accidental use of these, for the relative motions of the links of very different chains may be identical, and have, therefore, similar axoids. We might indeed treat chains by working through every possible combination of two, three, four, &c. pairs, and take systematically all the relative positions for the pairs in each combination. But the extreme unwieldiness of such a method, and the certainty that very many of the combinations so found would prove useless, unpractical, or altogether impractic- able makes it very desirable that some other treatment should be

V

adopted, even at the expense of external uniformity in our methods.

We shall adopt in general an inductive method, as is so frequently done in mathematical investigations, and choose for each series of problems the treatment which seems best to suit the special conditions of the case. It must be remembered, at the same time, that our object here is not to complete the synthesis of the chain, but merely to note its general direction. On these grounds we shall not begin with the general case of compound chains, but with the simple ones, the essential characteristics of which we have already examined somewhat closely. The conclusions arrived at in 128, where we found that the constrained chain took its place in the series of possible combinations of links between the unconstrained and the fixed chain, will be of great service to us here. If, that is to say, we find a chain to be fixed, we can convert it into a constrained closed chain by inductive addition of

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