A Treatise on the Theory of Screws

355] THE THEORY OF SCREW-CHAINS. 391 similar manner the ratio of the amplitudes of any other pair of twists can be found, and thus the whole problem has been solved. We are now able to decompose any given twist or wrench on a screw- chain into Gp, components on any arbitrary chains. The amplitudes or the intensities of these Gp, components may be termed the 6/z co-ordinates of the given twist or wrench. If the amplitude or the intensity be regarded as unity, then the 6/x quantities may be taken to represent the co-ordinates of the screw-chain. In this case only the ratios of the co-ordinates are of consequence. If the mass-chain have only n degrees of freedom where n is less than 6/z, then all the screw-chains about which the mass-chain can be twisted are so connected together, that if any n + 1 of these chains be taken arbitrarily, the system can receive twists about these n +• 1 chains of such a kind, that after the last twist the system has resumed the same position which it had before the first. In this case n co-ordinates will be sufficient to express the twist or wrench which the system can receive, and n co-ordinates, whereof only the ratios are concerned, will be sufficient to define any screw-chain about which the system can be twisted. 6p, — n screw-chains are taken, each of which is reciprocal to n screw- chains about which a mass-chain with freedom of the nth order can twist. The two groups of n screw-chains on the one hand, and 6/x — n on the other, may each be made the basis of a system of chains about which, a mechanism could twist with freedom of the wth order, or of the (6^i —?i)th older, re- spectively. These two systems are so related that each screw-chain in the one system is reciprocal to all the screw-chains in the other. 1 hey may thus be called two reciprocal systems of screw-chains. Whatever be the constraints by which the freedom is hampered, the reaction of the constraints upon the elements must constitute a wrench on a screw-chain. It is a fundamental point of the present theory that this screw-chain belongs to the reciprocal system. For, as no work is done against the constraints by any displacement which is compatible with the freedom of the mass-chain, it must follow, from the definition, that the wrench-chain which represents the reactions must be reciprocal to all possible displacement chains, and must therefore belong to the reciprocal system. For a wrench-chain applied to the mass-chain to be in equilibrium it must, if not counteracted by some other external wrench-chain, be counter- acted by the reaction of the constraints. Thus we learn that