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 6//. components on any arbitrary 6//, chains. The amplitudes or 

 the intensities of these G/A components may be termed the 6/i co-ordinates 

 of the given twist or wrench. If the amplitude or the intensity be regarded 

 as unity, then the 6/A 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 6/1, 

 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 4- 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. 



G/j, 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/i - 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 nth order, or of the (6/i - ?i)th order, 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. They 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 



