A Treatise on the Theory of Screws

345] THE THEORY OF SCREW-CHAINS. 375 rigorous construction. Any fifth point in one plane being indicated, the fifth point corresponding thereto in the other plane can be determined. It therefore follows that when four given screws on one three-system are the correspondents of four indicated screws on the other system, then the correspondence is completely established, and any fifth screw on one system being given, its correspondent on the other is determined. 345. Freedom of the third order. We are now enabled to study the small movements of any mass-chain which has freedom of the third order. Let such a mass-chain receive any three displacements by twists about three screw-chains. It will, of course, be understood that these three screw-chains are not connected in the specialized manner we have previously discussed in freedom of the second order. In such a case the freedom of the mass-chain would be of the second order only and not of the third. The three screw-chains now under consideration are perfectly arbitrary; they may differ in every con- ceivable way, all that can be affirmed with regard to them is that the number of primary screws in each chain must of course be equal to /x, i.e. to the number of material elements of which the mass-chain consists. It may be convenient to speak of the screws in the different chains which relate to the same element (or in the case of the intermediate screws, the same pair of elements) as homologous screws. Each set of three homologous screws will define a three-system. Compounding together any three twists on tile screw-chains, we have a resultant displacement which could hav© been effected by a single twist about a fourth screw-chain. The first theorem to be proved is, that each screw in this fourth screw-chain must belong to the three-system which is defined by its three homologous screws. So far as the primary screws are concerned this is immediately seen. Each element having been displaced by three twists about three screws, the resultant twist must belong to the same three-system, this being the im- mediate consequence of the definition of such a system. Nor do the inter- mediate screws present much difficulty. It must be possible for appropriate twists on the four screw-chains to neutralize. The four twists which the first element receives must neutralize: so must also the four twists imparted to the second element. These eight twists must therefore neutralize, however they may be compounded. laking each chain separately, these eight twists will reduce to four twists about the four intermediate screws: these four twists must neutralize; but this is only possible if the four intermediate screws belong to a three-system. On each of g primary three-systems, and on each of g -1 intermediate three-systems four screws are now supposed to be inscribed. We are to