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 fj,, 

 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 the screw-chains, we have a resultant displacement which could have 

 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. Taking 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 fj, primary three-systems, and on each of yu, 1 intermediate 

 three-systems four screws are now supposed to be inscribed. We are to 



