350] THE THEORY OF SCREW-CHAINS. 385 



order we may, by trial, determine five screw-chains about which the system 

 can be twisted. Each set of five homologous screws will determine a 

 five-system. In this method of proceeding we need not pay any attention 

 to the intermediate screws : it will only be necessary to inscribe one Dyname 

 (in this case a twist) in each of the homologous five-systems so that 

 the group of six shall be homographic. The set of twists so found will form 

 a displacement which the system must be capable of receiving. This is 

 perhaps the simplest geometrical presentment of which the question 

 admits. 



One more illustration may be given. Suppose we have a series of planes, 

 and three arbitrary forces in each plane. We insert in one of the planes 

 any arbitrary force, and its parallel projection can then be placed in all 

 the other planes. Suppose a mechanical system, containing as many distinct 

 elements as there are planes, be so circumstanced that each element 

 is free to accept a rotation about each of the three lines of force in the 

 plane, and that the amplitude of the rotation is proportional to the intensity 

 of the force ; it must then follow that the system will be also free to accept 

 rotations about any other chain formed by an arbitrary force in one plane and 

 its parallel projections in the rest. 



We may, however, also examine the case of a mass-chain with freedom 

 of the fifth order by the aid of the screw correspondence without intro 

 duction of the Dyname. We find, as before, five independent screw-chains 

 which will completely define all the other movements which the system 

 can accept. To construct the subsequent screw-chains, which are quadruply 

 infinite in variety, we begin by first finding any sixth screw-chain of the 

 system by actual composition of any two or more twists about two or more 

 of the five screw-chains. When a sixth chain has been ascertained the 

 construction of the rest is greatly simplified. Each set of six homologous 

 screws lie in a five-system. Place in each of these five-systems another 

 screw which, with the remaining six, form a set which is homographic with 

 the corresponding set in each of the other five-systems. These screws 

 so determined then form another screw-chain about which the system must 

 be free to twist. 



In the choice of the first screw with which to commence the formation 

 of any further screw-chains of the five-system we have only a single condition 

 to comply with : the screw chosen must belong to a given five-system. 

 This implies that the chosen screw must be reciprocal merely to one given 

 screw. On any arbitrary cylindroid a screw can be chosen which is reciprocal 

 to this screw, and consequently on any cylindroid one screw can always be 

 selected wherewith to commence a screw-chain about which a mass-chain 

 with freedom of the fifth order must be free to twist. 



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