A Treatise on the Theory of Screws

ggaMUBgjRS________ 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 tive-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. b. 25