A Treatise on the Theory of Screws

342] THE THEORY OF SCREW-CHAINS. 369 341. The definition of a Screw-chain. It will be convenient to have a name which shall concisely express the entire senes of p original screws with the p - 1 intermediate screws whose function in determining the amplitudes has just been explained. We may call it a screw-chain. A twist about a screw-chain will denote a displace- ment of a mass-chain, produced by twisting each element about the corresponding screw, through an amplitude whose ratio to the amplitudes on the two adjacent screws is indicated by the intermediate screws. The amplitude of the entire twist will, as already mentioned, be most conveniently expressed by the twist about the first screw of the chain. We hence have the following statement:—■ The most general displacement of which a mass-chain is capable can be produced by a twist about a screw-chain. 342. Freedom of the first order. Given a material system of p elements more or less connected inter se, or related to fixed points or supports : let it be required to ascertain the freedom which this material system or mass-chain enjoys. The freedom is to be tested by the capacity for displacement which the mass-chain possesses. As each such displacement is a twist about a screw-chain, a complete examina- tion of the freedom of the mass-chain will require that a trial be made to twist the mass-chain about every screw-chain in space which contains the right number of elements p. If in the course of these trials it be found that the mass-chain cannot be twisted about any screw-chain, then the system is absolutely rigid, and has no freedom whatever. If after all trials have been made, one screw-chain, and only one, has been discovered, then the mass-chain has freedom of the first order, and we have the result thus stated:— When a mass-chain is so limited by constraints, that its position can be expressed by a single co-ordinate, then the mass-chain is said to have freedom of the first order, and its possible movements are solely those which could be accomplished by twisting about one definite screw-chain. By this method of viewing the question we secure the advantage of eliminating, as it were, the special characters of the constraints. The essential moving parts of a steam-engine, for example, have but one degree of freedom. Each angular position of the fly-wheel necessarily involves a definite position of all the other parts. A small angular motion of the fly- wheel necessarily involves a definite small displacement of each of the other parts. Complicated as the mechanism may be, it is yet always possible to construct a screw-chain, a twist about which would carry each element from its original position into the position it assumes after the displacement has been effected. b. 24