A Treatise on the Theory of Screws

343] THE THEORY OF SCREW-CHAINS. 373 screws, can be uniquely determined. If, however, the intermediate screw between 8, and 82 be given, the entire chain 3 is also determined, yet it is not immediately obvious that that determination is unique. We can, however, show as follows that this is generally the case. Let 812 denote the given intermediate screw, and let this belong, not only to the chain 81; S3, &c., but to another chain 8/, 8/, &c. We then have 8lt S12, 8., co-cylindroidal, and also 8/, 312, &/ co-cylindroidal. Decom- pose any arbitrary twist of amplitude 6 on S12 into twists on 3, and 82, and a twist of amplitude — Ö on the same S12 into twists on and ö2. Then the four twists must neutralize; but the two twists on and compound into a twist on a screw on the first cylindroid of the system; and 32 and o2 into a twist on the second cylindroid of the system; and as these two resultant twists must be equal and opposite, it follows that they must be on the same screw, and that, therefore, the cylindroids belonging to the first and second elements of the system must have a common screw. It is, however, not generally the case that two cylindroids have a common screw. It is only true when the two cylindroids are themselves included in a three-system, this could only arise under special circumstances, which need not be further con- sidered in a discussion of the general theory. It follows from the unique nature of the correspondence between the intermediate cylindroids and the primary cyl indroids that one screw on any cylindroid corresponds uniquely to one screw on each of the other cylindroids; the correspondence is, therefore, homographic. We have now obtained a-picture of the freedom of the second order of the most general type both as to the material arrangement and the character of the constraints: stating summarily the results at which we have arrived, they are as follows:— A mass-chain of any kind whatever receives a small displacement. This displacement is under all circumstances a twist about a screw-chain. If the mass-chain admits of a displacement by a twist about a second screw- chain, then twists about an infinite number of other screw-chains must also be possible. To find, in the first place, a third screw-chain, give the mass- chain a small twist about the first chain; this is to be followed by a small twist about the second chain: the position of the mass-chain thus attained could have been reached by a twist about a third screw-chain. Lhe system must, therefore, be capable of twisting about this third screw-chain. When three of the chains have been constructed, the process of finding the re- mainder is greatly simplified. Each element of the mass-chain is, in each of the three displacements just referred to, twisted about a screw. Ihese three screws lie on one cylindroid appropriate to the element, and there are just so many of these cylindroids as there are elements in the mass-chain.