A Treatise on the Theory of Screws

346] THE THEORY OF SCREW-CHAINS. 377 If the mass-chain cannot be twisted about any screw-chain except those we have now been considering, then the mass-chain is said to have freedom of the third order. If, however, a fourth screw-chain can be found, about which the system can twist, and if that screw-chain does not belong to the doubly infinite system just described, then the mass-chain must have freedom of at least the fourth order. 346. Freedom of the fourth order. The homologous screws in the four screw-chains about which the mass- chain can twist form each a four-system. All the other chains which can belong to the system must consist of screws, one of which lies on each of the four-systems. It will facilitate the study of the homography of two four-systems to make use of the analogy between the homography of two spaces and the homography of two four-systems as already we had occasion to do in § 317. A screw in a four-system is defined by four homogeneous co-ordinates whereof only the ratios are significant. Each screw of such a system can therefore be represented by one point in space. The homography of two spaces will be completely determined if five points, a, b, c, d, e in one space, and the five corresponding points in the other space, a, b', c', d', e are given. From the four original screw-chains we can construct a fifth by com- pounding any arbitrary twists about two or more of the given chains. When five chains have been determined, then, by the aid of the principle of homo- graphy, we can construct any number. That each set of six homologous screws is homographic with every other set can be proved, as in the other systems already discussed. With respect to the intermediate screws a different proof is, however, needed to show that when one of these screws is given the rest of the chain is uniquely de- termined. The proof we now give is perhaps simpler than that previously used, while it has the advantage of applying to the other cases as well. Let a, ß, 7, 8 be four screw-chains, and let ej«, an intermediate screw of the chain e, be given. We can decompose a twist on e]2 into components of definite amplitude on a12, ß12, y12, 8]2. The first of these can be decomposed into twists on a, and a2; the second on ß2 and ß2, &c. Finally, the four twists on alt Ä, ylr can be compounded into one twist, e,, and those on a2, ß2, y2, compounded into a twist on e2. In this way it is obvious that when e12 is given, then ex and e2 are uniquely determined, and of course the same reasoning applies to the whole of the chain. We thus see that when any screw of the chain is known, then all the rest are uniquely determined, and therefore the principle of homography is applicable.