A Treatise on the Theory of Screws

 370 THE THEORY OF SCREWS. [343 343. Freedom of the second order. Suppose that after one screw-chain has been discovered, about which the mass-chain can be twisted, the search is continued until another screw-chain is detected of which the same can be asserted. We are now able to show, without any further trials whatever, that there must be an infinite number of other screw-chains similarly circumstanced. For, compound a twist of amplitude a on one chain, a, with the twist of amplitude ß' on the other, ß. The position thus attained could have been attained by a twist about some single chain 7. As a and ß' are arbitrary, it is plain that 7 can be only one of a system of screw-chains at least singly infinite in number about which twisting must be possible. The problem to be considered may be enunciated in a somewhat more symmetrical manner, as follows:— To determine the relations of three screw-chains, a, ß, y, such that if a mass-chain be twisted with amplitudes a', ß', y, about each of these screw- chains in succession, the mass-chain will regain the same position after the last twist which it had before the first. This problem can be solved by the aid of principles already laid down (Chap. 11.). Each element of the mass-chain receives two twists about a and ß; these two twists can be compounded into a single twist about a screw lying on the cylindroid defined by the two original screws. We thus have for each element a third screw and amplitude by which the required screw-chain y and its amplitude y' can be completely determined. A mass-chain free to twist to and fro on the chains a and ß will therefore be free to twist to and fro on the chain y. These three chains being known, we can now construct an infinite number of other screw-chains about which the mass-chain must be also able to twist. Let 8 be a further screw-chain of the system, then the screws ßlt ylt which are the four first screws of the four screw-chains must be co- cylindroidal; so must a2) ß2, y.2, 82 and each similar set. We thus have cylindroids determined by the two first chains, and each screw of every chain derived from this original pair will lie upon tho corresponding cylindroid. We have explained (§ 125) that by the anharmonic ratio of four screws on a cylindroid we mean the anharmonic ratio of a pencil of four lines parallel to these screws. If we denote the anharmonic ratio of four screws such as «i, ßi, 7i, S, by the symbol [“1, ßi, y1; 8J, now demonstrated is that 7a, SJ = &c. = [aM, ß», yM, then the first theorem to be [ai> ßi> Yi, SJ = [s2, or that the anharmonic ratio of each group is the same.