A Treatise on the Theory of Screws

376 THE THEORY OF SCREWS. (345, determine a fifth screw about which the system even though it has only freedom of the third order, must still be permitted to twist. To begin with we may choose an arbitrary screw in any one of the three- systems. In the exercise of this choice we have two degrees of latitude; but once the choice has been made, the remainder of the screw-chain is fixed by the following theorem :— If each set of five homologous screws of five screw-chains lies on a three- system, and if a mass-chain be free to twist about four of these screw-chains, it will also be free to twist about the fifth, provided each set of homologous screws is homographic with every other set. Let 3 denote the fifth screw-chain. If be chosen arbitrarily on the three-system which included the first element, then a twist about 8, can be decomposed into three twists on a,, ßlt yv By the intermediate screws these three twists will give the amplitudes of the twists on all the other screws of the chains a, ß, y, and each group of three homologous twists being compounded, will give the corresponding screws on the chain 8. We thus see that when is given, 82, &c., are all determinate. It is also obvious that if S2, or any other primary screw of the chain, were given, then all the other screws of the chain would be determined uniquely. If, however, an intermediate screw, S12, had been given, then, although the conditions are, so far as number goes, adequate to the determination of the screw-chain, it will be necessary to prove that the determination is unique. This is proved in the same manner as for freedom of the second order (§ 343). If there were two screw-chains which had the same inter- mediate screw, then it must follow that the two primary three-systems must have a common screw, which is not generally the case. We have thus shown that when any one screw of the chain 8, whether primary or intermediate, is given, then all the rest of the screws of the chain are uniquely determinate. Each group of five homologous screws must therefore be homographic. It is thus easy to construct as many screw-chains as may be desired, about which a mass-chain which has freedom of the third order must be capable of twisting. It is only necessary, after four chains have been found, to inscribe an arbitrary screw on one of the three-systems, and then to construct the corresponding screw on each of the other homologous systems. In choosing one screw of the chain we have two degrees of latitude: we may, for example, move the screw chosen over the surface of any cylindroid embraced in the three-system: the remaining screws of the screw-chain, primary and intermediate, will each and all move over the surface of corre- sponding cylindroids.