A Treatise on the Theory of Screws

374 THE THEORY OF SCREWS. [343- Between each two screws of a chain lies an intermediate screw, introduced for the purpose of defining the ratio of the amplitudes of the two screws of the chain on each side of it. In the three chains two consecutive elements will thus have three intermediate screws. These screws are co-cylindroidal. We thus have two series of cylindroids: the first of these is equal in number to the elements of the mass-chain (^u.), each cylindroid corresponding to one element. The second series of cylindroids consists of one less than the entire number of elements (u. - 1). Each of these latter cylindroids corre- sponds to the intermediate screw between two consecutive elements. An entire screw-chain will consist of /z primary screws, and /x—1 intermediate screws. To form such a screw-chain it is only necessary to inscribe on each of the 2/z, — 1 cylindroids a screw which, with the other three screws on that cylindroid, shall have a constant anharmonic ratio. Any one screw on any one of the 2/z — 1 cyliudroids may be chosen arbitrarily; but then all the other screws of that chain are absolutely determined, as the anharmonic ratio is known. The mass-chain which is capable of twisting about two screw-chains cannot refuse to be twisted about any other screw-chain con- structed in the manner just described. It may, however, refuse to be twisted about any screw-chains not so constructed; and if so, then the mass-chain has freedom of the second order. 344. Homography of Screw-systems. Before extending the conception of screw-chains to the examination of the higher orders of freedom, it will be necessary to notice some extensions of the notions of homography to the higher orders of screw systems. On the cylindroid the matter is quite simple. As we have already had occasion to explain, we can conceive the screws on two cylindroids to be homo- graphically related, just as easily as we can conceive the rays of two plane pencils. Ihe same ideas can, however, be adapted to the higher systems of screws the 3rd, the 4th, the 5th—while a case of remarkable interest is presented in the homography of two systems of the 6th order. Ihe homography of two three-systems is completely established when to each screw on one system corresponds one screw on the other system, and conversely. We can represent the screws in a three-system by the points in a plane (see Chap. XV.). We therefore choose two planes, one for each of the three-systems, and the screw correspondence of which we are in search is identical with the homographic point-correspondence between the two planes. We have already had to make use in § 317 of the fundamental property that when four pairs of correspondents in the two planes are given then the correspondence between every other pair of points is determined by