374 THE THEORY OF SCREWS. [343- 



Betvveen 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 (/it), each cylindroid corresponding to one 

 element. The second series of cylindroids consists of one less than the 

 entire number of elements (^ - 1). Each of these latter cylindroids corre 

 sponds to the intermediate screw between two consecutive elements. An 

 entire screw-chain will consist of fju primary screws, and //, 1 intermediate 

 screws. To form such a screw-chain it is only necessary to inscribe on each 

 of the 2yu, 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/^1 cylindroids 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. The 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. 



The 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 



