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 /3 on the other, /?. 

 The position thus attained could have been attained by a twist about some 

 single chain 7. As a and /3 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, /3, 7, such that if 

 a mass-chain be twisted with amplitudes of, ft , 7 , 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. H.). Each element of the mass-chain receives two twists about 

 a and /3 ; 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 7 and its amplitude 7 can be completely determined. 



A mass-chain free to twist to and fro on the chains a and /3 will therefore 

 be free to twist to and fro on the chain 7. 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 OL I} /3 1( 7^ 8 1 

 which are the four first screws of the four screw-chains must be co- 

 cylindroidal ; so must 2 &amp;gt; A, y 2 , 8 2 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 the 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&amp;gt; &, 7i&amp;gt; Si b y tne symbol 



[i, fii, 71, $1], 



then the first theorem to be now demonstrated is that 



[i, &, 71, S,]=[ 2 , &, 72&amp;gt; &&amp;gt;] = &c. = [a M , $1, 7 M , M ], 

 or that the anharmonic ratio of each group is the same. 



