346] THE THEORY OF SCREW-CHAINS. 377 



If the mass-chain cannot be twisted about any screw- chain except those 

 we have now been considering, then the mass-chain is said to have freedom 

 of the third order. If, however, a fourth screw-chain can be found, about 

 which the system can twist, and if that screw-chain does not belong to the 

 doubly infinite system just described, then the mass-chain must have freedom 

 of at least the fourth order. 



346. Freedom of the fourth order. 



The homologous screws in the four screw-chains about which the mass- 

 chain can twist form each a four-system. All the other chains which can 

 belong to the system m.ust consist of screws, one of which lies on each of the 

 four-systems. 



It will facilitate the study of the homography of two four-systems to 

 make use of the analogy between the homography of two spaces and the 

 homography of two four-systems as already we had occasion to do in 317. 

 A screw in a four-system is defined by four homogeneous co-ordinates 

 whereof only the ratios are significant. Each screw of such a system can 

 therefore be represented by one point in space. The homography of two 

 spaces will be completely determined if five points, a, b, c, d, e in one space, 

 and the five corresponding points in the other space, a, b , c , d , e are 

 given. 



From the four original screw-chains we can construct a fifth by com 

 pounding any arbitrary twists about two or more of the given chains. When 

 five chains have been determined, then, by the aid of the principle of homo 

 graphy, we can construct any number. 



That each set of six homologous screws is homographic with every other 

 set can be proved, as in the other systems already discussed. With respect 

 to the intermediate screws a different proof is, however, needed to show 

 that when one of these screws is given the rest of the chain is uniquely de 

 termined. The proof we now give is perhaps simpler than that previously 

 used, while it has the advantage of applying to the other cases as well. Let 

 a, /3, 7, 8 be four screw-chains, and let e I2 , an intermediate screw of the 

 chain e, be given. We can decompose a twist on e 12 into components of 

 definite amplitude on a I2 , &,, 7,3, 8 K . The first of these can be decomposed 

 into twists on c^ and a. 2 ; the second on & and /3 2 , &c. Finally, the four 

 twists on !, &, 7!, 8 1 can be compounded into one twist, e,, and those on 

 2, &, 7 2 , 8-2 compounded into a twist on e 2 . In this way it is obvious that 

 when e 12 is given, then ej and e 2 are uniquely determined, and of course the 

 same reasoning applies to the whole of the chain. We thus see that when 

 any screw of the chain is known, then all the rest are uniquely determined, 

 and therefore the principle of homography is applicable. 



