343] THE THEORY OF SCREW-CHAINS. 373 



screws, can be uniquely determined. If, however, the intermediate screw 

 between S x and 8 2 be given, the entire chain 8 is also determined, yei it is 

 not immediately obvious that that determination is unique. We can, however, 

 show as follows that this is generally the case. 



Let S 12 denote the given intermediate screw, and let this belong, not 

 only to the chain S 1} 8 2 , &c., but to another chain &/, 8 2 , &c. We then 

 have S 1} &amp;lt;5 12 , 8. 2 co-cylindroidal, and also S/, 8^, 8 2 co-cylindroidal. Decom 

 pose any arbitrary twist of amplitude 6 on S 12 into twists on 8 L and 2 , and a 

 twist of amplitude 6 on the same 8 12 into twists on 8/ and S 2 . Then the 

 four twists must neutralize ; but the two twists on S/ and S t compound into 

 a twist on a screw on the first cylindroid of the system ; and / and 6 2 into 

 a twist on the second cylindroid of the system ; and as these two resultant 

 twists must be equal and opposite it follows that they must be on the same 

 screw, and that, therefore, the cylindroids belonging to the first and second 

 elements of the system must have a common screw. It is, however, not 

 generally the case that two cylindroids have a common screw. It is only true 

 when the two cylindroids are themselves included in a three-system, this 

 could only arise under special circumstances, which need not be further con 

 sidered in a discussion of the general theory. 



It follows from the unique nature of the correspondence between the 

 intermediate cylindroids and the primary cylindroids that one screw on any 

 cylindroid corresponds uniquely to one screw on each of the other cylindroids; 

 the correspondence is, therefore, homographic. 



We have now obtained a picture of the freedom of the second order of the 

 most general type both as to the material arrangement and the character of 

 the constraints : stating summarily the results at which we have arrived, 

 they are as follows : 



A mass-chain of any kind whatever receives a small displacement. This 

 displacement is under all circumstances a twist about a screw-chain. If 

 the mass-chain admits of a displacement by a twist about a second screw- 

 chain, then twists about an infinite number of other screw-chains must also 

 be possible. To find, in the first place, a third screw-chain, give the mass- 

 chain a small twist about the first chain ; this is to be followed by a small 

 twist about the second chain : the position of the mass-chain thus attained 

 could have been reached by a twist about a third screw-chain. The system 

 must, therefore, be capable of twisting about this third screw-chain. When 

 three of the chains have been constructed, the process of finding the re 

 mainder is greatly simplified. Each element of the mass-chain is, in each 

 of the three displacements just referred to, twisted about a screw. These 

 three screws lie on one cylindroid appropriate to the element, and there are 

 just so many of these cylindroids as there are elements in the mass-chain. 



