390 THE THEORY OF SCREWS. [354, 



A screw-chain is defined by 6/4 1 data ( 353). It follows that a finite 

 number of screw-chains can be determined, which shall be reciprocal to 

 6/i 1 given screw-chains. It is, however, easy to prove that that number 

 must be one. If two chains could be found to fulfil this condition, then 

 every chain formed from the system by composition of two twists thereon 

 would fulfil the same condition. Hence we have the important result 



One screw-chain can always be determined which is reciprocal to 6//- 1 

 given screw-chains. 



This is of course only the generalization of the fundamental proposition 

 with respect to a single rigid body, that one screw can always be found 

 which is reciprocal to five given screws ( 25). 



355. Twists on 0/^ + 1 screw-chains. 



Given 6/^+1 screw-chains, it is always possible to determine the ampli 

 tudes of certain twists about those chains, such that if those twists be 

 applied in succession to a mass-chain of /u, elements, the mass-chain shall, 

 after the last twist, have resumed the same position which it had before the 

 first. To prove this it is first necessary to show that from the system formed 

 by composition of twists about two screw-chains, one screw-chain can always 

 be found which is reciprocal to any given screw-chain. This is indeed the 

 generalization of the statement that one screw can always be found on a 

 cylindroid which is reciprocal to a given screw. The proof of the more 

 general theorem is equally easy. The number of screw-chains produced by 

 composition of twists about the screw-chains a and /3 is singly infinite. 

 There can, therefore, be a finite number of screw-chains of this system 

 reciprocal to a given screw-chain 6. But that number must be one ; for if 

 even two screw-chains of the system were reciprocal to 6, then every screw- 

 chain of the system must also be reciprocal to 8. The solution of the original 

 problem is then as follows : Let a. and /3 be two of the given 6/z, + 1 chains, 

 and let 6 be the one screw-chain which is reciprocal to the remaining 

 6/i 1 chains. Since the 6/* + 1 twists are to neutralize, the total quantity 

 of work done against any wrench-chain must be zero. Take, then, any 

 wrench-chain on 6. Since this is reciprocal to 6/4 1 of the screw-chains, the 

 twists about these screw-chains can do no work against a twist on 6. It 

 follows that the amplitudes of the twists about a and @ must be such that 

 the total amount of work done must be zero. For this to be the case, the 

 two twists on a and /3 must compound into one twist on the screw-chain v, 

 which belongs to the system (a/3), and is also reciprocal to 0. This defines 

 the ratio of the amplitudes of the two twists on a and /3. We may in fact 

 draw any cylindroid containing three homologous screws of a, /3, and 7, then 

 the ratio of the sines of the angles into which 7 divides the angle between 

 a and @ is the ratio of the amplitudes of the twists on a. and /8. In a 



