356] THE THEORY OF SCREW-CHAINS. 393 



The first point to be noticed is, that the correspondence is unique. To 

 the instantaneous chain a one impulsive screw-chain 6 will correspond. There 

 could not be two screw-chains 6 and 6 which correspond to the same instan 

 taneous screw-chain a. For, suppose this were the case, then the twist 

 velocity imparted by the impulsive wrench on 6 could be neutralized by the 

 impulsive wrench on 6 . We thus have the mass-chain remaining at rest 

 in spite of the impulsive wrenches on 6 and 6 . These two wrenches must 

 therefore neutralize, and as, by hypothesis, they are on different screw- 

 chains, this can only be accomplished by the aid of the reactions of the 

 constraints. We therefore find that 6 and 6 must compound into a 

 wrench-chain which is neutralized by the reactions of the constraints. 

 This is, however, impossible, for 6 and 6 can only compound into a wrench 

 on a screw-chain of the original system, while all the reactions of the 

 constraints form wrenches on the chains of the wholly distinct reciprocal 

 system. 



We therefore see that to each instantaneous screw-chain a only one 

 impulsive screw-chain 6 will correspond. It is still easier to show that to 

 each impulsive screw-chain 6 only one instantaneous screw-chain a will 

 correspond. Suppose that there were two screw-chains, a and a. , either of 

 which would correspond to an impulsive wrench on 6. We could then give 

 the mass-chain, first, an impulsive wrench on d of intensity X, and make 

 the mass-chain twist about a, and we could simultaneously give it an im 

 pulsive wrench on the same screw-chain 6 of intensity X, and make the 

 mass-chain twist about a . The two impulses would neutralize, so that as 

 a matter of fact the mass-chain received no impulse whatever, but the 

 two twist velocities could not destroy, as they are on different screw-chains. 

 We would thus have a twist velocity produced without any expenditure of 

 energy. 



We have thus shown that in the w-system of screw-chains expressing the 

 freedom of the mass-chain, one screw-chain, regarded as an instantaneous 

 screw-chain, will correspond to one screw-chain, regarded as an impulsive 

 screw-chain, and conversely, and therefore linear relations between the 

 co-ordinates are immediately suggested. That there are such relations can be 

 easily proved directly from the laws of motion (see Appendix, note 7). We 

 therefore have established a case of screw-chain homography between the 

 two systems, so that if 1} ...0 n denote the co-ordinates of the impulsive 

 screw-chain, and if a a , ... a n denote the co-ordinates of the corresponding 

 instantaneous screw-chain, we must have n equations of the type 



0, = (11) ttl + (12) a 2 + (13) 3 . .. + (1 ) a B&amp;gt; 



(nn) B , 



