359] THE THEORY OF SCREW-CHAINS. 397 



is reciprocal to &amp;lt;/&amp;gt;, ft will be reciprocal to 6. This, it will be observed, is a 

 generalization of a property of which much use has been previousl)&amp;lt;jfcnade 

 ( 81). The proof is as follows. 



The co-ordinates of the instantaneous chains are 



&.-.. 



The co-ordinates of the corresponding impulsive chains are 



Pi Pn 



and 



V& Ujfin 



Pi Pn 



If the chain a be reciprocal to the impulsive chain which produces /3, then 

 we have 



&amp;lt;!& + ... + u n 2 nl3 n = ; 



but this being symmetrical in a and /3 is precisely the same as the condition 

 that the impulsive chain corresponding to a. shall be reciprocal to /9. Following 

 the analogy of our previous language we may describe two screw-chains so 

 related as conjugate screw-chains of Inertia. 



359. Harmonic screw-chains. 



We make one more application of the theory of screw-chains to the 

 discussion of a kinetical problem. Let us suppose that we have any material 

 system with n degrees of freedom in a position of stable equilibrium under 

 the action of a conservative system of forces. If the system receive a small 

 displacement, the forces will no longer equilibrate, but the system will be 

 exposed to the action of a wrench on a screw-chain. We thus have two 

 corresponding sets of screw-chains, one set being the chains about which the 

 system is displaced, the other set for the wrenches which are evoked in 

 consequence of the displacements. 



By similar reasoning to that which we have already used, it can be shown 

 that these two corresponding chain systems are homographic. We can 

 therefore find n screw-chains about which, if the system be displaced, a 

 wrench will be evoked on the same screw-chain, and (the forces having a 

 potential) it can be shown that this set of n screw-chains are co-reciprocal. 



If after displacement the system be released it will continue to make 

 small oscillations. The nature of these oscillations can be completely 

 exhibited by the screw-chains. To a chain a, regarded as an instantaneous 

 screw-chain, will correspond the screw as an impulsive screw-chain. To the 

 chain a, regarded as the seat of a displacing twist, will correspond a wrench 



