357] THE THEORY OF SCREW-CHAINS. 395 



systems of screw-chains by choosing n screws quite arbitrarily as the double 

 screws of the two systems, and then appropriating to them n factors (11), 

 (22), (33), &c., also chosen arbitrarily. In the case of impulsive and instant 

 aneous chains, the n double chains are connected together by the relation 

 that each pair of them are reciprocal, so that the whole group of n chains 

 form what may be called a set of co-reciprocals. 



To establish this we may employ some methods other than those 

 previously used. Let us take a set of w-co-reciprocal chains, and let the 

 co-ordinates of any other two chains, 6 and &amp;lt; of the same system, be 1&amp;gt; ... 6 n 

 and fa, ...fa,,. Let 2p 1} *2p 2 , &c., 2p n be certain constant parameters 

 appropriated to the screws of reference. 2pj is, for example, the work done 

 by a twist of unit amplitude on the first screw-chain of reference against a 

 wrench of unit intensity on the same chain. The work done by a twist O l 

 against a wrench fa on this chain is 2p 1 1 fa. As the chains of reference are 

 co-reciprocal, the twist on 6 l does no work against the wrenches fa 2 , &amp;lt; 3 , ... &c. ; 

 hence the total work done by a twist on 6 against the wrench on &amp;lt;/&amp;gt; is 



and hence if 6 and &amp;lt; be reciprocal, 



The quantities p 1 , . . . p n are linear magnitudes, and they bear to screw-chains 

 the same relation which the pitches bear to screws. If we use the word 

 pitch to signify half the work done by a unit twist on a screw-chain against 

 the unit wrench on the screw-chain, then we have for the pitch p 6 of the 

 chain 6 the expression 



The kinetic energy of the mass-chain, when animated by a twist velocity of 

 given amount, depends on the instantaneous screw-chain about which the 

 system is twisting. It is proportional to a certain quadratic function of the n 

 co-ordinates of the instantaneous screw. By suitable choice of the screw- 

 chains of reference it is possible, in an infinite number of ways, to exhibit 

 this function as the sum of n squares. It follows from the theory of 

 linear transformations that it is generally possible to make one selection of 

 the screw-chains of reference which, besides giving the energy function the 

 required form, will also exhibit p e as the sum of n squares. This latter 

 condition means that the screw-chains of reference are co-reciprocal. It only 

 remains to show that the n screw-chains of reference thus ascertained must 

 be the n principal screw-chains to which we were previously conducted. 



We may show this most conveniently by the aid of Lagrange s equations 

 of motion in generalized co-ordinates ( 86). 



