A Treatise on the Theory of Screws

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 n-co-reciprocal chains, and let the co-ordinates of any other two chains, 3 and of the same system, be (f, ... 0n and </>!, ...fan. Let 2pn 2p2, &c., 2pn be certain constant parameters appropriated to the screws of reference. 2pt 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 against a wrench fa on this chain is 2p101fa. As the chains of reference are co-reciprocal, the twist on f does no work against the wrenches fa,, <fa,... &c.; hence the total work done by a twist on 0 against the wrench on is 2p101fa+ ... + 2pn6nfan ■, and hence if 3 and <£ be reciprocal, pl </>i + • • • + pn 0n fan, = 0- The quantities p2, ...pn 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 pe of the chain 6 the expression Pi öi + p20* + ... + pn 0f. 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 p6 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).