A Treatise on the Theory of Screws

366] THE THEORY OF PERMANENT SCREWS. 405 the metric element must be essentially the amplitude of the twist about the screw-chain. We have thus proved the following theorem :— The co-ordinates must be twists about n screw-chains of reference whenever the identical equation, satisfied by T, assumes the form a dT + . dT 01 dør 0,1 døf ~ °’ 365. Transformation of the Vanishing Emanant. Suppose that the position and movement of a mass-chain were represented by the co-ordinates 0,', 02', ... 0f; 0lt 02, ... 0n when referred to one set of n screws of reference, and by <b,', <p2, ... cj>f; 4>ls <jx,, ... </>„ when referred to another set of screws of reference. Then of course these sets of co-ordinates must be linearly connected. We may write 0r (11) ^ ... + (ln)K. 0f = (nl)^... + (uni) (j>f. Then, by differentiation ^ = (11)^ ...+(ln)^t. Thus the two sets of variables are co-gredients, and by the theory of linear transformations we must have 0 dT 4 dT g dT W' + nd0f d^' ; dT • + <Pn 1, 7 d<j>n The expression on either side of the equation is of course known in algebra as an emanant (§ 261). We could have foreseen this result from the fact that whatever set of n independent screw-chains belonging to the system was chosen, the identical equation must in each case assume the standard form. 366. The General Equations of Motion with Screw chain Co-ordinates. The screw-chain co-ordinates of a mass-chain with n degrees of freedom are 0fi ... Øf-, the co-ordinates of the velocities are 0lt ... 0n. Let y be the wrench-chain which acts on the system. Let the components of the wrench- chain, when resolved on the screw-chains of reference, have for intensities V1", vf'- Let pi> ••• pn be the pitches of the chains of reference, by which is meant that 2p, is the work done on that screw-chain by a twist of unit amplitude against a wrench of unit intensity on the same screw-chain. Then the screw-chains of reference being supposed to be co-reciprocal, we have,