A Treatise on the Theory of Screws

402 THE THEORY OF SCREWS. .. xn, these circumstances T has obviously not altered, so that we Let us Under have also —f l (#1 > • • • xn > Let the mass-chain be stopped in the position A. Let it then be dis- placed to an adjacent position B defined by the following variations of the co-ordinates dT * □. dT s ^dT dT v n ■f Ö«! + •.. j OXn + -jr + • • • jV" OXn = 0. dx1 dx± dx^i [362, been adopted. We call this screw-chain 9, and 0 denotes the twist velocity with which the system moves round 0. d/f . + < 8xn + < &Z, + ... ^4 8xn = 0, assume, for brevity, the symbol A, such that . . d . d A = XA y- + ... + Xny— . dx^ docfi ^8x,...+ + +... dk =o. dx-^ dxn dd-y doc^i Choose any n independent screw-chains, about each one of which the mass-chain is capable of twisting. Then 6 can be decomposed into n com- ponents which have twist velocities 0n about the several screw-chains of reference (§ 355). Since everything pertaining to the position or the movement of the mass-chain must necessarily admit of being expressed in terms of the co- ordinates of the mass-chain, and since the quantities ... f)tl are definitely determined by the position and movements of the mass-chain, it follows that these quantities must be given by a group of formulae which may be written 8x^ — xf^, ... 8x% — XfiSs) where 8e is a small quantity. From this new position B let the mass-chain be started into motion so that it shall have the same twist velocity as it had just before being stopped in A and about the same screw-chain. This condition requires that each of the quantities 01,... 9n shall resume its original value unaltered by the stoppage and subsequent restarting of the mass- chain from a new position. There must accordingly be an adjustment of ... 8xn to satisfy the equations