A Treatise on the Theory of Screws

363] THE THEORY OF PERMANENT SCREWS. 403 Then we obtain, by elimination of ... and 8e, A T, dT df ’ dT dxn = 0. A/., dfi df ’ dfi dxn AA, dfn dfn df ’ ’ " dxn Such is the general condition which must be satisfied by the kinetic energy of any material arrangement whatever. But the equation is so complicated when expressed in ordinary rectangular co-ordinates that there is but little inducement to discuss it. 363. The Identical Equation in Screw-chain Co-ordinates. The Theory of Screw-chains exhibits this equation in a form of special simplicity. For, suppose that 0/ = «i, = On —■ Xn > 0n — > then the equation of the last article reduces to A7’=0. We thus have the following theorem:— If the co-ordinates, 0,, ... Of, of a mass-chain be n twists about n screw- chains, belonging to the system of screw-chains which express the freedom of the mass-chain, and if 0lt... 0n be the twist velocities of the mass-chain about these same screw-chains, then the kinetic energy T satisfies the equation å dT >dT eidoi- + dndof-Q- I have thought it instructive to exhibit the origin of this equation as a special deduction from the case of co-ordinates of the general type. For a brief demonstration the following simple argument suffices:— If the mass-chain be displaced through SOf... 80f while the velocities are unaltered, the change of kinetic energy is ^^'4. 4.dT def91 +- +dofWn- If the change of the position be due to a small twist 3e around the screw- chain with co-ordinates 0u ... 0n, then Ofe — 80 f 0n8e = 80„', 26—2