A Treatise on the Theory of Screws

133, 134] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 121 of ue upon the several screws of the cylindroid (§ 128). The representative circle (§ 50) will give a convenient geometrical construction. Let 0, and 0-, be the two co-ordinates of 0 relatively to any two screws of reference on the cylindroid. Then the components of the twist velocity will be 00, and 002. The actual velocity of any point of the body will necessarily be a linear function of these components. The square of the velocity will contain terms in which 0- is multiplied into 0,2, 0,0,, 0-?, respec- tively. If, then, by integration we obtain the total kinetic energy, it must assume the form J/02(XÖ13+ 2p,0,0.,+ v0A), whence, from the definition of ue, = X02 4- 2(10,02 + V0-21- The three constants, X, p, v, are the same for all screws on the cylindroid. They are determined by the material disposition of the body relatively to the cylindroid. We have taken the two screws of reference arbitrarily, but this equation can receive a remarkable simplification when the two screws of reference have been chosen with special appropriateness. Fig. 18. Let the lengths AX and BX (fig. 18) be denoted by p, and p2, and if e be the angle subtended by AB, we have from § 57, Xp,2 + 2/zp,p, + vp23 - ue2 (p,- - 2p,p., cos e + p22) = 0. Let us now transform this equation from the screws of reference A, B to another pair of screws A', B'. Let p/, p2 be the distances of X from A', B', respectively; then, from Ptolemy’s theorem, we have the following equations:— p,. AB — p2 • A A — p, .AB, p2.A'B' = p2.A'B-p,'.BB'.