A Treatise on the Theory of Screws

_____________________ _________ ____________________________ __________ _____ ______ ^00 THE THEORY OF SCREWS. [200- moves over the cylindroid and the corresponding point moves over the straight line. Hence we obtain the following important result:__ The several screws on a cylindroid correspond to the points on a straight line. Ill general two cylindroids have no screw in common. If, however, the two cylindroids be each composed of screws taken from the same three-system, then they will have one screw in common. This is demonstrated by the fact that the two straight lines corresponding to these cylindroids necessarily intersect in a point which corresponds to the screw common to the two surfaces. Three twist velocities about three screws will neutralize and produce rest, provided that the three corresponding points lie in a straight line, and that the amount of each twist velocity is proportional to the sine of the angle between the two non-corresponding screws. rhiec wrenches will equilibrate when the throe points corresponding to the screws are collinear, and when the intensity of each wrench is propor- tional to the sine of the angle between the two non-corresponding screws. 201. The Screws of the Three-system. In any three-system there are three principal screws at right angles to each other, and intersecting in a point (§ 173). It is natural to choose these as the screws of reference, and also as the axes for Cartesian co-ordinates. The pitches of these screws are pu p2, p3, and we shall, as usual, denote the screw co-ordinates by 3lt 32, 03. The displacement denoted by this triad of co-ordinates is obtained by rotating the body through angles 3lt 02, 33 around three axes, and then by translating it through distances p^, p..3.,, p3d3 parallel to these axes. As these quantities are all small, we have, for the displace- ments produced in a point x, y, z, Sx =p131+z3.2 - y 3. ^y = p-ß-1 + x3s-z31, & = pA + y31 - x32; these displacements correspond to a twist about a screw of which the axis has the equations pA + z32-y33 = pA + «A- zA _ p333 + y3, - x3„ 3, 3.2 3S while the pitch p is thus given : ,n _ PA + pA2 + p-A2 1 fff + ØJ + ØJ ■ ___________ ___________________________