A Treatise on the Theory of Screws

414 THE THEORY OF SCREWS. [377, 377. Different Screws on the same axis. Let the body be displaced from a standard position to another position defined by the co-ordinates 0/, (?./, ... (9/, and let it then be set in rotation about a screw of zero pitch with a twist velocity whose co-ordinates are 0i, 02, ... Let the kinetic energy of the body in this condition be T. Suppose that in addition to the rotation about 0 the body of mass M also received a velocity v of translation parallel to 0. Then the kinetic energy of the body would be T', where T'=T + ^Mv\ It is obvious that the position of the body, i.e. the co-ordinates 0/, 0.t,... 0e', can have no concern in ^Mv'2, whence dT' dT d0’ ~ d0' ’ and s™^ar equations. But a body rotating about 0 with an angular velocity 0 and translated parallel to 0 with the velocity v is really rotating about a screw on the same axis as 0 and with a pitch v+p. As v may have any value we obtain the following theorem :— All instantaneous screws lying on the same axis have the same restraining screw. 378. Co-ordinates of the Restraining Wrench for a free rigid body. Suppose the body to have a standard position from which we displace it by small twists 0/,... 0e' around the six principal screws of inertia. While the body is in its new position it receives a twist velocity of which the co-ordinates relatively to the six principal screws of inertia are 0lt... To compute the kinetic energy we proceed as follows:—Let a point lie initially at x, y, z, then, by the placing of the body at the starting position the point is moved to X, Y, Z, where X = a (01 — ft/) + y (05' + 06') — z (0i + 04') + x, Y = b (0.' - 0') + z (0' + 02') - x(0; + 0^ + y, Z = c (0/ - 06') + x (03 + 0/) - y (0i + 6>2') + z, m which a, b, c are the radii of gyration on the principal axes. The six principal screws of inertia lie, of course, two by two on each of the three principal axes, with pitches + a, — a on the first, -\-b, — b on the second, and + c, — c on the third. In consequence of the twist velocity with the components 0,,... 0„, each point X, Y, Z receives a velocity of which the components are