A Treatise on the Theory of Screws

200] PLANE REPRESENTATION OF THE THIRD ORDER. 199 If we fix our glance upon the screws about which the body is free to twist, the principle of classification will be obvious. Take an arbitrary triad ø„ et, and then form the infinite group of triads /A, p0„ p03 for every value of p from zero up to any finite magnitude: all these triads will correspond to the positions attainable by twisting about a single screw. We may therefore regard 03, 03 as the co-ordinates of a screw, it being understood that only the ratios of these quantities are significant. We are already familiar with a set of three quantities of this nature in the well-known trilinear co-ordinates of a point in a plane. We thus see that the several screws about which a body with three degrees of freedom can be twisted correspond, severally, with the points of a plane. Each of the points in a plane corresponds to a perfectly distinct screw, about which it is possible for a body with three degrees of freedom to be twisted. Accordingly we have, as the result of the foregoing discussion, the statement that— To each screw of a three-system corresponds one point in the plane. To develope this correspondence is the object of the present Chapter. 200. The Cylindroid. A twist of amplitude 6' on the screw 0 has for components on the three screws of reference 0'0i, 0'0„ Ø’Ø.p, a twist of amplitude </>' on some other screw </> has the components </>'</>!, When these two twists are compounded they will unite into a single twist upon a screw of which the co-ordinates are proportional to 0'0i + </>'</>!, 0'02 + </>/</>2, 0'03 + If the ratio of </>' to 0' be X, we see that the twists about 0 and </> unite into a twist about the screw whose co-ordinates are proportional to o, + Xc/h, 03 + X<f>2, 03 + X(f>3. By the principles of trilinear co-ordinates this point lies on the straight line joining the points 0 and </>. As the ratio X varies, the corresponding screw