A Treatise on the Theory of Screws

54] THE REPRESENTATION OF THE CYLINDROIU BY A CIRCLE. 49 The circular representation of the cylindroid is now complete. We see how the pitch of each screw is given, and how the perpendicular distance and the angle between every pair of screws can be concisely represented. We may therefore study the dynamical and kinematical properties of the cylindroid by its representative circle. We commence by proving a funda- mental principle very analogous to an elementary theorem in Statics. 54. The Triangle of Twists. It has been already shown (§ 14) that any three screws on the cylindroid possess the following property:— If a body receive twists about three screws, so that the amplitude of each twist is proportional to the sine of the angle between the two non- corresponding screws, the body, after the last twist, will be restored to where it was before the first. With the circular representation of the cylindroid we transform this theorem into the following:— If any three screws, A, B, C (Fig. 8), be taken on the circle, and if twists be applied to a body in succession, so that the amplitude of each twist is proportional to the opposite side of the triangle ABC, then the body will be restored by the last twist to the place it had before the first. From the analogy of wrenches, and of twist velocities to twists, we are also able to enunciate the following theorems:— If wrenches upon the three screws A, B, C be applied to any rigid body, then these wrenches will equilibrate, provided that the intensity of each is proportional to the opposite side of the triangle. B, 4