A Treatise on the Theory of Screws

14] THE CYLINDROID. 21 z\ = (^a — pß) sin I cos I, z2 = (pa — Pß) sin m cos m, h = Zy — z2, I = £ (a + tan-1 .- \ ill with similar values for m and z2. Pe = Pa cos21 + pß sin2( ?>./, — Pa cos2wi + pß sin2?n, A = I — m, _____________ I h? + (pe - p^f sin A Pa + Pß— pe + p<i> — h cot A, ) , *1 = 1 («?ß - Pa) cot A + I, It is therefore obvious that the cylindroid 18 determined, and that the solution is unique. It will often be convenient to denote by (0, <^>) the cylindroid drawn through the two screws P and </>. On any cylindroid there are in general two but only two screws which like a and ß intersect and are at the same time at right angles. These two important screws are often termed the principal screws of the surface. 14. General Property of the Cylindroid. If a body receive twists about three screws on a cylindroid, and if the amplitude of each twist be proportional to the sine of the angle between the two non-corresponding screws, then the body after the last twist will have regained the same position that it held before the first. The proof of this theorem must, according to (§ 12), involve the proof of the following;—If a body be acted upon by wrenches about three screws on a cylindroid, and if the intensity of each wrench be proportional to the sine of the angle between the two non-corresponding screws, then the three wrenches equilibrate. Ihe former of these properties of the cylindroid is thus proved:—Take any three screws 6, </>, y/r, upon the surface which make angles I, m, n, with the axis of x, and let the body receive twists about these screws of amplitudes Each of these twists can be decomposed into two twists about the screws a and ß which lie along the axes of x and y. The entire effect of the three twists is, therefore, reduced to two rotations around the axes of x and y> and two translations parallel to these axes. The rotations are through angles equal respectively to ff cos I + </>' cos m + cos n ajid 0' sin I + cfy' sin m + y(r' sin n. The translations are through distances equal to pa (0' cos I + <ß' cos m + cos n) and pp (0' sin Z + </>' sin m + sin n).