A Treatise on the Theory of Screws

58] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 51 If the amplitudes a and ß had opposite signs, then the point I should have divided AB externally in the given ratio. 57. Screw Co-ordinates. We have developed in the last chapter the general conception of Screw Co-ordinates. In the case of the cylindroid, the co-ordinates of any screw X, with respect to two standard screws A and B, are found by resolving a wrench of unit intensity on X into its two components on A and B. These components are said to be the co-ordinates of the screw. If we denote the co-ordinates of X by X{ and X2, we have v V A'~AB' 3 AB' The co ordinates satisfy the identical relation, Xf — 2X)X2 cos e + Xf = 1, where e denotes the angle between the two screws of reference, that is, the angle subtended by the chord AB. 58. Reciprocal Screws. Every screw A on the cylindroid has one other reciprocal screw B lying also on the cylindroid (§ 26). Denoting as usual A and B by their corre- sponding points on the circle, we may enunciate the following theorem :— The chord joining a pair of reciprocal screws passes through the pole of the axis of pitch. The condition that two screws shall be reciprocal is (/>« + PA cos # ~ d*f> sin 0 = where pa and pß are the pitches, 6 is the angle between the two screws, and daß their shortest distance. It is easy to show that this condition is fulfilled for any two screws A and B (Fig. 10), whose chord passes through 0, the pole of the axis of pitch PQ. 4—2