A Treatise on the Theory of Screws

184 THE THEORY OF SCREWS. [182- 182. Four Screws of the Screw System. Take any four screws a, ß, y, 8 of the screw system of the third order. Then we shall prove that the cylindroid (a, ß) must have a screw in common with the cylindroid (7, 3). For twists of appropriate amplitudes about a, ß, y, 8 must neutralise, and hence the twists about a, ß must be counter- acted by those about y, 8; but this cannot be the case unless there is some screw common to the cylindroids (a, /?) and (7, 8). This theorem provides a convenient test as to whether four screws belong to a screw system of the third order. 183. Geometrical notes. The following theorem may be noted: Any ray y which crosses at right angles two screws a, ß of a three-system is the seat of a screw reciprocal to the system. For, draw the cylindroid a, ß, then of course y. whatever be its pitch, is reciprocal to all the screws on this cylindroid. Through any point P on y there are two screws of the system which lie on the cylindroid, and there must be a third screw y of the system through P, which, certainly, does not lie on the cylindroid. If, therefore, we give y a pitch — py, it must be reciprocal to the three-system. In general, one screw of a three-system can be found which intersects at right angles any screw whatever y. For y must, of course, cut each of the quadrics containing the screws of equal pitch in two points. Take, for example, the quadric with screws of pitch p. There are, therefore, two screws, a and ß of pitch p belonging to the system, which intersect y. The cylindroid a, ß must belong to the system, and from the known property of the cylindroid the ray y, which crosses the two equal pitch screws (§ 22), must cross at right angles some third screw y on this cylindroid; but this belongs to the three-system, and therefore the theorem has been proved. 184. Cartesian Equation of the Three-System. If we are given the co-ordinates of any three screws of a three-system with reference to six canonical co-reciprocals, we can calculate in the following manner the equation to the family of pitch quadrics of which the three-system is constituted. Let the three given screws be a, ß, y, with co-ordinates respectively