A Treatise on the Theory of Screws

226 THE THEORY OF SCREWS. [215- This equation involves the co-ordinates of cf> in the second degree. If this equation stood alone it would merely imply that <£ belonged to the quadratic five-system (§ 223) which included all the screws that intersected at right angles any one of the screws of the given cylindroid. If we further assume that <f> is to be a screw on the given cylindroid, then we have ... + A0<f>6 = 0, ... + — 0, = 0, Bi </>i ... + Be <j)6 = 0. From these five equations two sets of values of ø can be found. Thus among the system of screws which satisfy four linear equations there must be two screws which intersect at right angles. These are of course the two principal screws of the cylindroid. 216. Application to the Three-System. The equations of the three-system can be also deduced from the principle employed in § 214 which enunciated for this purpose is as follows. If 0 be a screw of stationary pitch in a three-system P then there is a cylindroid belonging to P such that every screw of the cylindroid intersects 6 at right angles. It is obvious that this condition could only be complied with if 6 lies on the axis of the cylindroid, and as the cylindroid has two intersecting screws at right angles we have thus a proof that in any three-system there must be one set of three screws which intersect rectangularly. Let their pitches be a, b, c, then on the first we may put a screw of pitch - a, on the second a screw of pitch - b, and on the third a screw of pitch - c. Thus we arrive at a set of canonical co-reciprocals specially convenient for the particular three-system. We have therefore learned that whatever be the three linear equations defining the three-system it is always possible without loss of generality to employ a set of canonical co-reciprocals such that the 1st, 3rd and 5th screws shall belong to the system. These three screws will define the system. Any other screw of the system can be produced by twists about these three screws. Hence we see that for every screw of the system we must have 02 = O; 04 = O; 06 = O.