A Treatise on the Theory of Screws

204 THE THEORY OK SCREWS. [204, 204. The Pitch Conics. The discussion in § 203 will prepare us for the plane representation of the screws of given pitch p, for we have p W + 6/ + 032) - pA2 ~ pA2 - pA2 = 0. This, of course, represents a conic section, and, accordingly, we have the following theorem:— The locus of points corresponding to screws of given pitch is a conic section. A special case is that where the pitch is zero, in which case the locus is given by pA2 + P-A2 + pA32 = 0. This we shall often refer to as the conic of zero-pitch. Another important case is that where p is infinite, in which case the equation is ^ + ^+^=0. The conic of zero-pitch and the conic of infinite pitch intersect in four points, and through these four points all the other conics must pass. The points, of course, correspond to the screws of indeterminate pitch: we may call them If, P2, P3, Pt. Any conic through these four critical points will be a conic of equal- pitch screws. As a straight line cuts a conic in two points, we see the well-known theorem, that every cylindroid will contain two screws of each pitch. The two principal screws on a cylindroid are those of maximum and minimum pitch ; they will be found by drawing through Plt P2, P3, Pit the two conics touching the straight line corresponding to the cylindroid. The two points of contact are the screws required. If a and ß are the two principal screws on a cylindroid, then any pair of harmonic conjugates to a and ß represent a pair of screws of equal pitch. For if S + kS' = 0 be a system of conics, then it is well known that the pairs of points in which a fixed ray is cut by this system form a system in involution. The double points of this involution are the points of contact of the two conics of the system which touch the line. 205. The Angle between Two Screws. From the equations of the screw given in § 201, we see that the direction