A Treatise on the Theory of Screws

208 THE THEORY OF SCREWS. [208, and Y must therefore be mutually perpendicular and reciprocal. If pe and p^, be the pitches of these screws; if a> be the angle between them, and d their perpendicular distance, then the virtual coefficient is one-half of Oo + ft>) cos ® sin w; as they are reciprocal, this is zero, and as w is a right angle, we must have d — 0; in other words, the screws corresponding to X and Y must intersect at right angles. The same may be proved of either of the pairs X and Z or Y and Z. The points X, Y, Z must therefore correspond to three screws of the system mutually perpendicular, and intersecting at a point. But in the whole system there is only a single triad of screws possessing these properties. They are the axes in the equations of § 201, and are known as the principal screws of the system. The three points X, Y, Z being the vertices of a self-conjugate triangle with respect to both the conics A and B, and hence to the whole system, we have the following theorem : — The vertices of the conjugate triangle common to the system of pitch conics correspond to the three principal screws of the three-system. 209. Expression for the Pitch. Each conic drawn through the four points of indeterminate pitch, Plt P,, Ps, Pt, is the locus of screws with a given pitch belonging to the system. We are thus led to connect the constancy of the pitch at each point of this conic with another feature of constancy, viz. that of the anharmonic ratio subtended by a variable point of the conic with the four fixed points. The connexion between the pitch and the anharmonic ratio will now be demonstrated. Let 0,, 02, 0S be the co-ordinates of any point on the conic, and let a, ß, y be the co-ordinates of one of the four points, say P2; then if </>,, <£„, </>3 be the current co-ordinates, the equation of the line joining 0 to Py is <£>1, </>2> </>3 = 0. 0„ 0-2, 02 \ a, ß, y As we are dealing with the anharmonic ratio of a pencil, we may take any section for the calculation of the ratios, and, accordingly, make </>3=0; and we have for the co-ordinates of the point in which the line joining 0 and Pi intersects </>,. = (), the conditions, </>i _ _ 03a_- 0iy fa d2y — 03ß