A Treatise on the Theory of Screws

202] PLANE REPRESENTATION OF THE THIRD ORDER. 201 We have now to investigate the locus of the screws of given pitch, and as p is presumed to be a determinate quantity, we have (pi-p) 0i + z02 -y03 = O, -z01+(p2- p) 02 + x03 = 0, + y01 - x02 + (p3-p) 03 = 0, whence, by eliminating 0lt 02 , 03 we obtain, as the locus of the screws of pitch p, the quadric otherwise found in the previous chapter (pi-p) & + (p3-p) y1 + (p3-p) z2 + (pi-p) (p3-p)(p3 ~p) = 0. According as p varies, this family of quadrics will exhibit all the screws of the three-system which possess a definite pitch. 202. Imaginary Screws. To complete the inventory of the screws it is, however, necessary to add those of indefinite pitch, i.e. those whose co-ordinates satisfy both the equations M2+M2+M2=°> 0* + e.? + e,3 ■= o. There are four triads of co-ordinates which satisfy these conditions, and, remembering that only the ratios are concerned, the values of 0lt 02, 03 may be written thus: + ( p-2 ~ p3)h, +(p3~Pi)i, + ( pi - P$, -(Pt-Psft, +(ps-pi)i, +(pi~P2)i, + ( P2 - Ptfi, -(p3~pi)h, + (pi~ p2)'2, + (p2~p3)i, H-Cl’s-Pl)4, -(Pl-prf- The equations of the axis written without p are x (0/ + Øfi) - yØfi. - zØfi3 + (p2 - p3) 0203 = 0, y (0/ + 0f) — z0..03 - x0.fi, + (p3-pi) 0-fii = 0, z (0;2 + Øf) - x030, - y0302 + (pi- P2) 0i03 = 0, of which two are independent. If we substitute the values of 0,, 02, 03 for the first indeterminate screw, the thi-ee equations just written reduce to ® (Pi - prf +y(p3-Pi)i + 2(Pi~ -(P-2- P-^ ( P3 ~ P$ ( Pi ~ P^ = °-