A Treatise on the Theory of Screws

APPENDIX I. 493 we form the polar conic of the point 0 by changing separately <8 to P and P to S' and we have its equation as 28,'S^ + PPPx-S'xS]>-0, which as Sp' = 0 becomes ppp„ -S'A = o- Repeating this operation we have for the equation of the polar of 0 or the tangent to the cubic the vanishing expression PvS'x-S'xPp=G. This proves the duplicity of the point. Therefore PPPX -8'xSp = 0 represents the tangents at the point and these accordingly pass through the intersection of Px = 0 and Sp = 0. I may also add that the principles here laid down will enable us to investigate the various relations between the screws of a 3-system which intersect. Let us seek for example the number of screws of the system which are common transversals to two screws which also belong to the system and which are represented by 0 and O'. If we draw two cubics of the class just considered from 0 and O' as double points, they will in general intersect in nine distinct points. Of these, four will of course be the points common to all these cubics on the conic of infinite pitch. We have thus five remaining intersections each of which corresponds to a screw of the system, whence we deduce the theorem that any two screws of a 3-system will in general be both intersected by five other screws of the 3-system. NOTE VI. Remarks on § 224 by Professor G. J. Joly. If there is no speciality the nodal curve of the sextic ruled surface of the quadratic 2-system is of the tenth degree with four triple points on the surface. Of course every generator of the surface meets four other generators ; this follows from the plane representation. An arbitrary section is a unicursal sextic having therefore Jr (6 - 1) (6 - 2) = 10 double points. A section through a generator is the generator plus a unicursal quintic, and a section through two generators consists of the generators and a trinodal quartic. When the director cone of the surface breaks into a pair of planes, the nodal curve rises to the eleventh degree and consists of the two double lines, the common generator and the remaining curve of intersection of the two cylindroids into which the surface degrades. The four triple points are those in which the double lines of one cylindroid meets the other __not on the common generator. We should expect to find four triads of con- current axes belonging to the quadratic system. The locus of the feet of perpendiculars from an arbitrary origin is a twisted quartic. The quartic is not the intersection of two quadrics. Only one quadric