APPENDIX I. 493 



we form the polar conic of the point by changing separately S to P and P to S 

 and we have its equation as 



which as S^ = becomes 



V.-J .A-o. 



Repeating this operation we have for the equation of the polar of or the 

 tangent to the cubic the vanishing expression 



ivar*-* .p,sa, 



This proves the duplicity of the point. 



Therefore PpP^ & X S P ~ represents the tangents at the point and these 

 accordingly pass through the intersection of P x = Q and S p = 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 and 

 . If we draw two cubics of the class just considered from and 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 J (G - 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 



