( 258 ) 



If we nllow ;. to revolve round /, then the curve )} describes a 

 surface on which / is a single line. For through each point of / 

 passes one q\ and one of the planes through the tangent of that point 

 contains /. From this ensues that the locus of the points in which 

 curves (>■* can be touched bj planes of a pencil with axis / is a 

 surface of order six. 



8. The curves (5", touching the plane X in the points of the curve 

 ;/, intersect ). each in two points; the locus of those pairs of points 

 is a curve of order fourteen ^) (branchcurve) with quadruple points 

 in the points of intersection of the base-curves i:{^ and /3'\ The surface 

 containing the curves (>^ meant here, has thus with )• an intersection 

 of order twenty-four, is therefore a surface A'-'^ with quadrisecant 

 curves /?* and /?'". 



With a plane [i the surface A^"^ has a curve ft^^ in common con- 

 taining quadruple points in the points of intersection with /i^ and ^i' ^; 

 these eight points lie on the curve (L«^ which is the locus of the points of 

 contact of ft witii curves of r. The two curves have 24X5 — 8X4=88 

 points in common besides the base-points. So there are 88 curves ^'^ 

 touching two planes. 



9. To r belong ao^ curves (V which contain a nodal point, because 

 they are the intersections of two surfaces touching each other. 

 According to a wellknown property") the locus of the points of 

 contact of two quadrics belonging to two given pencils is a twisted 

 curve j)^\ cutting each of the two base-curves in 16 points. 



On an arbitrary surfiice Q" lie therefore 2X^4 — 16 = 12 points 

 of intersection with as many surfaces Q". The locus of the curves 

 d^ is therefore generated by two quadratic pencils in correspondence 

 (12,12^; consequently it is a surface A"^ on which /i^ and /3'^ are 

 twelvefold curves. 



10. The intersection of Q- and Q- breaks up into a line and a 

 q\ when they pass through a common bisecant of ^^ and i3'\ The 

 bisecants of these cur\'es forming two congruences (2,6) the number 

 of the common bisecants is 2 X 2 + 6 X 6 = 40. So to r belong 



orty figures consisting of a cubic curve with one of its bisecants. 



Through a ^^ can be laid four cones belonging to the quadratic 



pencil having q'^ as basis. The pencils determined by the 00" curves 



1) G. loc. p. 83. 



~) See a,o. Mineo, Bendiconti del Circolo matematico di Palermo, XVII, 297. 



