44 



On ö' lie consequeDtly .36 pair's of poinls, eacli Ijearing 'x' ciirveb 

 Q^ ; in other words, the net contains 3(5 dimonoids, of whicli the 

 two nodes are lying on a\ The coiignience further contains 24 

 curves q\ which osculate the singular curve a*. 



The curves ^' lying on the monoid -S', are, by central projection 

 out of <S, represented by a pencil of plane curves </''', with two double 

 base-points and eight single base-points ; to it belong the images of 

 the live fundamental points. The remaining three are the intersectiojis 

 of three singular hisecants h ; through each point of such a straight 

 lin« passes a p' of ^'. The two nodes are the intersections of two 

 dmfiihv trlsecaali t ; eacii straight line t is moreover intersected 

 in two points by each q^ of the monoid ; for two 9* the line t is 

 a tangent. The three straight lines 1), and the two straight lines t 

 lie of course on i" ; the sixth straight line passing through ,S' is a 

 trisecant d of n\ It is component part of a degenerate q^ ; for all 

 «P" passing througii an arbitrary point of (/ contain this straight line 

 and have moreo\er another elliptic curve 9^ in common. 



3. The locus of the straight lines (/ is the hi/perholoid L'', which 

 may be laid through <5\ The latter has with a monoid -S" the 

 singular curve 0'^ and two trisecanfs d in common. Consequently ü" 

 contains a straight line (7 not passing throngh 5; thacurve ()^ coupled 

 to this straight line must contain the point S. It is represented by 

 a curve f/", containing the intersections of the straight lines ^, 6 and 

 the images of the points F, while the line connecting the intersections 

 of the two singular trisecants is the image of the straight line d 

 belonging to this q\ 



The locus of the curves q* has in common with S^ the curves 

 a* and two curves p" ; so it is a surface of order four, L\ With 

 A^ the surface A^ has in common the curve o* ; the remaining 

 section is a rational curve d^ being the locus of the point Z) ;£^ (rf, 9^). 

 As the trisecants of 6^ form the second system of straight lines of A', 

 Ó* and (3^ have ten points in common. This is confirmed by the 

 observation that the pairs d, q* determine on 0^ a correspondence 

 (7, 3), which has the said ten points as coincidences. 



4. The locus of the pairs of points which the curves q^ have 

 in common with their chords drawn through a point P is a surface 

 /7', with a quadruple point P. The tangents in Pform the cone if-, 

 which projects the curve q^ laid through P. the two trisecants < of 

 this curve are nodal edges of that cone and at the same time nodal 

 lines of W. The cone, which projects 0* out of P has in common 



