62 



The complex curves in an arlntmry plane tlirongli such a slraiglit 

 line (ouch each other al the point in <pieslion, so that tlie two 

 complex curves have 5 coincidini'>- tangents in common in the /^ 

 of their plane. Now we have split oft the straight lines of l^^^ as 

 4-foid rajs of the congruence of the intersection of the two com- 

 plexes, hence : 



V^ is a suupdar plane of the order 6. 



^ 4. We can also anive at this last result in the following way. 

 The quadratic surfaces thiongh 5 points form a linear system of 

 00'' individuals; these cut F^ in a {k-)^; the conic of the double 

 straight lines of this (^'), belongs to the parabolical cylinders of 

 {0*)^ Let C be such a cylinder, T its vertex, c the line along which 

 C touches F . 



CO 



The polar plane of T relative to C is indefinite, hence T has a 

 fixed polar plane relative to all 0*'s of the pencil through A^ Bj, 

 which touch y* in its points of intersection with c. This fixed polar 

 plane is at the same time the |)lane of the centres of the individuals 

 of the pencil; it passes Ihiough the polar line p of T relative to 

 yV In the null system [2,1] belonging to Tj the pole /^ofc relative 

 to 7* is associated to />. 



As the fixed polar plane of T relative to the 0^'s through A^ Bj 

 that touch y" at its points of intersection with c, pass likewise through 

 p, in the two null systems corresponding to F^ and I\ the point P 

 is associated to p. 



P was the pole of c relative to y', hence the locus of P is a 

 conic. The order of the null systems is three; accordingly the locus 

 of the straight line /> is a curve of the sLctJi class. 



We remark also that to each pai'abolical cylinder one axis in 

 V^ remains associated ((^*'s through 4 points, § 2), namely the 

 polar line of its vertex lelative to y\ 



§ 5. If six points are given I consider a group of 4 and a group 

 of 5 of these points which have 3 points in common. To the 

 group of 4 points there belongs a complex ^^ to that of 5 points 

 a congruence 6^>2. The axes in question are part of the common 

 rays of complex and congruence; however, we must split off: the 

 tangents of three curves of the 3''d class and twice the tangents of 

 a curve of the sixth class, so that we arrive at a ruled surface of 

 the order 3 (7 + 2) — 3.3 — 2.6 = 6, hence : 



the axes of the 0^'s through si.r points form a ruled surface of 

 the sixth order, q\ 



