432 



remaining intersections lie three by three on the 42 trisecanfs of<>', 

 resting on a. From this it follows that 31 is a ruled surface of order 

 32. As an arbitrary point bears eleven straight lines /, a is elevenfold 

 straight line of 51", and a plane passing through a contains more- 

 over 21 straight lines /. The singular trlsecaids form therefore a 

 congruence (11, 21). 



In order to investigate whether the congruence \_(/\ possesses 

 other singular bisecants besides, we consider the surface IJ, which 

 contains the points of support of the chords, which the curves p' 

 send through a given point P. A straight line r passing through P, 

 is, in general, chord of one q\ therefore contains two points of U 

 lying outside P. One of those points of support comes in P, as soon 

 as r becomes chord of the q\ passing through P. The cone .S' pro- 

 jecting this p' out of P, is therefore the cone of contact of the 

 octuple point P and 77 is of order 10. The 11 straight lines t 

 passing through P are nodal edges of .Si' and at the same time 

 nodal lines of 77'°. The complete section of these two surfaces 

 consists of the 11 double lines mentioned, the curve pt, and the 27 

 straight lines PP^. From this it ensues that the straight lines ƒ are 

 the only singular bisecants. 



With an arbitrary <*' 77'° has the points of support of the 18 

 chords in common, which the curve sends through F; the remaining 

 54 intersections lie in the points F; consequently 77'° has nodes 

 in the 27 fundamental points. 



2. If two surfaces *' touch each other, the point of contact D 

 is node of their section d' and at the same time node of a surface 

 belonging to the net. The locus of D is a curve d'^ In order to 

 find the locus of the nodal curves ö", we consider two pencils of 

 the net. Each surface of the first pencil has 72 points Z) in common 

 with ö-\ is therefore touched by 72 surfaces of the second pencil; 

 by this a correspondence (72,72) is determined between those pencils. 

 The intersections of homologous surfaces with a straight line / are 

 homologous points in a correspondence (216,216); and both pencils 

 produce therefore a figure of order 432. But the surface that the 

 pencils have in common has been assigned 72 times to itself; the 

 real product is therefoi'e of order 216 only. From this it appears 

 that the nodal curves d' form a ■ surf ace of order 216, A"°. 



An arbitrary y° can intersect this surface in the points F only ; 

 consequently A"'" has the fundamental points as 72-fold points. 



3. The pencils mentioned above are brought in a correspondence 



