438 



passes through 5. Consequent!}' there lie on ^' two curves ()', 

 which pass through S. The locus A of the curves d' has consequently 

 three nodal lines q,q',q"; its section with a 2' consists further of 

 three curves <P, is therefore of order 21. Hence Zi is a surface of 

 order seven. 



The figures (</, f/') determine on q a correspondence (3,2) ; so there 

 lie on q five points D^{d,ö^). The locus of the points Z) is therefore 

 a tnnsted curve {D)\ intersecting each of the straight lines q, q, (f 

 five times. 



Now 35' and A' have the three straigiit lines q and the curve 

 {DY in common, consequently another figure of order two. This 

 figure must consist of two straight lines (/, hence there is a figure 

 of [(./'] consisting of two straight lines (/ and a curve d*. This curve 

 has q, q', q" as bisecants and intersects D' moreover in two points D. 



Through an arbitrary' point P pass five singular trisecants; they 

 are nodal lines of /7' and ^t^ These surfaces have moreover in 

 common the curve q^ laid through P, the six parabolic bisecants 

 P£ and three straight lines b. The straight lines b are determined 

 by the points which the straight lines q outside the curve ^^ have 

 in common with the cone ?iV ; hence they are simjular bisecants. 



If P is supposed to be on q, If is replaced by the figure com- 

 posed of 2' and a cone (6)\ Tke shujular bisecants form consequently 

 three congruences (1, 4). 



The locus of the straight lines which intersect a ïigwr e {q\ q,q,q') 

 thrice, consists of the hyperboloid (</ (j' q"), three ruled surfaces :'i' 

 with nodal lines q, q' and the ruled surface of the trisecants of (>"; 

 this is therefore of order 16. 



From this it is now deduced, in the way followed before, that 

 the sinyuhir trisecants form a congruence (5, 6) possessing six singular 

 points F of order three. 



Tlie surface A' has three triple straight lines q, q', q''. In a plane 

 (f the congruence [p'] determines a sextuple involution with three singular 

 •points of order three, which are at the same time nodes of the curve of 

 coincidence r/". The curves t', touching (p, form a </'" with 12-fold 

 straight lines y, </', (/". There are 48 curves p", osculating one plane, 

 and 144 curves touching two planes. 



9. Let us now consider the case that all the surfaces of the net 

 [*"] have in common a conic a^ and a straight line q not inter- 

 secting it. Any two surfaces then determine a twisted curve q', 

 which rests in six points on -.', in four points on q. A third surface 

 intersects (,)" moreover in eight points. The congruence [9°] possesses 



