1189 



6. Tlie transformaüon {Q,Q' ) chjiiiges a conic into a curve of 

 order 16, with sextuple points in Si-. F(u- the conic t'^ passing through 

 the five points -S, tliis lignre degenerales into the (ive curves <f and 

 a line u,, which contains the triplets of points Q' , corresponding to 

 the points Q of t'' ; consequently u is a singular triseeant of [9'']. 

 On the other hand a bisecant u lying in (p is transformed inio a 

 figure of order 8, to which u itself belongs twice; as the coni|)leting 

 figure must be counted three times and must contain the points Sk, 

 it is the conic t\ Consequently (p bears onlj^ one line u, and the 

 singular trisecants of [(>''] form a congruence (1,1). 



The surface of trisecants of q'^ cuts r/ in a curve t*^ with 5 nodes 

 in Sk- With u, r^ has five points 7\. in common ; each of these 

 points determines a quadruple {Q*), of which one point li^s on t', 

 while the remaining two are situated on u. By means of the trans- 

 formation [Q,Q') T* is therefore changed into a curve of order 10, 

 T^". The latter is apparently the intersection of ff with the surface 

 formed by the twisted cubics t^ which with the trisecants t are 

 associated into degenerate curves of [r>^J. 



With öl^ T^ has, apart from the singular points S, three points in 

 common; for in S^ lie 4 intersections and in each of the remaining 

 S, two; therefore S^ is a triple point on the curve t^". 



The curves t^ form therefore a surface of order ten with three- 

 fold curve q\ 



Of the points of intersection of t^* with y", 5 X ^ X 2 — 20 lie 

 in the points S; in each of the remaining 10, a triseeant t is cut 

 by the corresponding cubic curve t'. From this it ensues that the 

 locus of the points {t, t^) is a twisted curve of order ten. 



7. The pairs of points Q,Q, which are collinear with a point P, 

 lie (§ 4) on a curve :7r^ which passes through the points Sk. If Q 

 describes the line /, QQ will envelop a curve of class 5. The points 

 Q describe then (^ 5) a curve A% which passes three times through 

 the points S, consequently has still 25 points in common with rr^ ; 

 5 of them connect a point Q of /' with a point Q of /; the rest 

 form 10 pairs Q',Q"; so that Q' Q" passes through P. From this it 

 ensues that the triplets of Ihe involution [Q'Y lying on r form 

 triangles which are circumscribed to a curve {curve of involution) 

 of class ten, (7)10. 



For a point *S'/^. rr* degenerates into the curve oj^ and two singular 

 lines sjc and .%>* (§ 4); such a line bears an involution /' of pairs 

 QM'- A pair is formed by Sk and the intersection of sk with u; 



