(578) 
can imply the condition that the surface S, shall contain five arbi- 
trarily chosen lines. That this conclusion is to be rejected is clearly 
shown in the following. 
The locus of the conics cutting twice a given line 7 and onee 
each of five given lines cj (4 = 1,2,3,4,5) is a surface S8 with six- 
fold line /.)!. So for n>>8 only four lines of S* crossing the (x—2)- 
fold line 2 can be chosen arbitrarily. 
The mentioned surface S$ contains the common transversals axis 
brim of the quadruples Lezer em. The planes (akin), (brinl) cut SS in 
right lines which ean be denoted by bnp and a. So the 20 pairs of 
lines, lying on SS according to § 1, can be represented by (akin, 
bnp) and (rum, anp). 
§ 3. A twisted curve ZW of order p cutting the sixfold line / 
of SS in p—1 points, has still 2p+6 more points in common with 
the surface. So the conics in planes through / meeting Rr and the 
four lines cj, co, e3,c4 generate a S%+6 with (2p+4)-fold line J. It 
RP cuts ej in P, the plane (P/) contains an infinite number of conics 
of the locus; in that case S?+6 breaks up into this plane and 
an 4 2p+5, 
Now we are able to construct an S» with (n—2)-fold line / passing 
through four lines ¢, crossing this line. For n=2m any twisted curve 
Rm-3, for n=2m-+1 any twisted curve Am? may figure as a 
fifth directing line. 
More generally we can suppose Zp to have successively in common 
with ej, eo, ca, c4 a number of 1, Ya. 73:74 points. Then the conics 
cutting 2 twice and each of the five directors ej. co, ¢3, cy, RP onee 
generate a surface of order 2p +6 — (7, + ya + 73 + 74). 
In order to obtain e. g. a surface S* with double line / and four 
simple lines cz crossing it, we put 2p = = yz — 2; for yx =1 we 
find p=1. So we can assume a fifth line e crossing 7 under the 
condition that it meets each of the four lines ¢,; then the five 
lines ¢ represent together 21 points®) (§ 2). 
S 4. The surface SS also breaks up if e; cuts /. For the plane 
(le;) contains an infinite number of conics each of which represents 
') See e. g. my communication in these Proceedings, Sept. 28, 1901 , p. 183. 
*) It is also possible to choose for ej, ca, ¢3,¢, the four sides of a skew quadrilateral 
in which case ce, can be taken arbitrarily ; then S° breaks up into S* and four planes, 
The five lines ej, ca, ¢s, 4,4 may form also a simple broken line, 
