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5. We finally regard five pairs of lines and determine liow many 

 points give five eomplanar transversals. 



The surface {P)abe l»as forty points in common with the twisted 

 curve Qahcd found above. Sixteen of these lie on the four quadrisecants 

 a, a' ; h, h' ; in each of those points ta, to, tc, ta are eomplanar, but 

 their plane does not contain 4- 



Then to those forty points belong the four points of intersection 

 of the bisecants tac, t'ab of q with that curve; in such a point the 

 plane Utj passes through the bisecant, but not through tc. 



Hence there are ticentij points for which the jive transversals lie 

 in one plane. 



This result can be confirmed as follows by applying the law of 

 the permanency of the number. 



If we substitute for each of the five pairs of lines a pair of inter- 

 secting lines^and if A, B, C, D, E are the five points of intersection, 

 «, ;3, y, d", 8 the five connecting planes, we then lind one of the points P 

 in the point of intersection of the plane ABC with the line 6e ; for 

 the lines PA^ PB, PC are to be considered as transversals ta, tb, t^, 

 the traces of ö and e as transversals td, te. Analogously the point «/? 

 satisfies the question ; td and 4 connect it with Daxi&E; ta, t^, tc are 

 the intersections of «, /?, y with the plane thi'ough «/?/, D and E. 

 In all we evidently lind twenty points P. 



6. In connection with § 2 we have still to notice that we can 

 bring a quartic surface through six arbitrarily chosen lines and four 

 of the thirty quadrisecants which they possess four by four. But 

 such a surface will contain in general not more than these ten lines. 



We can determine quartic surfaces also passing through a bised'- 



iuple of a cubic surface. For, each 0' through the thirty points of 



intersection of the two sextuples must contain the twelve lines, as 



each line contains five points of 0\ Thus through a bisextuple pass 



OD^ surfaces 0\ 



So we can find surfaces with thirteen lines; the thirteenth line 

 must then intersect one of the lines of the bisextuple. 



An 0^ with fourteen lines is found by drawing two lines, each 

 of which rests on three of the twelve given lines and by making 

 the surface to pass still through four points, two of which lie on 

 each of those transversals. 



If the lines of the bisextuple in wellknown notation are indi- 

 cated with aic, bjc and if / is a line in the plane {a^b.^) cutting b^, 

 then an 0* through two arbitrary points of / will contain not only 

 this line, but moreover a fourteenth line eomplanar to /, ^^i and b.^ 



