( 2G0 ) 



Such a surface can be regarded in two mays as surface (P). For 

 the six lines a,a' ; b,h' ; c.c' can be found out of the six tic, i' be; 

 taci'ac-, tab, i' ab , in the same way as the second six out of the first. 

 For, a,a' are the two transversals of tab, i' ab; tac, t'ac, etc. So the 

 surface is at the same time the locus of the points P, which send 

 out three complanar transversals to the pairs ^at» t' ah; tbc, t' bc; tac, t'ac- 



The points of intersection and the connecting planes of the 12 lines 

 form evidently a configuration (24-, 24^}. Each of tiiose planes inter- 

 sects (P) in a conic; so the surface contains 24 conies. 



3. The plane cr of the transversals t,„ tb, tc envelops a surface of 

 class four containing the same twelve lines. For, each plane through 

 c contains a tn and a ti; the line connecting the point fJi, with the 

 poiut in which c' meets the plane is the corresponding transversal 

 t,.. For a plane nr through tab the corresponding point P lies in the 

 point of intersection of tab with 4 lying in rr. 



The following confirms the fact that ct is a surface of class four. 

 If we let a plane r re\olve about the line /, the point of intersection 

 X of the lines ta,tb lying in v describe a twisted cubic which, con- 

 sidered as the intersection of I lie hyperboloids [laa') and {Ibb'), has 

 / as bisecant. 



The transversal tc lying in v describes a hyperboloid passing through 

 the points of intersection of the above mentioned twisted cubic with 

 the bisecant /. In each of the remaining four common points three 

 complanar lines ta, to, tc meet; hence the line / bears four planes rr. 



4. We now regard four pairs of lines a, a' ; b, b' ; c,c' ; d, d' 

 and we determine the locus of the points P for which the four 

 transversals tn, tb, tc, tj lie in 07ie plane. 



The surfaces {P)abc and {P)abd have evidently the six lines a, a' ■ 

 b, // ; tab, t'nb in common. 



For an arbitrary point of a the transversals tb, tc and td are not 

 complanar ; this is the case for the four points of intersection of {a) 

 and {P)bcd- Consequently a, a' ; b, b' are quadrisecants of the tAvisted 

 curve 9'" which {P)abc and {P)abd have still in common. 



Moreover, tab, t'ab are bisecants of ^"' ; for, on each line, hence 

 also on tab, be two points for which the plane tctd passes through 

 that line (see § 1). 



Hence we may conclude that the locus of the points bearing four 

 complanar transversals is a twisted curve of order six having the 

 four given pairs of lines as quadrisecants and their six pairs of 

 transversals as bisecants. 



