( ^95 ) 



part of pairs corresponding to the points of the nodal curve lying 

 on 1,1, whilst T., corresponding to the node of the curve s^ is 

 likewise node of the curve. So in a to the nodal curve h'^ coi'responds 

 a curve h'* {T^^, 1\^), having 2\ and 1\ as nodes and being of genus 

 unity. And to the rational space sections If of 0" correspond in a 

 curves h'^ {T^, 2V) through 1\ with T^ as triple point, which is 

 found immediately when we remember that an arbitrary space has 

 three poinls in common with each of the rational cubic curves of 0" 

 and one point with each generatrix of 0^ As is proper each of 

 those rational curves l:^ {7\, 2\^) has with the representation k'\T^*,T,^) 

 of the nodal curve k^ besides T^ and T^ four pairs of points in 

 common, corresponding to the four points of the nodal curve lying 

 in the selected space, whilst two curves k'* {2\, 2\^) cut each other 

 besides in T^ and 7\ in six points corresponding to the six points 

 of intersection of 0" with the plane of section of the two spaces. 



10. The locus of the bisecants of a tortuous curve of order four 

 is a curved space of order three ha\'ing the indicated curve as nodal 

 curve. For the twofold infinite number of bisecants furnishes a triple 

 intinite number of points and three of these lie on an arbitrary right 

 line /, because the curve projects itself out of / on to a plane not 

 intersecting / as a rational curve of order four and this plane curve 

 possesses three double points. If we apply this to the nodal curve 

 k* of 0\ taking into cousideration that the generatrices of this scroll 

 are all bisecants of k\ we lind : 



"The tortuous scroll 0^ with the nodal curve k^ is the complete 

 section of a quadratic conic space with a curved space of order three, 

 of ^vhich the fii-st passes once, the second twice through k\" 



Whilst the cubic space is the locus of the bisecants of k*, the 

 quadratic conic space with F^^ as vertex is the locus of the planes 

 containing three points of k'^ and passing through F^^. 



The tortuous surface 0^ with a nodal curve k"^ is determined by 

 this curve and the point F^,. As P^^ lies arbitraril}^ with respect to 

 k* each tortuous curve k'^ in Sp^ is nodal curve of a fourfold infinite 

 number of surfaces 0\ 



11. We observe that the case just considered of the correspon- 

 dence of the curves s^^ and s/, where a tortuous scroll with a 

 double curve k'^ is formed, is not the most particular one that one 

 can think of. If for instance — instead of starting from two rational 

 curves 6\^ and s^^ taken arbitrarily in a^ and «^ — we start by making 

 the pointfields «^ and «^ to be in projective correspondence and then 



