( 490 ) 



are two lines O, P cutting- llio coiTespoiKliiig- line /,, i.e. thai two 

 transversals / pass ihrongli O^. So the locus of ,S'i is a cubic curve 

 s ^ having ()^ as node, and so we find for the loci of ^S', and S^ 

 in «2 and a^ in the same way rational curves ^^^ and s^^ with 

 Ó2 and Oj as nodes. 



3. To determine the degree of the scroll of the liiies / we first 

 investigate what this scroll has in common with an arbitrary space 

 through «J. Each point ^2 Iving outside «^ which this space has in 

 common \vith \\\v scroll gi\ es a line / having two points in common 

 with that space, therefore lying entirely in that space. So that space 

 can contain besides .v^-' oidy a certain number of generatrices / of 

 the scroll. As the line of intersection of o.., with the assumed space 

 through ^1 has three |)oiiits in common with .v,'' the number of 

 generatrices to be found is three and the scroll, having a system of 

 lines of order six in common with the assumed space, must be a 

 tortuous surface < y'' of order six. So it is cut by an arbitrary space 

 according to a twisted cur\e of order six : this section in general 

 not degenerating is rational, its })oints corresponding one by one 

 to the lines / and llierefore to the rays of each of the pencils 

 (/j), (/J, (4). So the surface is of gemis zero. 



We call the locus just found — however not yet what was meant 

 in the title — a surface, to show by this that the number of points 

 is twofold intinite ; Ity the [)redicate "tortuous" we express that it 

 is not situated in a threedimensional space. 



■4. By considering the three projective series of points (/IJ, (^J, (^4,) 

 marlved by the three projective pencils of rays (/J, (4), (4) on «j, ^?2, r7, 

 we easily [)rove that the plane « contains three generatrices of ()\ 

 For it happens, we know, three times that three corresponding points 

 .li, .1.,, .1., of the projective series of points {A^),{A^),{Az) lie in a 

 same right line, which then becomes a generatrix / of (>" ; for, the 

 conies enveloped by the lines .4^ A^ and .4^ A^ comiecting each point 

 A^ Avith the corresponding points A.^ and .4, have besides a^ still 

 three common tangents. 



To the rule that the tangents in a point of O^ drawn to 0^ are 

 situated in a plane, the points of intersection of two non-successive 

 generatrices / form an exception. In such a point, through which 

 the surface passes twice, a tangential plane \\ill belong to each of 

 the two lines /; so it can be called a "bii)lanar node". From the 

 above is evident that ^>" possesses six biplanar nodes, the three 

 points 0^, 0^, ()^ and the lliree points of intersection of the genera- 



