24 



There 'are of course fwe more analogous groups represented by 

 the plane pencils having their vertices in B'\, B\, R\, B\, B'\. 



§ 3. A degeneration into three straight lines is represented by a 

 ray of (1,1), whicii cuts the singular lines twice. This is among 

 others the case with the bisecant d of a* which rests on ^i and t^ 

 (and differs from a^,a^,a^. Its image consists of the straight line 

 c/i, = *S\/S, and the two transversals t' and t'' that rest on (Z^, «j, ^/, 

 and a^ and that are the images of the points were (/ rests on ö'. 



The image of the ray B\ B'\ consists of the line of intersection 

 of the planes n\ and a\ corresponding to tlje planes a^^a^ B'\ 

 and a^^a,B\ and of the straight lines 6'„ and b'\^. Through com- 

 bination of the points B'k and B"i we find in this way six configu- 

 rations ()' formed by three straight lines. 



The straight line b\z lies on A*; together with *S, it defines a 

 plane; the straight line in this plane through *S, intersecting a^ 

 forms together with b\z and the straight line t resting on it a con- 

 figuration ()^ 



There are apparently five analogous configurations ; the congruence 

 [(>*] contains accordingly in all thirteen of those figures, each con- 

 sisting of three straight lines. 



\ 4. The curves of [(>'] resting on a straight line /, are repre- 

 sented by the straight lines r of the (1,1), which cut a curve A' 

 that has a^, a^, a, as chords and that meets ^jMwice. These straight 

 lines r form a scroll of the sixth order, (7•)^ with threefold direc- 

 trices /j, t, and double generatrices ak- The straight line i\ which 

 is a chord of A', hence a double generatrix of (r)% has for image 

 a curve ^/ that meets / twice and which is therefore a double 

 curve of the image of (I'Y. As therefore an arbitrary straight line is 

 cut twice by only one q\ [^'] is a bilinear congruence. 



The image fx* of a straight line 7)i has four points on a^ in common 

 with {ry, for this curve cuts the double straight line «i in two points. 

 Besides the straight lines ak ii^ and {ry have six more points in 

 common ; hence the image of (?')" is a surface of the sixth order, 

 A^; with three double lines, ajc, and the double curve ^,*. 



If fx* passes through a point of the line t^ (which is threefold on 

 (?■)% m contains only three points of A^ outside the singular lines; 

 here S^ and S^ are therefore threefold points. 



On A" there lie also the six lines b (§ 2) as component parts of 

 the degenerate figures of which the conies q^ rest on /. 



