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points, because bz, is connected to different conics by the two plane 
pencils (¢) and (m). The other ten points of intersection of the two 
curves lie in pairs collinear with Ms; the plane pencil (m) contains 
therefore five bisecants of curves 83° lying on ®*. In other words, 
the bisecants of the curves a* which have each a bisecant in common 
with a given plane pencil, form a complex of the fifth order. 
Now let p’ be the passage of the plane rt, p the homologous straight 
line. To every point P of p corresponds a point P/ of p’. The curve 
8° having the ray ¢ = TP’ as a chord, defines in « three bisecants 
t,, which cut p in three points P*. The complex {z}* of the bisecants 
of the curves 3° which have the rays ¢’ of the plane pencil (7) 
as chords, sends five straight lines wv through the point P*; to 
this point correspond consequently five rays ¢’ and therefore five 
points P. Whenever a point P* coincides with a bomologous point 
P, P carries a ray ts to which a ray through P’ is connected. 
The singular rays of the field [t,| are, accordingly, represented by the 
rays of a complex of the eighth order. 
The cone ( P’)? associated (§ 1) to the ray s; = Bz P, contains two 
rays t, Hach ray of the sheaf {| 4;| can, therefore, be considered 
_twice as a ray of the complex {¢’}*. Consequently this complex has 
the points B, as double principal points. 
Each straight line ¢’ in a is associated to two rays t,. For, if t’ is a 
chord Q'Q" of a B® cutting « besides in Q’, it appears that ¢’ 
is associated to each of the two rays Q'Q", QQ". The line ¢’, homo- 
logous with QQ'=t, in [ea], cuts ¢’ in a point P’ of which the 
homologous point P lies on Q'Q". Hence a is a double cardinal 
plane of the complex |}. 
5. There are still other sengular rays. The curve g° passing through 
a point /”/ of a, sends a bisecant s through the homologous point 
P. To the ray s are associated all the rays # of the quadratic cone 
which projects 83° out of P/. The rays s form a congruence, the 
corresponding rays ¢’ a complex. 
Any ray ¢’ of the plane pencil (7,1) is a chord of a 8°, and the 
pairs of points of intersection form the curve rf considered before. 
This curve defines on the straight line et four points P’; the plane 
pencil contains therefore four rays of the complex {t’}. The singular 
rays s are accordingly associated to the rays of a complex of the 
fourth order. 
This complex has the points Br as principal points and the plane 
a as a principal plane; any line ¢, is a generatrix of two cones 
(t°), and belongs therefore twice to the complex. 
