468 
3. Any straight line of the plane 9, is a singular bisecant for 
the congruence (98°, but at the same time a singular ray for (t,t). 
Indeed, 8,,, contains a pencil of conics #6’, each forming with the 
straight line 6,, a figure belonging to [8°]. The straight line ¢ of 
8,,, is a bisecant of each of these figures, hence it is associated to 
the bisecants £ which they send through the point P/ associated to 
the passage P of t. The plane pencils (¢’) in this way associated to 
the rays ¢ of the field [f,,,|, form evidently a bilinear congruence 
of which 6,, and the straight line p’,,, (homologous with the passage 
Piss Of Boss) are the directrices. But also the rays ¢ of this congru- 
ence are singular, for to a ray with passage P’ are associated the 
rays of a plane pencil with vertex in P. 
There are therefore ten fields of singular rays, each belonging to 
a bilinear congruence of singular rays. 
4. Any ray t, of a is singular; for any of its points may be 
considered as its passage, consequently also any point of ¢, as the 
passage P’ of a ray tf; this ray is a bisecant of the curve # cutting 
i, twice. For this reason the rays ¢ of the scroll (t)*, the locus of 
the bisecants of the §*® resting on the straight line fx, are associated 
to the singular ray 4 
When ¢, passes through A, hence coincides with #,, (t’)* degene- 
rates into the two quadratic cones which project the corresponding 
curve B® out of its intersections with ¢,. 
The scrolls (¢')* form a complex. In order to determine its order 
I consider the surface ® produced by the curves f* having the 
rays { of a plane pencil (7, r) as bisecants. To it belong ten figures, 
each composed of a straight line 6;; and a conic cutting it. The 
intersection of ® with g8,,, consists therefore of the straight lines 
b,., Dis Das and the conie connected with 6,,; consequently ® is a 
surface of the fifth order. 
The locus of the pairs of points defined by the curves p* of ® 
on the rays f, is evidently a curve tf with a double point 7. 
The plane rt has with ®° the curve rt‘ and also a straight line / 
in common; hence ®° is at the same time the locus of the curves 
8° intersecting the line /. 
Now let (J/,u) be an arbitrary plane pencil and u the curve 
analogous to t*, therefore the locus of the pairs of points in which 
the rays m of (M, u) are twice intersected by curves 8°. The curve 
u°, along which the surface ®° is cut by u, has with the curve u' 
the passages of the ten straight lines 6;; in common; but every ray 
m resting on one of these straight lines, cuts u‘ and u* in different 
