467 
2. The rays ¢’ associated to the rays s, of the sheaf [B,|, form 
a complex. In order to be able to determine the order of this com- 
plex, I consider a plane pencil of rays ¢’ with vertex 7, of which 
the plane rt has with the plane « the straight line p in common. 
The 6? which has one of these rays t/ as a bisecant, is projected 
out of B, in the conic «°; this conic defines on the straight line p 
homologous with p’, two points P* which may be associated to the 
passage /” of ¢’ and ‘also to the point P homologous with P’. In- 
versely a point P* of p defines by means of the straight line 5, P* 
a cone (B), hence a conic a’, and the homologous conic «°° 
yields on p’ two points /”; the corresponding points P may be 
associated to P*. As P* coincides four times with P, the plane 
pencil (7, rt) contains four rays /’, each associated to a singular ray s,. 
To each of the five sheaves [B] corresponds therefore a complex of 
the fourth order. The complex curve in the plane u has the passage 
p’ =au as a double tangent. For the curve 8° which has p’ as a 
chord, is projected out of B, in a conic a’, and the intersections of 
a’ with the line p define two rays s,, both associated to p. 
To a singular ray s,= 5,P’ a cone (P*) of rays ¢ is associated, 
which among others passes through B, and accordingly has the 
generatrix B,P in common with the cone (B,)’ defined by s,= B, P. 
Any ray s, belongs therefore to the complex {?’}*, corresponding to 
the sheaf [B,]. This complex has in other words the four principal 
points br (k F 1). 
If P’ lies on the intersection p's, Of Bes; With @, hence P on 
the homologous straight line p,,,; to the singular ray B,P the plane 
pencil (7, 8,,,) is associated, in this case a component of the cone 
(P’)?. The planes Bx, (4, l,m F1) are therefore principal planes of 
the complex {4}. 
Also a is a principal plane. For the ray fx in a is a bisecant of 
a B? and this is projected out of B, into an «° cutting the homo- 
logous ray f, in two points P for which the point P’ lies on ?’,. 
That B, is not a principal point appears in this way. The cones 
(B) form a pencil and cut therefore « in a pencil (a’). This is 
projective with the homologous pencil («/?) and the two pencils 
produce a figure of tbe fourth order. As two corresponding conics 
intersect each other on a, this figure consists of the straight line u 
and a cubic that is invariant with regard to the homology [e]. 
Any two points P, P’ of this curve furnish two associated singular 
rays t,t’, while the points of the axis a furnish a plane pencil of 
double rays through B,. The complex-cone of B, consists therefore 
of a plane pencil.and a cubical cone. 
