464 
When ¢ belongs to the sheaf [M |, the passages Pand Q form two 
projective fields. As in this case also P’ and Q’ correspond in 
projective fields, we find for the image of the sheaf a congruence (3,1), 
Of the three rays which this congruence sends through an arbitrary 
point, two are associated to each other in the involution (¢, t’), 
while the third is a double ray (§ 1). The ray ¢ which it has in 
an arbitrary plane u, is the image of the ray ¢ which the (1,1) asso- 
ciated to mu, sends through M. 
As the sheaf [M/] contains the plane pencil of which the rays inter- 
sect the straight line c, the scroll (c)* belongs to the image (3,1) of 
the sheaf. 
The sheaf [Jf] contains a plane pencil óf rays ¢ intersecting cg. 
This defines on the intersection m of the plane (Mc,) with « a range 
of points (P’). Any homologous point P’ defines with the point C 
corresponding to Cs one ray ¢,. Any plane pencil (t) with vertex C 
contains therefore one ray corresponding to a ray of the axial 
complex [ez] belonging to [M]. But also the line c belongs to the 
congruence (3,1), it being the image of the transversal through M 
to c, and cz. Consequently the images ¢, of the rays of the plane 
pencil in (Mes) envelop a conic. From this appears that « and B 
belong to the singular planes of the congruence (8,1); in other 
words, « and 8 are osculating planes of the twisted cubics of which 
the axes (intersections of two osculating planes) form the (3,1). 
5. The rays ¢ resting on the straight lines d, and d, and also 
on cg, form a quadratic scroll; their passages / lie therefore on a 
conic J?. The corresponding points 7?’ form on a conic d’* a range 
of points projective to the range of the points C,, hence also to the 
range of the points C. Consequently the ray ¢’ envelops a curve of 
the third class. Through a point N/ of « pass four lines ?¢’, the 
images of rays ¢ of the bilinear congruence with directrices d,, d,, 
namely three rays t, and besides the ray associated to the ray which 
the point MN sends to the (1, 1). 
The bilinear congruence representing the field of rays (ul, has 
two rays in common with the (1,1) mentioned above; the image 
of the latter has therefore two rays in the plane u. Consequently 
a bilinear congruence is represented by a congruence (4, 2). 
The latter has « and 8 as singular planes of the third class. 
The rays sent by the (4,2) through a point M/, are the images 
of the rays which the (1,1) has in common with the image (3, 1) 
of the sheaf {M7}. 
The images of two bilinear congruences have among others the 
