470 
The congruence [s| has singular points of the second order in A 
and in every point A* of the straight line a. The generatrices of 
each of these cones are associated to each other and at the same 
time they are donble rays of the involution; these cones belong 
apparently also to the complex {/’}*. The generatrices of the cones 
the vertices of which lie on a, are combined to la congruence 
(4,2). 
Each ray ¢, represents two rays s; indeed, if P’ and P’, are the 
points that ¢ has in common with the curve 8? of which ¢, is a 
chord, t« is a singular ray for each of the homologous points Pand 
P,. If a ray s is to lie in @ without passing through A, it must 
contain the points P" and P" where the 8? through P’ intersects 
the plane. If P’ describes the ray m through A, p’ = P" P" revolves 
round a point M; for the groups (P’, P", P'’) form polar triangles 
with regard to a definite conic. The plane pencil (p’) is apparently 
projective with the range of points (?) on m; therefore p’ passes 
twice through the corresponding point P and is then a ray s. Con- 
sequently « is a singular plane of the fourth order for the congru- 
ence [s|. As a point of « carries besides one ray s that does not lie 
in a, the sheaf-degree (order) of s is equal to five. ; 
In order to be able to determine the field-degree (the class) of [s], 
I assume a plane u. Let P be a point of the straight line p= au; 
the curves 28° cutting the rays ¢ of the plane pencil (Pu) twice, 
form the surface ®° considered in $ 4, and therefore define on the 
line p' five points Q’, consequently on p the homologous points Q 
which may be associated to P. Inversely a point Q yields a point 
Q’ and the curve #* through Q’ cuts u in three points, determines 
therefore in u three chords ¢ and consequently three points P. When- 
ever Q coincides with P, there passes through P a singular ray s, 
the corresponding cone (¢')? of which has its vertex in the homologous 
point P'. The field-degree amounts therefore to eight. The singular 
rays s form a congruence (5,8). 
The points B, are singular for [s]. This appears when we consi- 
der the rays s belonging to a plane pencil (Bz, u); let p be the 
intersection of u with «, P a point of p. The curves 9’ intersecting 
B P, are projected out of A, into the conic «@ and on p’ this conic 
defines two points Q’, which may be associated to P’. Inversely 
the 8° through Q’ intersects the plane u in two more points, defines 
accordingly two points P, and through them also two points P’. 
As apparently Q’ coincides four times with P’, the plane pencil 
(Br, u) contains four rays s. Bx is, therefore, a singular point of the 
fourth order for the congruence [s]. 
