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the congruence (1,1) having 6,, and p’,,, as directrices. Hence the 
congruence (8,5) contains ten plane pencils (P, Brun) and ten plane 
pencils in planes through the straight lines bs. 
The plane pencil (P,u) contains eight rays of the complex {4}; 
the corresponding rays ¢, passing through the point P’, the rays 
in « belonging to (8,5) form a system with index eight. a is there- 
fore a singular plane of the eighth order. 
9. Let A’ be the image of a bilinear congruence 4. The image 
of the sheaf [|M] has 15 rays in common with 4, hence [J/] contains 
15 rays of dA’. Analogously a field [|u|] appears to contain 13 rays 
af A’. 
The image of a congruence (1,1) is therefore a congruence (15,13). 
This congruence contains the ten principal rays bz/, for the point 
P’;; has one ray in the (1,1). 
The complex {¢},* associated to Bx, has a scroll of the eighth order 
in common with a (1,1). To its intersection with a corresponds in 
[a] a curve a°, containing the passages of the rays sp in the image 
of the (1,1). The congruence (15,13) has, therefore, the points Bz as 
singular points of the order eight. 
I now consider the plane pencil (P?,«@) and the homologous plane 
pencil (P’, a). The curve 8° which has a ray ¢, of the former as a 
chord, has four bisecants « in the congruence (1,1); their passages 
Q joined to 7?” furnish four rays g, whieh may be associated to the 
ray fe A ray q separates from (1,1) a quadratic seroll and this 
_seroll has ten rays « in common with the complex {w}> belonging 
to the plane pencil (P, 4) (§ 4). To q are therefore associated ten 
rays tz; whenever two associated rays q and f„ coincide, there 
rests on ¢, a chord of a 8° that meets f, twice. From this follows 
that the plane « is a singular plane of the order fourteen for the 
congruence (15,13). 
To the ray which a (1,1) has in the plane ~,,,, corresponds a 
plane pencil, the plane of which passes through 6,,; to each of the 
two rays of (1,1) resting on 6,, and p’,,,, a plane pencil in the 
plane #8,,, is associated. The congruence (15,13) contains consequently 
twenty plane pencils in the planes Bxim and ten plane pencils in planes 
through the straight lines bs. 
10. The image of an axial complex with directrix d is a complea 
of the ninth order. For d intersects nine rays of the scroll (£)° which 
is the image of a plane pencil. 
Two generatrices of the cone (P/)' associated to a ray sp, cut 
