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$ 2. If a plane p rotates around the line / its null-point F de- 
scribes a conic (/)?; for on account of y=1 there is one position 
of p for which F' lies on J. 
The null-points of the planes p passing through any point P lie 
on a quadratic surface (P)?; evidently it contains P and on account 
of y=1 one point more on any line through P. 
Evidently the null-plane of P touches (P)* in P and cuts it 
according to two lines y,g’. Any point # of one of these lines is 
- null-point of a plane g passing through # and also through P. 
Therefore these lines are null-rays of oo! pencils (1,7), i.e. singular. 
So the singular lines of a (1,1,1) form a congruence (2,2). 
All the other lines of (P)’ are characterized by the fact that the 
null-planes of their points concur in P; otherwise: P is the vertex 
of the quadratic cone enveloped by these planes. 
§ 3. Two surfaces (P,)’? and (P,)? have in common the conic 
(L)* corresponding to the line 7=P,P,. As any other common point 
S bears two and therefore co! null-planes, it is singular. The locus 
of this point S is a conic o* meeting (/)? in two points. 
The surfaces (P\? corresponding to the points P of / form a pencil; 
the surface passing through any point /’ is indicated by the point 
of intersection of 7 and the null-plane of /. The null-planes of any 
point S evidently form a pencil, the axis of which may be repre- 
sented by s*. 
As 5? contains two points of (/)’, the line / bears two null-planes 
the null-points of which lie on o°; therefore the locus of the axes 
st is a quadratic scroll or regulus. 
According to the laws of duality there is a quadratic cone >, 
any tangent plane of which is singular, as it contains oo! null-points 
lying on a line s,; these lines generate a second regulus. 
§ 4.. We now consider three surfaces (P)?. As any pencil of 
planes (s*) admits a plane passing through a point P,, the surface 
(P,)* also contains o°. Amongst the points common to (/,)? and 
the conic (l,,)? we find in the first place the points of intersection 
of (/,,) and o?. One of the two remaining points common to (P,)? 
and (/,,)? is the null-point of the plane P,P,P,, the other which 
may be denoted by 7’ lies in three null-planes which do not pass 
through a line, on account of the arbitrary position of the points 
P; so T bears ow’ null-points, i.e. 7’ is principal point. | 
Evidently all the surfaces (P)° form a complex with the singular 
conic 5 and the principal point T as common elements. This com- 
