1166 
§ 2. The bilinear null-system (1,1) has three singular points A, B, C. 
The line AB admits A and B as null-points and bears therefore 
o' null-points. So the sides a, b, c of triangle A B Care singular lines. 
If / describes any line /, the null-ray / envelops a conic touching 
a, b,c and 7 (the latter in its null-point). 
The conic (P)? degenerates if P lies on a singular line. If we 
assume P on a the null-points of the other lines through P lie on 
the line PA. 
Let f be a line cutting a,b,c in A’, B’, C’ and F its null-point. 
If f rotates around A’ the point # describes a line through A, and 
the cross ratio (A’ B’ C” F) remains constant = d. If f rotates around 
B’, the point F describes a line through B and (A’S’C’F) is once 
more =d. So this cross ratio has the same value for all the rays 
and is characteristic of the null-system. Now, according to a known 
theorem, we have also F ABCf)=d. 
So any null-system (1,1) consists of the pairs (F, f) connected 
with each other with respect to the singular triangle A BC by the 
relation F(A BC f) = const. 
In his “Lehre von den geometrischen Verwandtschaften” (vol IV, 
p. 461) M. R. Srurm proves that this construction furnishes a (1,1) 
but probably it has escaped him that we can get any (1,1) in this way. 
A pencil of conics touching each other in two points A, B deter- 
mine a (1,1) by its tangents. Then the singular points are A, B and 
the point C common to the common tangents in A and B. 
If in any collineation with the coincidencies A, B, C the point £” 
corresponds to F, the line f= FF’ admits / as null-point in a 
bilineair null-system *). 
§ 3. From a given linear null-system (#,f) we derive a new one 
(F, f*), if we replace f by the line f* normal to it in £. In this 
construction f and f* are harmonically related with respect to the 
absolute pair of points. By a harmonic transformation we will 
understand the transformation of a null-system in which f and /* 
are harmonically separated by the tangents from / to a given curve 
p? of class two. 
For any point # of g’, the null-ray f passes into the tangent 
f* of g’ in F; if f touches gy’ in F, we may assume for f* any 
line through F, and F, is-a singular point of the new null-system 
1) From yy, = C, Xz pad x, = 0, Stores we deduce pj = (Cg—C3) Latz, etc. 
Le gE Ly = C2— 03, etc. 
