1168 
points are collinear, the line bearing them is singular, as it contains 
three and therefore o' null-points. 
Though we can determine any (1,2) by a net of cubic curves we 
do not judge it superfluous to point out some null-systems (1, 2) 
which can be obtained otherwise. 
If the points # and £” correspond to each other in an involutory 
quadratic transformation (quadratic involution) they may be considered 
as null-points of the connecting line f. Then any line is cut by the 
conic into which it is transformed in its null-points. Then the figure 
of singularity contains the four points of coincidence and the three 
fundamentai points and consists therefore in the vertices and the 
co-vertices of a complete quadrangle, the sir sides of which are 
singular lines. 
The same figure of singularity is found in the case of the null- 
system, where any line has for null-points its points of contact with 
two conics of a ‘pencil. 
Another null-system (1,2) is determined by a pencil of cubic 
curves admitting three collinear points of inflexion B,, B,, B, with 
common tangents b,,b,,b,. The cubic involution determined by the 
curves of this pencil on any line f has a threefold point on the 
threefold line -b, = B,B,b,; so f is touched by two cubic curves 
only. We generate a (1, 2) by considering their points of contact as 
the null-points of 7. Three of the singular points coincide with the 
vertices of the triangle 6,5,6,, whilst B,, B,, B, are three others; 
the seventh is node of a non degenerating cubic curve. Evidently 
there are four singular lines. - 
By applying the harmonic transformation to a null-system (1, 1) 
with ABC= abe as singular triangle in such a way that the conic 
p? touches a,b,c respectively in A’, Bb’, C” we get a null-system 
(1,2) of which A,B,C, A’, B’,C’ are singular points whilst the 
seventh can be found by a linear construction. Here a,b,c are 
singular lines. 
§ 5. For any null-system (J,4) the curves P*+! form a net with 
the singular points as base points. Here any line / bears an involution 
of order k—1 and the second rank admitting a neutral group 
formed by the & null-points £. But for £>>2 the net is not more a 
general one; for this would cut any line in an involution with 
3h (k—1) neutral pairs. Indeed a general net of curves git! admits 
at most Zh (k+5) base points, whilst the curves (P)*t’ pass through 
(A?+4+1) fixed points and the latter number surpasses the former by 
4 (L—1) (k—2). 
