1167 
(14, 4*). As any singular point of (1,4) remains singular, £* must 
surpass /. 
In order to determine 4* we bear in mind that all the rays f, 
which pass into a definite ray /* by means of the transformation 
considered, must pass through the pole P* of f* with respect to 
yg’. So the null-points of f* lie on the curve (P*)F+! corresponding 
to P* in the null-system (1, 4). 
So a (1,4) passes into a (1, hk + 4) by the harmonic transformation. 
From these facts we can derive that 2 (#1) singular points of 
(1, A41) must lie on gy’. We can confirm this result as follows. Let G 
be the second point of intersection of y* with a ray / admitting a 
null-point #' on p*. Then the curve (G)'t' cuts p° in G and in 
2k +1 points / more. In any of the 2 (4-+1) coincidencies of the 
correspondence (/’, G), the ray f touches p? and f* can be taken 
arbitrarily through /’; then /’ is singular. 
By repeating the transformation (/’, f*) must pass reversely into 
the original null-system (1,4). The null-points of f lie on the curve 
(P)*+2 corresponding to the pole P of f in the null-system (1,4-+1). 
On this curve we also find the points of contact of gy? with the 
tangents passing through ; these points are null-points of f in the 
special null-system (0,2) of the pencils the centres of which lie on 
gy’. So the null-system (1,4-+1) is transformed into the combination 
of (4,4) and a (O, 2) admitting exclusively singular points (the points 
of p*). 
If a is a singular ray of a null-system (1, %), harmonic transfor- 
mation with respect to a pair of points lying on a generates once 
more a. (1,4). For in this case *) the pole P* of a ray f* lies on a, 
which implies that the locus (P*)*+! breaks up into a and a curve 
entting f* in & null-points #'*. 
$ 4. In the case of the null-system (1, 2) the curves (P)* form a 
net with 7 base points. Any net of cubic curves with 7 base points 
determines a null-system (1,2), in which any line f admits as null- 
points two base points of a pencil belonging to the net. For the 
curves of the net generate on f a cubic involution of the second 
rank, the neutral pair of which belongs to oo’ triples, i. e. consists 
of two base points of a pencil. 
The figure of singularity has no special characteristic, as we can 
choose the base points of the net arbitrarily. As soon as three singular 
1) So the null-system (1, 1) of the tangents of a pencil of conics in double con- 
tact passes by transformation with respect to the absolute pair of points into the 
null-system of the normals. 
76 
Proceedings Royal Acad. Amsterdam. Vol. XV, 
