1128 
At the same time S is singular point for the null system N33, 
for any straight line passing through S is a stationary tangent for 
a c*, which has S as point of inflection. 
If a straight line h is made to rotate round 7, its null points B 
describe now a curve (/)’, with septuple point P, which evidently 
passes through S. Through P pass now only 16 base tangents ¢; 
they are also tangents at the curve (P)’, which is determined by 
N33. But P must lie on 18 tangents of (P)' (§ 2); bence PS replaces 
two of those tangents, and is consequently stationary tangent, with 
S as point of inflection of (P)°. This is confirmed by the observa- 
tion that (P)* and (P)’ have in P 21, in the points of contact of 
the 16 straight lines ¢ 32 points in common, so that they must 
intersect in S. In consequence of this the possibility that (P)° should 
have a node in S is excluded. 
The null rays 4 of the points of a straight line p now envelopa 
curve of class 9, which has p as bitangent. Let us consider the 
tangents it sends through S. Three of them are indicated by the 
points that p has in common with d*. The remaining six must be 
component parts s* of eompound figures c°. Of the 21 straight lines 
s, sie pass consequently through S. On each of those six straight 
lines the net determines an involution /, of associated base points 
B, B*; such a singular straight line is consequently simple tangent 
of (p)*, while the remaining singular straight lines are now also 
bitangents. ; 
The curves (p)’ and (q)’ have consequently in common the 7 null 
rays of the point pg, the 8 null rays, which each have one null 
point on p, the other on .g, the 6 singular null rays s* and the 15 
singular null rays which are bitangents for the two curves. 
§ 7. If the net has two base points S, and S,, their connector 
is really component part of a figure (c’,s), consequently singular 
for N33, but not a singular null ray of Nez. Each of the two 
singular null points S,,.S, bears 5 singular null rays s, and the null 
systems Ve. and N33 have moreover 10 singular null rays s. 
Let us now suppose that the net has / base points S. The variable 
base points B of the pencils (c°) determine a null system Ns 7,2. 
Each singular point S bears (7—A) singular null rays s*; for of the 
(10—%) tangents which the curve (p)!®-* sends through S, three 
are again indicated by the intersections of p with the curve c’, 
which has a node in S. The straight lines that connect the points 
two by two, are not singular for Nsg—z2 (they are for N3,s). The 
number of singular null rays s, therefore, amounts to 
