1126 
From this it ensues that the curves (p)' and (q)*, indicated by 
the straight lines p and q, have 12 tangents 7 in common, on which 
every time one null point lies on p, the other null point on q. 
Moreover the three null rays of the point pg are common tangents. 
The remaining 21 common tangents can only arise from figures 
c? composed of a conic c° and a straight line s. The number of 
those figures amounts therefore to 21. The 21 straight lines s are 
singular rays of N33; for each point of s is to be considered as 
point of inflection /, consequently as null point of s. 
The curve (p)° is of order 24, is therefore intersected in 18 points 
by its triple tangent.. For each of those points two null rays 7 
coincide; the points that have this property form therefore a curve 
y'*. On this curve lie of course the 24 cusps and the 42 triple 
base points B®) (§ 2). . 
§ 4. The straight lines s are at the same time singular null rays 
for the null system Ng». For on s the net curves determine a cubic 
involution, of which each group confains three base points belonging 
to one and the same pencil. For each of the four coincidences D 
of this /, s is a base tangent ¢; these points, therefore, lie on A’. 
The remaining intersections of s and A* are found in the nodes of 
the figure (c’,s). 
If the base point B describes the straight line p its eight null 
rays envelop a curve (p)"*, of class 10, with bitangent p. At the 
same time the base points B* associated to B describe a curve of 
order 8, 2°, which has the two null points of p and 6 points D 
in common with p. From this it ensues that the curves (p)'° and 
(q)* have eight tangents in common, which each possess one null 
point on p and the second null point on g. Those curves have 
moreover the eight null rays of the point pq in common. The 
remaining common tangents are procured by the 21 singular null 
rays s; they are consequently bitangents of the curve (p)’®. 
From this it ensues that (p)'° is of order 90—22 2 or 46, so 
that p contains 42 points B, for which two of the associated base 
points B* have coincided in a point D. The groups of seven base 
points, which are associated to the double base points J, lie therefore, 
on a curve of order 42; it is the branch curve B** of the involution, 
the groups of which consist of 9 base points of a pencil (c®). This 
result may also be arrived at by the following consideration. The 
curve „° has with A*. six points of the line p in common; the 
remaining 42 intersections of those curves are double base points 
D, for which one of the associated base points lies on p. 
