1127 
The curve p** touches A° in each of the 42 triple base points 
B®); for such a point may be considered in two ways as coincidence 
of a double base point 5°) with one of the base points associated 
to it. The temaining intersections of 3** and A° form 84 pairs of 
double: base points. The net contains therefore 84 pencils, each 
possessing two base tangents t, of which the curves c’, therefore, 
touch each other in two base points. : 
§ 5. The curve (4), enveloped by the base tangents tf, is, as well 
as A°, of genus 10; its singular tangents must therefore be equiva- 
lent to 126 bitangents. They are evidently represented by the 21 
singular rays s, which are quadruple tangents of (/)'8. The order of 
('* is consequently 54. 
If a point / is made to describe the singular straight line s, its 
null rays # envelop a curve (s); for the intersections of s with a 
eurve (P)*° send each a null ray through P. But through each point 
of s pass. but two other null rays, as s is null ray to each of its 
points. Consequently s is quadruple tangent of (s)°, and s contains 
four points S, for which two null rays coincide with s. 
Analogously is s quadruple tangent of the curve (s)'°, which is 
enveloped by the groups of six null rays 6 belonging to the points 
B of s (two of them always coincide with s). The four points of 
contact of s are easily indicated: they form the two pairs of base 
points, which are associated in the /, to the nodes of the figure 
(c’, s). For through each of those nodes D passes a base tangent 4, 
for each of the base points lying on s and belonging to D three 
null rays 5 have consequently coincided with s. 
The two base tangents ¢ just mentioned are at the same time 
common tangents of (s)° and (s)'*; the remaining ones are repre- 
sented by s (which replaces 16) and by the remaining 20 singular 
null rays which are bitangents of (s)*°. 
§ 6. We shall now suppose that all the curves of [c*] pass through 
a point S. The net then contains a curve 6*, which has a node in 
S and determines with every other c° of the net a pencil, the curves 
of which touch each other in |S. Any straight line passing through 
S is therefore a base tangent of a pencil, and d® is the locus of the 
groups of seven base points belonging to S. A° too has a node in S. 
The variable base points B now form a null system N72, which 
has a singular point in S. For, any straight line 5 passing through 
S contains two null points: the point S and the third intersection 
of 4 with d°, 
