910 
are the tangents at the points w; the others are tangents of 0, con- 
taining two associated points. 
g. The parabolic curve of W, is of the order 20 and corresponds 
in JT to a curve of the 12! order c,, with quadruple points in 
A,, A,, A. A,; ¢,, and c, cut each other 28 times; 12 of the points 
of intersection are in the 6 points of contact of c,, and c,; 16 lie 
in the 8 points w, where c,, has double points. 
2. Any point in M is double point of one curve of S. The 
envelope of the double point tangents of the nodal curves which 
have their double points on a straight line /, is of the 7" class‘); 
it has 14 tangents in common with A, hence on / lie 14 points 
for which the nodal curve with a double point in one of these 
points, has a double point tangent touching A. It is easy to see, 
that to these 14 points also belong the 5 intersections of 7 and c,. 
The locus of the remaining points is therefore a curve c, of order 9 
with nodes in A,, A,, A,, A,; it is the image of the locus of the 
points of inflexion of the plane cubics on W,, represented by the 
tangents of A*). The locus is a curve o,, of the 19 order. 
3. In connection with the last remark it ensues from this, that 
in the 8 points w c, has tangents which are at the same time 
tangents of A. In these 8 points c, and @ touch. The tangents in 
the nodes A,, A,, A,, A, are the tangents from these points to A. 
Each plane cubic of Y, has 3 points of inflexion B’,, B’,, B’,, 
lying on one line 6’. Each 6’ cuts ¥, in 2 more points P’,, P’,, 
so that the ruled surface, formed by these straight lines, intersects 
the surface ¥Y,, besides in the locus of the points of inflexion, in 
a complementary curve o, the locus of the points P’,, P’,. 
In J the images B,, B, B, of the points B’, B’, B’, lie on 
one tangent to A, the images P,, P, of the points P’,, P’, on the 
corresponding conic through A,, A,, A,, A, On any such a conic 
there cannot lie more than two points P. If the locus & of the 
points P is of order & and passes 7 times through each base point Ag, 
we have 2&£—4y = Jor §—2y — 1 (1). . 
4. The curves c, and ec, intersect, besides in the base points, in 
29 points, to which the 8 points w belong. 
1) Jay pe Vries, Null Systems Determined by Linear Systems of Plane 
Algebraic Curves. These Proc. Vol. XXII, No. 3, p. 156. 
2) See my paper: Versl. Kon. Akad. v. Wet. XXVII, p. 791. 
