911 
Now a conic &, and a plane cubic 4, of W, ina plane V intersect 
in 6 points, 4 of which belong to the double curve H; the other 
two are represented in ZZ by a pair of points Q,, Q, of c,. If now 
e.g. Q, is at the same time a point of ¢,, it means, that on W, the 
corresponding point Q’,, intersection of 4, and &,, is either a point 
of inflexion of 4, in V or a point of inflexion of the second cubic 
k’, (lying in a plane W), passing through Q,. We can distinguish 
the following cases: 
a. @, is a point of inflexion of &, in V. The three points of 
inflexion of &, lie on a straight line through Q’,, which cuts /:, in 
1 more point. Q’, is therefore also a point of e, in other words: 
there are a number of points in ZZ through which the curves, c,, 
c, and & all pass. 
b. Q’, is a point of inflexion of 4’, in W. As V is tangent plane 
at Q’,, the inflexional tangent of 4’, must lie in V and there form 
one of the principal tangents; it is therefore a tangent either of 4, 
or of &,. 
a. If the inflexional tangent is also tangent of 4, at Q’,, it inter- 
sects k, as well as &, in one more point. The conic in W must cut 
V in these two points, but this is impossible, because two conics 
of , do not intersect. 
8. If the tangent at the point of inflexion Q', is also tangent to 
k,, &, has in @, two points in common with /,; that means that 
in JT the tangent at Q, to the conic through Q,, Az is again a 
tangent to A. 
If we draw out of a point O the tangents to the conics of the 
pencil (A,, A,, A,, A), the points of contact lie on a curve of the 
3d order, passing among others through the base points of the 
pencil. This cubic intersects c, in 15—4.2 = 7 points; consequently 
the envelope of the tangents to the conics at their intersections with 
ec, is of class 7 and has 14 tangents in common with A. 
To them belong the 6 tangents in the 6 points where the conics 
and the corresponding tangents of A touch. (See 15). There are 
therefore B points of intersection of c, and c, that are not points of 
k. Accordingly 13 of the 21 points of intersection also belong to 4, 
while £ can have no further intersections with c,. 
From this follows 5&€—8y = 13 (2), which equation in combination 
with (1) gives £—=9 andy =4. The curve g is therefore represented 
by a curve k, of order 9 with quadruple points in Aj. It is itself 
of order 11. 
5. The conics through Az cut £, in a g',, so that k, (of genus 4) 
60 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
