912 
is hyperelliptic; this is also the case with 9,, and the joins of the 
corresponding points form a ruled surface FR, of which the plane 
sections are rational; for if we project g,, out of an arbitrary point 
on a plane, there arises a hyperelliptie c,, of genus 4. The joins 
of the g', on a hyperelliptic curve of order m, genus p, envelop a 
rational curve of class m—p—1, hence in our case a curve of 
class 6 *). The points of this envelope correspond one for one with 
the points of a plane section of A; these curves are therefore also 
rational. At the same time it appears from the projection that the 
order of R is si; R cuts W, in o,, and g,,. The double curve 
D of R, is of the 10" order. 
6. The intersection of ®, and &, has double points among 
others in the 30 points of intersection of H with R,. Now @,, has 
8 double points in the pinch points of #; the remaining 22 are 
intersections of g,, and 9,, on HL; in Ic, and &, cut the curve O 
each in 22 points, which are associated points of @. 
The intersection of WW, and R, has also double points in the 13 
points which are represented as common points of c,, c, and ky. 
In these points the two surfaces touch. Finally double points arise 
in the 50 intersections of D with W,. Now c, and &, in #, hence 
Q,, and o,, on W,, intersect in 36 more points. The other 14 must 
be real double points of @,,, therefore also of c,. Through each of 
these latter points pass 2 straight lines of R,; they are the points 
where the two cubics of W, of the same system have a point of 
inflexion. 
On the surface Ws for each system of cubics 14 points can be 
found where the two cubics through these points have points of 
inflexion; or: 
There are 14 points where the principal tangents are the inflexional 
tangents of two cubics of the same system through that point; or: 
In WD there are 14 points such that in the net of the curves 
out of S through one of these points X, the degenerations 
NA, XA,, KA, XA,, belong to the same system. 7) 
1) Bertini, La geometria delle serie lineari sopra una curva piana secundo 
il metodo algebrico. Annali di Mat., (2), XXII, p. 894, p. 1. 
2) See my paper: Versl. Kon. Akad. v. Wet., XXVII, p. 797 and 798. 
