803 
— 4 Ber + (— a, 6, ¢, —a, b,c, + a, 6, c,) r—a, be, = 0 ne =) 
this equation appears to be the sameas the equation for 7 found above. 
Our auxiliary proposition reads therefore: 
When in a net of cubics with five base points the lines joining 
one of them to the other four are parts of degenerations belonging 
to the same pencil, the two other straight lines through that base point 
also parts of degenerations, are the nodal tangents of the curve of 
the net that has a double point in that base point. 
Some consequences are easily derived from this proposition. 
All the curves of the pencil containing the degenerations have 
according to § 2 a point of inflexion in Panda common harmonica! 
polar line /. 
Any straight line through P, hence also m and n, is cut besides 
in P in two more points lying harmonically with respect to A and 
the point of intersection with /. There is therefore a curve of the 
pencil touching m, resp. mn, in the point (/,m) resp. (/,7), and a 
curve having m resp. n as inflexional tangent at P. 
By a complex of cubics S,‘ with four base points A,,..., A, a 
surface ®, of the 5 order with a double curve of the 5% order 
is represented.*) The point P corresponds to a point P’ of ®,, the 
pencil of curves containing the degenerations PA,,..., PA, to the 
intersections of ®, with a pencil of planes of which the axis passes 
through P; this axis cuts ®, in the points B',,..., B', corresponding 
to the points B,,..., 6, in the image. The straight line m corre- 
sponds to a plane cubic c,” (lying in a plane VV) through P. This 
c‚” has a double point in one of the points of intersection of V 
with the double curve o,. Any curve of #, lying in a plane of the 
pencil (P’, B;’) cuts cy in 2 more points on the same straight line 
through /’. As appears from the image it must happen once that 
these two points of intersection coincide in P’, in other words P’ 
is a point of inflexion for c,”. For the same reason P’ is also a 
point of inflexion for the plane cubic c,” represented by the straight 
‘line n. We find therefore: 
The points defining a net out of S,* where the joins of these points 
and the base points of the same system are parts of degenerations 
belonging to the same pencil, are the images of those points of ®, 
where two curves belonging to one of the five systems of plane cubics 
on this surface, have both a point of inflewion; or 
1) Gaporau, „Sulla superficie del quinto ordine dotata d'una curva doppia 
del quinto ordine", Annali di Matematica, Ser. Il, t. VIL, 1875, p. 149. 
52 
Proceedings Royal Acad. Amsterdam. Vol XXIII. 
