Mathematics. — “Complexes of Plane Cubics with Four Base 
Points” By Dr. K. W. Rurerrs. (Communicated by Prof. 
JAN DE VRIES.) 
(Communicated at the meeting of January 31, 1920). 
1. The image of surface ¥, of the 5 order with a double 
curve £ of the 5 order on a plane MZ, is formed by the above 
mentioned complex S of plane cubics. By the aid of this surface 
P, the following properties are derived *): 
a. The triple point U of W, is represented in 7 by 3 points 
O., O,. Oy; together these define a net out of S. 
b. The double curve £ corresponds in 7 to a curve @ of the 
6 order with double points in the base points A,, A,, A,, A, and 
in the points O,, O0,,0,. The points of © are associated to each 
other two for two; @ is therefore hyperelliptie. On Z lie 8 pinch- 
points, corresponding to 8 points w on @. 
ce. The envelope of the joins of the associated points of @ is a 
conic 4, inseribed in the triangle O,, O,, O, and intersecting @ in 
6 points. 
d. On YW, lie five systems of conics and five systems of plane 
cubies, which form together with the conics complete plane sections 
of WY. The curves of one of these latter systems are represented in 
IH by the straight lines joining two associated of @, hence by the 
tangents of 4. 
e. The bitangent planes (containing a conic and a plane cubic) 
of one system envelop a surface of class 3 and order 4. The 
contact curve of this is of order 7 and passes through the 8 pinch 
points of £. To this curve corresponds in JJ a curve of the 5 
order, c,, the locus of the points of intersection of the tangents to 
“4 with the corresponding conie through the base points. The curve 
c, has double points in A,, A,, A,, A, and passes through the 8 
points w. 
Str conies through A,, A,, A,, A, touch the corresponding tangents 
of Z; the points of contact are images of parabolical points of yr. 
f. The curve @ has 32 tangents in common with 4; 8 of them 
') Caporatut, Sulla superficie del quinto ordine dotata di una curva doppia 
1 1 PI 
del quinto ordine, Annali di Mat. (2), 7 or Memorie di geometria, p. 1. 
