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counted twice, a curve g'* in common possessing four sextuple points 
in S*, Sy. Apart from the multiple points, g° and g'* have moreover 
6 x 18—4xX2%* 6 points in common; from this it ensues that 
each plane is osculated by thirty curves @’. 
Mathematics. — “Some particular bilinear congruences of twisted 
cubics.” By Prof. JAN DE VRIES. 
(Communicated in the meeting of March 27, 1915). 
The bilinear congruences of twisted eubies o° may principally be 
brought to two groups.') The congruences of the first group may 
be produced by two pencils of ruled quadries, the bases of which 
have a straight line in common; the congruences of the second 
group consist of the base-curves of the pencils belonging to a net 
of cubic surfaces, which have in common a fixed point and a 
twisted curve of order six and genus three. Reryn’s congruence 
formed by the 9° passing through five given points /;, belongs to 
both groups; it may be produced by two pencils of quadratic cones; 
the straight lines, connecting each of two points /,, /, with each 
of the remaining four, are base-edges. We shall now consider some 
other particular cases of congruences of the first group, which may 
also be produced by two pencils of quadratic cones. 
1. We consider the curves 9* passing through the fundamental 
points F,, F,, F,, #, and having the lines s, (passing through /’) 
and s, (passing through #’,) as chords. Each y* is the partial inter- 
section of a quadratic cone passing through the lines (s,,/, 4,4, 7,,/,/,); 
(s,, EF, FF, FF); the congruence is consequently bilinear. 
Apparently s, and s, are singular bisecants. Any point S, of s, is 
singular; the y* passing through S, lie on the cone of the second 
pencil passing through .S,. Consequently s,, as well as s,, is a 
singular straight line of order tivo. 
The figures of the congruence consisting of a straight line d and 
a conic d*, may be brought to four groups. 
A. The straight line d,, = FF, may be combined with any d? 
of the system of conies passing through /’, and F, and resting on 
\) Vereroni, Rendiconti del Circolo matematico di Palermo, tomo XVI, 209— 
229. In a short communication in vol. XXXVII, 259, of the Rendiconti del Ist. 
Lombardo, Veneront has added to these two main types a third which by the 
way may be considered as a limit case of the first type. This congruence may be 
produced by a pencil of quadrics and a pencil of quartic surfaces, one surface of 
which is composed of two quadrics of the first pencil. The bases of the pencils 
have a straight line in common, which is nodal line for the second pencil, 
