( 738 ) 
touches the plane V; the third point of intersection of this curve 
with V is a point of the branch curve forming with the point of 
contact a group of mutually conjugate points of the involution. A 
triple point of the involution is a point, in which a twisted curve 
out of ( B) is osculated by F. From this ensues: 
1. All twisted cubics passing through 5 given points and touching 
a given plane V form a surface F 10 of order ten, which touches 
V in a conic and cuts V moreover according to a rational curve 
of order six. 
2. There are 6 twisted cubics passing through five given points 
and having a given plane as osculating plane. 
As a special case of this last theorem we have still: through five 
given points pass six twisted parabolae. 
Through the pencil (B) of twisted cubics with P x , P % , P t , P A and 
P t as base points a plane V is cut according to a cubic involution 
of the first rank. If a is a curve out of this pencil cutting V in 
A iy A t and A t , then « is projected out of A x by a cone cutting V 
according to the lines A x A % and A x A 9 . If however a curve y out 
of {B) touches a plane V in a point G lt and if moreover it cuts 
V in a point G s , then y is projected out of G 12 by a cone cutting 
V according to 6r 12 G, and the tangent in G tt to y; y is projected 
out of G t by a cone touching V according to G 9 G l9 . We have 
seen that G lt must lie on the double curve and G s on the branch 
curve of the involution, whilst G s G lt touches the former; if there¬ 
fore a quadratic cone is to pass through the base points of the pencil 
(5) and to touch V moreover, then its vertex must lie on the branch 
curve and the tangent with V must touch the double curve. 
The number of quadratic cones passing through five given points, 
and touching a given plane is singly infinite-, the tangents envelope 
a conic. The vertices of the cones form a rational curve of order six. 1 ) 
The tangential planes of all these cones whose number is cp' 
envelope a surface of which we wish to determine the class and 
which for the present we will call *P n . If K t is one of these cones 
and G % its vertex, then through a line l drawn in V through G 9 
one more tangential plane to A, will pass; as / has with the branch 
curve 6 points of intersection, it lies still in 6 tangential planes 
of except in V. Furthermore V is a trope of <f>„ (that is a 
tangential plane touching (y,) in the points of a conic) to be counted 
double; the surface <P n is therefore of class eight. 
The tangential planes of these cones envelope a surface of class 
tight 1 ) 
x ) C. F. Ghser: *Uber Systeme von Kegeln zweiten Grades”. 
