( 757 ) 
8. A point whose conjugate points coincide we call a branch pointy 
the locus of these points the branch carve. If we let a point G describe 
the conic y 2 , then the curve of order eight, generated by the points 
conjugate to G, must degenerate into 2 parts, of which one is y 2 itself 
and the other the branch curve. From this ensues that the latter is 
of order six and possesses nodes in the 10 singular points; so it is 
rational as it should be, as it corresponds point for point to a conic. 
Also in an other way we can easily deduce the order of the 
branch curve; if a point describes a line a lt then the conjugate points 
describe a curve a 4 having with y, eight points of intersection, of 
which two coincide with the points of intersection of a , and y 2 , 
whilst the others point to 6 points of intersection of a x with the 
branch curve. 
If G 12 is a point of the double curve y 2 and g the tangent in that 
point to y 2 , then g will intersect the branch curve in 6 points of 
which one G t forms with the double point G l9 a group of conjugate 
points; so in the triple points of the involution y a and the branch 
curve will have to touch each other. 
The■ branch curve is a rational curve of order six, having double 
points in the singular points and touching the double curve in the 
triple points of the involution. 
Observation. A rational curve of order six has 10 double points; 
of which however only 8 can be taken arbitrarily 2 ); from the pre¬ 
ceding follows however that 10 points determining a Cf (10„ 10,) 
can always be double points of a rational curve of order six. 
In an other form C. F. Geiser (see his paper quoted in the fol¬ 
lowing number) makes the same observation. 
9. We shall now apply the preceding to some problems out of 
Threedimensional Geometry. To that end we regard the pencil 
(P) of twisted cubics which can be brought through 5 fixed points 
P x , P 2 , P s , P 4 , and P g . These determine on an arbitrary plane V 
a cubic involution of rank one; the lines Pi Pj cut V in the sin¬ 
gular points Sij, the planes Pk Pi Pm cut V in the singular lines 
Sij of the involution. Through an arbitrary point of V passes only 
one curve out of this pencil, through a singular point S$ however 
pass an infinite number of curves, which have all degenerated into 
the fixed line P, Pj and a variable conic; these conics form a pencil 
with Pi, P, P m and the point of intersection of P, Pj with the 
plane Pk Pi P m as base points. Each double point of the involution in 
V is now a point, in which a twisted curve out of the pencil (B) 
!) Salmon-Fiedler : Hohere ebene Kurven, Zweite Auflage, p. 42. 
