( 635 ) 
powers of 7, (see $ 8), it is evident, that (2) and (3) possess in D, a 
double point, whilst (1) has in D, a single point; farthermore they 
have all three in D, a common tangent with the equation 
w, (ab'—a'b) + w, (bb'— ac) = 0. 
So in each of the base points of the pencil (9) lie three common 
points of intersection of (1), (2) and (3); besides D,, D,...D, the 
curves (1), (2) and (3) have 39 — 8 X 9 — 12 more common points 
So there are really 12 curves, possessing besides D,, D,...D, still 
a tenth double point. 
We can directly indicate those twelve points. Each of those points 
must lie on the curve j,, determined by D,, D,...D, and likewise 
on the curve j’,, which is determined in the same way by D,, D,.. 
D,. These two curves have 81 points of intersection of which, however, 
nine lie in each of the points D,, D,...D, and three in each of 
the points D, and D,. The remaining points are those indicated. 
To a c,-pencil with nine double points belong twelve curves possessing 
still a ah double point. 
§ 11. I wish to draw attention to another property of these points. 
If P (a',, a’,,2',) is an arbitrary point then the polar lines of P with 
respect to the curves out of the ies (8) are represented by 
dw ; du sik dw a te dw „du 0 
AH == N= U u v a == ‘ 
a dx A i da, P dea, de, . dx, ne dz, P 
We shall now put the question whether it is possible to give P 
such a position that the polar line of / with respect to each curve 
out of the pencil is the same. Evidently for that it is necessary that 
the coordinates of P satisfy the equation w= 0, or the equations, 
dw dw dw 
de, _ de, _ de, 
oo —_—— . . . . . ar . zE i 
du du BU eae (Z) 
de, da, de, 
So P must be on w‚ or — as the system of equations (I) is the 
same as the system wnich we came across in $ 10 — P must be 
one of the 12 points found there. Hence: 
If D,,D,..D,, are the double points of a rational: curve of order 
sta, then the polar line of one of these points with respect to the curves 
out of the Cr pencil possessing the other mine as double points, isa 
fixed line. 
