1115 
Mathematics. — “On the nodal-curve of an algebraic surface’. 
By Dr. J. Worrr. (Communicated by Prof. Hx. pr Vries). 
(Communicated in the meeting of September 30, 1916). 
1. We consider a surface F of order n with a nodal curve A, 
and without any other singularity. Suppose that we represent # by 
means of a birational transformation on another surface /’*, in such 
a way that A passes into a non-singular curve A* of /’*, A* may 
then be one single curve or consist of two parts. The former occurs 
if the developable surface 2 of the pairs of tangent planes along 
A forms one whole, the latter if 2 consists of different parts. We 
shall occupy ourselves with the first case. The deficiency a* of A* is 
then equal to that of @, for the points of A* correspond one for 
one with the planes of 2. In whatever way F is birationally trans- 
formed into a surface in which A gets a non-singular curve as 
image, that image will always have the the same deficiency 2*. The 
value of 2* has been calculated by CreBscH in case of / being a 
rational surface, in other words, may be birationally represented 
in a plane. He finds a* = d(n—4) +1, in which d is the order 
of A*. This is deduced analytically. By means of a geometrical 
wording the proof is to be simplified. We shall start with this and 
then prove the proposition for an arbitrary surface, consequently 
also if it is not rational. 
2. Let F"(2,2,2,¢,)—=O0 be a rational surface of order n,, which, 
by means of the formulae 
or, 05. 5, 5) 
vr, Sane 5. 5) 
Ox, =——W (EE) 
Ov, Fahey 5. Ei 
‘ 
is represented in a plane /’*, in which §,, §,,§, stand for the homo- 
geneous coordinates of a point, while the /, are homogeneous functions 
of a certain degree ». Let / have no other singularities but a nodal 
curve A of order d, and let its image on /’* be one single curve 
A*, The plane sections C of F are represented as curves C* of 
order v, forming a linear system on /’*. The sections of / with 
the o? planes. passing through a point P represent themselves as 
the ew? curves C* of a net. The Jacobian J* of that net, locus of 
the’ nodus of the ov! curves provided with them contained in the 
net, is the image of the curve of contact / of the cone of contact 
1) Math. Ann. Bd. 1, bl. 270. 
