1246 
if it has a (p+q—l)-fold point in each point of intersection in which 
F, possesses a p- fold and F, a q-fold-point, and if no tangents at 
F, and F, coincide in the points of intersection” 
After the simplest case‘), in which #, and F, have only | 
points of intersection, had been treated by Nörnrer before, he gives 
in the above mentioned article a proof of the general case. Further 
proofs have appeared from the hands of HaArPHeN*) and Voss*). Yet 
the importance of the theorem may justify the attempt made here 
to deduce it once more in a most simple way. 
$ 2. We understand by #, and F,, curves respectively of order 
m and n, which may also be degenerate, however, not in such 
a way that /, and #, possess a common divisor. We suppose that 
zm oeeurs in /,, 2 in #,; further that no intersections of the two 
curves are at infinity and no points of intersection are connected by 
a line parallel to one of the axes of coordinates. By these suppo- 
sitions which are always to be arrived at by a linear transformation 
and a fit supposition of the axes, nothing is done to diminish 
the universality. 
The curves are represented by 
Fey.) ay Sea Eef 2%), Ji ae ee 
F(a, y)=b,a° +b, 4....+5=0... 2 
From these two equations we find by elimination of w the resultant 
DNO rn at a nd oh ee 
in which 
a, a, 4,--0 Cte a ds ode ea 
| 
Oa zere Oa ADE ENE ST 
0 0 0 Gin BAO Gi Ax F, 
9 (y) — = 
bbs 0 OR MERK Pe 
Dh bre OBD eer a 
os ged Se ° Re ass S = | 
000 De | 000 Ee | 
1) The proof for this case occurs in J. BacuaracH’s dissertation: “Ueber Schnitt- 
punktsysteme algebraischer Curven” (Erlangen 1881) and in a paper of the same 
author in the Math. Annalen, 26 (p. 275). 
*) Bulletin de la Société Math. de France, V (p. 160): ‘Sur un theorème 
d’Algebre.” 
5) Math. Annalen, 27 (p.527): ‘Ueber einen Fundamentalsatz aus der Theorie 
der algebraischen Funktionen.” 
