1249 
no common tangent in 0. /’, is supposed to pass through all the 
points of intersection of /’, and /’, and to possess in Oa (p-+-g—1)- 
fold point; it is to be seen at once that the curve /’,’ determined 
by (5) satisfies the same requirements. 
Let us again consider the identity 
OF = ee eros BE NEN) ie he 
The’ resultant g contains the factor y’7; PF, has no terms of a 
lower degree than (p + q — 1). 
Let us write the equations of #, and #, thus: 
PF, =(y — ae) (y — aya)... (y — apt) + uti + ute +... + un = 0 
fF, =(y — Be) y — 82)... (y — Bot) + Vga + rgpet+.--ttr=0, 
in which a; = Br, 
then it appears from (7) that the terms of the lowest degree in R, 
must at least be of degree , those in S at least of degree p. For 
if R or S were of a lower degree the terms of the lowest degree 
in RI, and SF, could not neutralize eaeh other and in RF, + SF, 
terms of a lower degree than (pg) must consequently occur. So 
R has a q-fold, S a p-fold point in 0. R passes moreover through 
all the points which /’, — to the number of (n—g) — has in 
common with the X-axis apart from 0; the function R contains 
therefore the factor y. At the same time y is a factor of S and so 
we may divide both members of (7) by y. After that we may 
however, follow the same reasoning once more and going on in 
that way, prove that both members of (7) are divisible by 777. 
Consequently all the factors of @ are again divisible on R and S 
and in this case too we find again 
i ith A AN EV UR A eR ed ae ee) 
To wind up with we may suppose that /’, possesses in 0 a p- 
fold point, /, a q-fold, that they have moreover in 0 at one of 
the branches contact of an arbitrary order. Reasoning in the same 
Way as above, we find, that even now the identity (9) remains in 
force, if only #, has a (p+-q—1)-fold point in O and moreover in 
0 contact with /’, and /’, of the same order as they have between 
them. 
Observation. We have supposed that in the points of intersection 
of /, and /, neither curve has a multiple point with coinciding 
tangents. Nörner has already shown how that case may be reduced 
to one of those treated here. 
§ 5. If F, is a curve of degree », we can observe that the 
