1248 
Further has /, apart from 0 moreover (m— 1) points of intersection 
with the X-axis, which intersections do not lie on #,, so they do 
on SS; in the same way the points of intersection of /, with the 
X-axis lie to the number of (n—1) on R. 
Now the X-axis has already n points of intersection with FR, and 
m points of intersection with S all situated in the finite while R 
and S are respectively of degree (n—1) and (m—1) in 2. So Rand 
S are both divisible by y. We may, however, prove in the same 
way that Zand S are divisible by all the other factors of #, so 
that we find : 
Fl 2 RR SFY fen ee 
From (5) it further ensues 
PSS Ag BR SS ee 
§ 4. The preceding proof undergoes only a slight change if #, 
and /, show contact in one or more of their points of intersection, or 
possess multiple points there. We suppose in the first place that 4, 
and /*, touch each other in a point 0, which we again take as 
origin of the system of coordinates ; moreover that /’, too bas in 
that point the same tangent / as /’, and #5, 
Let us now again consider the identity 
PFU RE BRK Yn) Ate eeN 
If we suppose that the curves # and S do not both pass through 
0, we might determine by RF, and SF, which have in 0a tangent 
/ in common,- a pencil of which one of the curves A has a node 
in 0; AK would however not be touched by / in that case. 
For in that case A would have one point of intersection more 
there with RF, or SF, than these two possess there between them. 
Now ef, is a curve, however, from the pencil determined by AF, and 
SF, and one of its tangents in O coincides with the common tangent 
of F, and F,. Consequently R and S must pass through 0. 
As in $ 3 it appears further that R and S are divisible by 4 
and by all the other factors of vg. Consequently the identities (8) 
and (9) remain in force. 
In the same way it appears that (9) remains in force if #, and 
F, have in any point contact of higher order, provided that they 
show there contact of the same order with #, as well. 
Let us finally suppose that in a point 0, which we again take as 
origin of the system of coordinates, the curve /, possesses a p-fold, 
#, a g-fold point; we provisionally suppose that /’, and F, have 
