The second determinant is deduced from the first by multiplying 
the terms of the first column by mdr! , those of the second by 
omm? and by adding them afterwards to those of the last 
column. At the same time appears from this the well-known identity: 
OSE ENE BA On, SY aD) 
in which P and Q are of no higher order than (n—l) and (m—1) in «. 
Let now /’; be an arbitrary integral function of « and y of degree 
r in & (r2m, r2n); we arrange F,, #, and F, according to the 
descending powers of « and divide /’, by FF, let us call ‘the 
quotient q and the rest /,', then 
ENDE AI MERANER) 
The function /', is in 2 of no higher order than (m + n—1). 
From (4) follows 
EERE PB A QR 0e 8) 
The terms in the second member whose degrees in « are higher 
than vm +- 72 — 1) must disappear. 
If we therefore divide PF, and Qs", by F,F,, after arrangement 
according to the descending powers of a, the quotients will be each 
other's opposites. 
Hence: 
Pe Re RE 
QF F, =—qF.F, + SF, 
By this (6) is reduced to 
es sm DA en ata lay OR 
in which #& and S are of no higher degree than (n—1) and (7—1) in w. 
From this identity Nörner’s theorem is simply and generally to 
‘be deduced. 
§ 3. We suppose now provisionally that all the points of inter- 
section of /’, and /’, are simple ones. Is /, =O the equation of a 
curve passing through all the points of intersection of /, and /,, 
the same is true according to (5) for the curve represented by /,’ = 0. 
We will prove that now in the identity 
Gh SS, GO Shige ea ed 
the functions # and S are divisible by v. 
We take for convenience sake one of the points of intersection 0 
as origin of our system of coordinates, y is then a factor of g. As 
falso passes through 0, off, has a node in 0; while #, and /, 
possess only simple points there with different tangents. This is only 
possible if # and S also pass through 0. 
han | 
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