🚀 go-pugleaf

I have been investigating a plane birational transformation specified
in a paper by Max Noether in {\it Mathematische Annalen.}  Noether claims
several properties for it which relate to resolving the singularities 
of a curve into ordinary multiple points.  It is an extension of the 
standard method of quadratic transformations described by Fulton, for instance.

Noether neither justified his claims nor gave any reference to other works
which describe this transformation.  I hope that someone might be able to 
tell me a reference that might relate to this particular transformation.  
Here's the definition in homogeneous coordinates as he gave it. The notation 
might seem a little awkward.  It transforms a curve $f(x_1: x_2: x_3)=0$, 
and the new coordinate system is in coordinates $\xi_i$.  Subscripts 
denote degree.

$$
\xi_1:\xi_2:\xi_3 = x_1 \cdot S_{s-1} : x_2 \cdot S_{s-1} : S_s,
$$
where
$$
S_{s-1}  =  x_3 U_{s-2}(x_1,x_2) + U_{s-1}(x_1,x_2), 
$$
and
$$
S_s      =  x_3 V_{s-1}(x_1,x_2) + V_s(x_1,x_2),
$$
with $U_h,$ $V_h$ homogeneous polynomials of degree $h$.

Thanks in advance!
\bye
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