  Let $f(x,y)$ be a complex irreducible formal power series without constant
term. One may solve the equation $f(x,y)=0$ by choosing either $x$ or $y$ as
independent variable, getting two finite sets of Newton-Puiseux series. In 1967
and 1968, Abhyankar and Zariski published proofs of an \emph{inversion
theorem}, expressing the \emph{characteristic exponents} of one set of series
in terms of those of the other ones. In fact, a more general theorem, stated by
Halphen in 1876 and proved by Stolz in 1879, relates also the
\emph{coefficients} of the characteristic terms of both sets of series. This
theorem seems to have been completely forgotten. We give two new proofs of it
and we generalize it to a theorem concerning equations with an arbitrary number
of variables.
