  It is well known that the Painlev\'e equations can formally degenerate to
autonomous differential equations with elliptic function solutions in suitable
scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou
steepest-descent techniques to a Riemann-Hilbert representation of a family of
solutions. This method leads to an explicit approximation formula in terms of
theta functions and related algebro-geometric ingredients that is difficult to
directly link to the expected limiting differential equation. However, the
approximation arises from an outer parametrix that satisfies relatively simple
conditions. By applying a method that we learned from Alexander Its, it is
possible to use these simple conditions to directly obtain the limiting
differential equation, bypassing the details of the algebro-geometric solution
of the outer parametrix problem. In this paper, we illustrate the use of this
method to relate an approximation of the algebraic solutions of the
Painlev\'e-III (D$_7$) equation valid in the part of the complex plane where
the poles and zeros of the solutions asymptotically reside to a form of the
Weierstrass equation.
