  The analysis of the large-$N$ limit of $U(N)$ Yang-Mills theory on a surface
proceeds in two stages: the analysis of the Wilson loop functional for a simple
closed curve and the reduction of more general loops to a simple closed curve.
In the case of the 2-sphere, the first stage has been treated rigorously in
recent work of Dahlqvist and Norris, which shows that the large-$N$ limit of
the Wilson loop functional for a simple closed curve in $S^{2}$ exists and that
the associated variance goes to zero.
  We give a rigorous treatment of the second stage of analysis in the case of
the 2-sphere. Dahlqvist and Norris independently performed such an analysis,
using a similar but not identical method. Specifically, we establish the
existence of the limit and the vanishing of the variance for arbitrary loops
with (a finite number of) simple crossings. The proof is based on the
Makeenko-Migdal equation for the Yang-Mills measure on surfaces, as established
rigorously by Driver, Gabriel, Hall, and Kemp, together with an explicit
procedure for reducing a general loop in $S^{2}$ to a simple closed curve. The
methods used here also give a new proof of these results in the plane case, as
a variant of the methods used by L\'{e}vy.
  We also consider loops on an arbitrary surface $\Sigma$. We put forth two
natural conjectures about the behavior of Wilson loop functionals for
topologically trivial simple closed curves in $\Sigma.$ Under the weaker of the
conjectures, we establish the existence of the limit and the vanishing of the
variance for topologically trivial loops with simple crossings that satisfy a
"smallness" assumption. Under the stronger of the conjectures, we establish the
same result without the smallness assumption.
