  The Euler--Plateau problem, proposed by \cite{gm}, concerns a soap film
spanning a flexible loop. The shapes of the film and the loop are determined by
the interactions between the two components. In the present work, the
Euler--Plateau problem is reformulated to yield a boundary-value problem for a
vector field that parameterizes both the spanning surface and the bounding
loop. Using the first and second variations of the relevant free-energy
functional, detailed bifurcation and stability analyses are performed. For
spanning surface with energy density $\sigma$ and a bounding loop with length
$2\pi R$ and bending rigidity $a$, the first bifurcation, during which the
spanning surface remains flat but the bounding loop becomes noncircular, occurs
at $\sigma R^3/a=3$, confirming a result obtained previously via an energy
comparison. Other bifurcation solution branches, including those emanating from
the flat circular solution branch to nonplanar solution branches, are also
shown to be unstable.
