  We examine existence and stability of relative equilibria of the $n$-vortex
problem specialized to the case where $N$ vortices have small and equal
circulation and one vortex has large circulation. As the small circulation
tends to zero, the weak vortices tend to a circle centered on the strong
vortex. A special potential function of this limiting problem can be used to
characterize orbits and stability. Whenever a critical point of this function
is nondegenerate, we prove that the orbit can be continued, and its linear
stability is determined by properties of the potential. For $N\geq 3$ there are
at least three distinct families of critical points associated to the limiting
problem. Assuming nondegeneracy, one of these families continues to a linearly
stable class of relative equilibria with small and large circulation of the
same sign. This class becomes unstable as the small circulation passes through
zero and changes sign. Another family of critical points which is always
nondegenerate continues to a configuration with small vortices arranged in an
$N$-gon about the strong central vortex. This class of relative equilibria is
linearly unstable regardless of the sign of the small circulation when $N\geq
4$. Numerical results suggest that the third family of critical points of the
limiting problem also continues to a linearly unstable class of solutions of
the full problem independent of the sign of the small circulation. Thus there
is evidence that linearly stable relative equilibria exist when the large and
small circulation strengths are of the same sign, but that no such solutions
exist when they have opposite signs. The results of this paper are in contrast
to those of the analogous celestial mechanics problem, for which the $N$-gon is
the only relative equilibrium for $N$ sufficiently large, and is linearly
stable if and only if $N\geq 7$.
