  Morse theoretical ideas are applied to the study of relative equilibria in
the planar $n$-vortex problem. For the case of positive circulations, we prove
that the Morse index of a critical point of the Hamiltonian restricted to a
level surface of the angular impulse is equal to the number of pairs of real
eigenvalues of the corresponding relative equilibrium periodic solution. The
Morse inequalities are then used to prove the instability of some families of
relative equilibria in the four-vortex problem with two pairs of equal
vorticities. We also show that, for positive circulations, relative equilibria
cannot accumulate on the collision set.

Dynamical Systems;Classical Analysis and ODEs