Symmetric Relative Equilibria in the Four-Vortex Problem with Three Equal Vorticities

We examine in detail the relative equilibria of the 4-vortex problem when three vortices have equal strength, that is, $\Gamma_{1} = \Gamma_{2} = \Gamma_{3} = 1$,and$\Gamma_{4}$ is a real parameter. We give the exact number of relative equilibria and bifurcation values. We also study the relative equilibria in the vortex rhombus problem.