  We study self-similar solutions of the point-vortex system. The explicit
formula for self-similar solutions has been obtained for the three point-vortex
problem and for a specific example of the four and five point-vortex problems.
We see that the families consisting of these self-similar collapsing solutions
are described by one-parameter families, and their collapse time and
Hamiltonian are also expressed by functions of the same parameter. Then, the
configurations at limit points of the parameter are in relative equilibria. For
the many-vortex problem, we investigate the point-vortex system with the help
of numerical computations. In particular, considering the case that $N - 1$
point vortices have a uniform vortex strength, we show that families of
self-similar collapsing solutions continuously depend on the Hamiltonian and
the self-similar solutions asymptotically approach relative equilibria as the
Hamiltonian gets close to certain values. In addition, we prove the existence
of relative equilibria for the four point-vortex system. We also investigate an
example of seven point vortices with non-uniform vortex strengths and give
numerical results for it.
