We offer a new method of reduction for a system of point vortices
on a plane and a sphere. This method is similar to the classical node elimination
procedure. However, as applied to the vortex dynamics, it requires substantial
modification. Reduction of four vortices on a sphere is given in more detail.
We also use the Poincaré surface-of-section technique to perform the reduction
a four-vortex system on a sphere.