The dynamics of three vortices revisited

The dynamics of three vortices was studied by Synge (1949) using the length of the sides of the triangle formed by the vortices as prime variables. The critical states at which the lengths of the sides remain fixed throughout the motion were found to be either equilateral triangles or collinear configurations. The equilateral configurations were either stable or unstable depending on whether the sum of the products of strengths K was greater or less than zero, respectively. In the case of K = 0, a one-parameter family of solutions of contracting and another of expanding similar triangles were found. It is shown here that, for this special case, the family of contracting similar solutions is always unstable while the family of expanding ones is stable. The critical states for collinear configurations in the general case are studied where K is greater than or less than zero. It is shown that there are either six or four critical states depending on the strengths of the vortices. The properties of these states are discussed.