J. L. Synge [Can. J. Math. 1, 257-270 (1949; Zbl 0032.223)] has studied the dynamics of three vortices, using the length of the sides of the triangle formed by the three vortices as prime variables. The critical states at which the lengths of the sides remain fixed were found to be either equilateral triangles or collinear configurations. The equilateral configurations are stable or unstable depending on whether the sum K of the products of strengths was greater or smaller than zero. When \(K=0\), a one-parameter family of solutions of contracting and of expanding similar triangles are found. In this paper it is shown that the former family is unstable and the latter is stable. The authors also study the critical states for the collinear configurations where \(K>0\) or \(K<0\).