 Motion of two point vortices

Take vortex strength to mean circulation and $J(x,y)=(-y,x)$. Each point vortex moves with the velocity induced by the other, so for $\mathbf{d}={\mathbf{x}}_1-{\mathbf{x}}_2$ the displayed equation holds. Since $\mathbf{d}\cdot J\mathbf{d}=0$, separation is constant; weighting the two velocities also gives $d({\kappa }_1{\mathbf{x}}_1+{\kappa }_2{\mathbf{x}}_2)/dt=0$. Equal strengths at opposite points of a circle of radius $a$ rotate with angular speed $\kappa /(4\pi a^2)$.