= Motion of two point vortices
{title2=$\dot{\mathbf d}=(\kappa_1+\kappa_2)J\mathbf d/(2\pi|\mathbf d|^2)$}
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 $\kappa/(4\pi a^2)$.
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