Potentials having two orthogonal families of curves as trajectories

We study, using the tool of Joukovsky’s orthogonal coordinates, the determination of the potentials having two families of orthogonal trajectories. We show for compatible cases the existence and the uniqueness, up to a constant factor, of the solution. We note the importance of the ‘isothermal’ nets of curves. We study as an example the net of geometrically similar conic curves and orthogonal trajectories. This revised version was published online in July 2006 with corrections to the Cover Date.