Pencils of circles of equal potentials (Q1331277)

In a point \(P\) of the Euclidean plane \(\mathbb{R}^ 2\), the potential of a mass \(m_ 1\) being situated at a point \(P_ 1\) is \(m_ 1/| P - P_ 1|\). The equipotential of two masses \((m_ 1,P_ 1)\), \((m_ 2,P_ 2)\) is the circle of Apollonius \(c_{12} : m_ 2 \cdot | P - P_ 1| = m_ 1 \cdot | P - P_ 2|\). The authors prove several fundamental properties of these ``circles of equal potential'', for instance: For three masses \((m_ i,P_ i)\), \(i = 1,2,3\), the three circles \(c_{12}\), \(c_{23}\), \(c_{13}\) belong to a single pencil of circles; its radical axis contains the circumcentre of \(\Delta P_ 1 P_ 2 P_ 3\) (Prop. 2). Given a pair of conjugate pencils of circles with focal points \(F\) and \(G\), a mass distribution in the plane \(\mathbb{R}^ 2\) with the mass \(m_ i\) at each point \(P_ i\) proportional to \((| P_ i - F| \cdot | P_ i - G|)^{1/2}\) has the property that the circle of equal potential of two masses belongs to one of the pencils if and only if the two masses lie on an arc of a circle in the other pencil; these are the only mass distributions with this property (Theorem 1).