A new mathematical example shows that a conjecture attributed to J. C. Maxwell about the number of equilibria of an electric field is false. The authors construct five positive point charges in three‑dimensional space whose electrostatic potential has at least 24 non‑degenerate critical points. For five charges the Maxwell conjecture would predict at most (5−1)^2 = 16 such points, so this example violates that bound.

What the paper studies are the equilibria of the electric field. Equilibria are points where the field vanishes, equivalently critical points of the electrostatic potential. “Non‑degenerate” here means isolated equilibria whose second‑derivative matrix (the Hessian) is invertible, so they are robust to small changes. The authors start with three unit charges placed at the corners of an equilateral triangle. That triangular configuration has four equilibria: one at the center and three nearer the edges. They then add two very small positive charges on the triangle’s symmetry axis, one slightly above and one slightly below the plane, making a shallow triangular bipyramid. In this configuration the three edge equilibria persist, while the central equilibrium splits into a family of 21 new non‑degenerate equilibria, giving at least 24 in total.

The construction is proved using elementary analytic tools and careful expansions. The authors rescale the coordinates around the triangle center and write a deformed potential Φε that captures the leading behavior as the small axial charges shrink. They compute the limit polynomial Φ0 explicitly and show it has exactly 21 non‑degenerate critical points by writing it in cylindrical coordinates and checking the derivatives and second derivatives. The implicit function theorem then shows these 21 critical points persist for sufficiently small axial charges. The paper gives explicit formulas for the small axial charge strengths and reports computer algebra checks (Mathematica and Maple) used to verify the computations and to produce visualizations.