In this paper we consider a model for the electric potential
Φ
arising from the field due to a coronating electrode. This results in the third-order nonlinear partial differential equation ∇. (∆
Φ
∇
Φ
) = 0, which is of mixed type and we study here in two spatial dimensions. We describe a new way of analysing this equation by constructing an orthogonal coordinate system (
Φ
,
Ψ
) and recasting the equation in terms of
x
,
y
and ∆
Φ
as functions of (
Φ
,
Ψ
). It may be regarded as a synthesis of the method of characteristics for a hyperbolic equation and a hodograph method for Laplace’s equation. This results in an effective algorithm for a numerical solution with finite differences.