Note on Jackson's formalism of gauge transformation

An outline is given of how Jackson may have obtained the inhomogeneous wave equations for the auxiliary functions $\mathrm{\Psi }$ and $\mathbf{V}$ in his influential 2002 AJP paper on the transformation from the Lorenz gauge to other electromagnetic gauges. It clarifies the roles of these functions in the calculation of the Coulomb-gauge vector potential ${\mathbf{A}}_C$ by showing that while ${\mathbf{A}}_C$ is given directly $\mathrm{\nabla }\times \mathbf{V}$, only the subtraction of $\mathrm{\nabla }\mathrm{\Psi }$ from the Lorenz-gauge vector potential ${\mathbf{A}}_L$ yields ${\mathbf{A}}_C$.