  We discuss the Kirchhoff gauge in classical electrodynamics. In this gauge
the scalar potential satisfies an elliptical equation and the vector potential
satisfies a wave equation with a nonlocal source. We find the solutions of both
equations and show that, despite of the unphysical character of the scalar
potential, the electric and magnetic fields obtained from the scalar and vector
potentials are given by their well-known retarded expressions. We note that the
Kirchhoff gauge pertains to the class of gauges known as the velocity gauge.
