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In a calculation that directly parallels the derivation of the Thomasprecession, the first time derivative of the retarded potentials is derived.The solutions have to be integrated in time to obtain the potential solution.The Thomas precession vanishes when the acceleration and velocity vectors areparallel, causing the solution for the dipole antenna to be the same as for theLienard-Wiechert solution, and those solutions are in turn always solutions tothe Maxwell equations. The solution for the current loop antenna is not asolution to the Maxwell equations. Field equations are obtained byrestructuring the Proca equations that are commensurate with the low orderretardation solutions. The solutions are not in the Lorentz gauge and they arenot solutions to the unmodified Proca equations. The high order terms are notsolutions to the equations. In representing angular relationships, an argumentis developed that derivatives beyond the first will be required for morecomplete solutions. The calculations are not in tensor form, but the tensorsrepresent angular relationships, and the inference is based on the tensorirreducibility theorem. In being linear equations expressing angularrelationships, the theorem implies that exact retardation equations do notexist unless the contravariant tensor of rank n+1 is reducible.

A low order extension the Lienard-Wiechert retardation equations to include the Thomas precession