  In a calculation that directly parallels the derivation of the Thomas
precession, 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 are
parallel, causing the solution for the dipole antenna to be the same as for the
Lienard-Wiechert solution, and those solutions are in turn always solutions to
the Maxwell equations. The solution for the current loop antenna is not a
solution to the Maxwell equations. Field equations are obtained by
restructuring the Proca equations that are commensurate with the low order
retardation solutions. The solutions are not in the Lorentz gauge and they are
not solutions to the unmodified Proca equations. The high order terms are not
solutions to the equations. In representing angular relationships, an argument
is developed that derivatives beyond the first will be required for more
complete solutions. The calculations are not in tensor form, but the tensors
represent angular relationships, and the inference is based on the tensor
irreducibility theorem. In being linear equations expressing angular
relationships, the theorem implies that exact retardation equations do not
exist unless the contravariant tensor of rank n+1 is reducible.
