It becomes evident that $\textbf{E}$ and $\textbf{B}$ are retarded fields. The system formed by the *coupled* four first-order equations (13)-(16) imply a system formed by two *uncoupled* second-order equations. To find the latter system we apply the d’Alembertian operator $\Box^2$ to (17) and (18), use (7), and integrate the resulting expressions over all space to get the wave equations $$\Box^2 \textbf{E}=\frac{1}{\epsilon_0}\nabla\rho +\mu_0\frac{\partial\textbf{J}}{\partial t},\quad   \Box^2\textbf{B} =-\mu_0\nabla\times\textbf{J}.$$ Our task will be complete if we identify $\epsilon_0$ and $\mu_0$ with the vacuum permittivity and the vacuum permeability. With this identification, the potentials $\Phi$ and **A** are the electromagnetic scalar and vector potentials and the fields $\textbf{E}$ and $\textbf{B}$ are the electric and magnetic fields.

We have obtained two equivalent versions of electromagnetic field equations. The first one is represented by equations (6) and (8) which are expressed in terms of the retarded scalar and vector potentials defined in (5) and the second one is represented by equations (13)-(16) which are expressed in terms of the retarded fields defined by equations (17) and (18). This second version of the equations is identified with the familiar Maxwell’s equations.

Let us emphasize that the fundamental elements of our axiomatic-heuristic approach to find the Maxwell equations were 