given in (5) constitute the causal solution of the set formed by equations (6) and (8). This solution is shown to be unique [@21].

Equations (8) form a set of *second order* equations connecting the potentials $\textbf{A}$ and $\Phi$ with their sources $\textbf{J}$ and $\rho$. A question arises: could there be a set of *first order* field equations equivalent to the set of equations in (8)? Let us investigate this possibility. Using the identity $\nabla^2\textbf{A}\equiv  \nabla (\nabla\cdot \textbf{A}) -\nabla \times (\nabla \times \textbf{A})$ and (6) and (8) we get the equivalent system of equations $$\begin{aligned}
\nabla\cdot\bigg\{\!-\nabla\Phi-\frac{\partial\textbf{A}}{\partial t}\bigg\} = \frac{\rho}{\epsilon_0}, \\
\nabla\times\big\{\nabla\times\textbf{A}\big\} -\frac{1}{c^2}\frac{\partial}{\partial t}\bigg\{\!-\nabla\Phi-\frac{\partial\textbf{A}}{\partial t}\bigg\} =\mu_0 \textbf{J}.\end{aligned}$$ We realise that the quantity $\{-\nabla\Phi-\partial\textbf{A}/\partial t\}$ appears in both (10) and (11), and this does not seem to be a fortuitous coincidence. This quantity together with its partner $\{\nabla\times\textbf{A}\}$ could be physically significant. Let us introduce the fields $\textbf{E}$ and $\textbf{B}$ through the equations $$\begin{aligned}
\textbf{E}=-\nabla\Phi-\frac{\partial\textbf{A}}{\partial t},\quad \textbf{B}= \nabla\times\textbf{A}.\end{aligned}$$ This justifies the name of potentials to the functions **A** and $\Phi$. According to (12) these potentials determine the fields $\textbf{E}$ and $\textbf{B}$. In terms of **E** and **B**, (10) and (11) take the compact form $$\begin{aligned}
\nabla\cdot \textbf{E} = \frac{\rho}{\epsilon_0}, \\
\nabla\times \textbf{B} 