leads to the electric and magnetic fields satisfying Maxwell’s equations. In other words, from a conceptual point of view our procedure could (and should!) be formulated as an existence theorem. Let us enunciate this theorem. *Existence Theorem*. Let $\cJ(\textbf{x},t)$ and $\mathscr{G}(\textbf{x},t)$ be vector and scalar functions which are spatially localised and satisfy the continuity equation $$\nabla\cdot \cJ+\frac{\partial\mathscr{G}}{\partial t}=0.$$ If this equation is evaluated at the source point $\textbf{x}'$ at the retarded time $t'=t-R/\mathscr{C}$ with $\mathscr{C}$ being a constant with units of velocity, then *there exist* the retarded scalar and vector functions: ${\mA}(\textbf{x},t)$ and $\mathscr{P}(\textbf{x},t)$ defined by $$\mA=\frac{1}{4\pi}\!\int\!\frac{[\cJ]}{R}d^3x', \quad \mathscr{P}=\frac{1}{4\pi}\!\int\! \frac{[\mathscr{G}]}{R}d^3x',$$ that satisfy the equation $$\nabla\cdot \mA +\frac{\partial \mathscr{P}}{\partial t}=0,$$ where the retardation symbol $[\;\;]$ indicate that the enclosed quantity is to be evaluated at the source point at the retarded time.

*Corollary 1.* The functions $\mathscr{P}$ and ${\mA}$ in (37) satisfy the wave equations $$\Box^2\mathscr{P} =-\mathscr{G}, \quad \Box^2 \mA=-\cJ,$$ where $\Box^2\!\equiv\!\nabla^2\!-\!(1/\mathscr{C}^2)\partial^2/\partial t^2$. *Corollary 2.* There exist retarded fields: ${\cE}(\textbf{x},t)$ and ${\cB}(\textbf{x},t)$ defined by $$\begin{aligned}
{\cE}=-\nabla\mathscr{P}-\frac{1}{\mathscr{C}^2}\frac{\partial \mA}{\partial t},\quad {\cB}= \nabla\times \mA,\end{aligned}$$ that satisfy the field equations $$\begin{aligned}
\nabla\cdot {\cE} =\mathscr{G} , \quad \nabla\times {\cE} +\frac{1}{\mathscr{C}^2}\frac{\partial {\cB}}{\partial t} =0,\\
\nabla\cdot {\cB}=0, \quad \nabla\times {\cB} -\frac{\partial {\cal E}}{\partial t} =\cJ.\end{aligned}$$ The proof of 