Proof of the Invariance under the Lorentz Transformation of the Equation of Motion of the Electron

The equation of motion of the electron relative to an inertial system $S^{\prime }$ in which it is momentarily at rest is $e{\mathbf{E}}^{\prime }=m{\mathbf{f}}^{\prime }-n{\dot{\mathbf{f}}}^{\prime }\dots +a_n{\mathbf{f}}^{\prime \mathrm{}\left(n\right)\mathrm{}}\dots$ where E' is the electric intensity, f' the acceleration, etc. To obtain the equation of motion relative to a system $S$ with respect to which the electron has a velocity V, it is necessary to substitute for the components of E', f', etc. their values in terms of the components of E, H, V, f, etc. measured in $S$ as specified by the usual transformations. In this paper it is shown that the equation of motion thus obtained is invariant under the Lorentz transformation. The method of proof consists in writing down an equation, all the terms of which are shown to be four-dimensional-vectors, which reduces to (1) when referred to the system $S^{\prime }$ in which the instantaneous velocity V' of the electron is zero.