If we apply the transformations in §3 to these equations, and if we refer the electromagnetic processes to the co-ordinate system moving with velocity _v_, we obtain, $$ \frac {1}{c} \frac {\partial}{\partial \tau} [X, \beta (Y - \frac {v}{c} N), \beta (Z + \frac {v}{c} M)] = \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} L & \beta(M + \frac {v}{c} Z) & \beta(N - \frac {v}{c} Y) \end{vmatrix} and $$ \frac {1}{c} \frac {\partial}{\partial \tau} [L, \beta(M + \frac {v}{c} Z), \beta(N - \frac {v}{c} Y)] = - \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} X & \beta(Y - \frac {v}{c} N) & \beta(Z + \frac {v}{c} M) \end{vmatrix} $$ ... (2) where $$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$ The principle of Relativity requires that the Maxwell-Hertzian equations for pure vacuum shall hold also for the system k, if they hold for the system K, _i.e._, for the vectors of the electric and magnetic forces acting upon electric and magnetic masses in the moving system k, which are defined by their pondermotive reaction, the same equations hold, ... _i.e._ ... $$ \frac {1}{c} \frac {\partial}{\partial \tau} (X', Y', Z') = \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} L' & M' & N' \end{vmatrix} $$ , $$ \frac {1}{c} \frac {\partial}{\partial \tau} (L', M', N') = - \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} X' & Y' & Z' \end{vmatrix} $$ ... (3) Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these systems are equivalent to the Maxwell-Hertzian equations for the system K. Since both the systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor ψ(_v_), which depends only on _v_, and is independent of (ξ, η, ζ, τ). Hence the relations, _v_ _v_ [X′, Y′, Z′] = ψ (_v_) [X, β(Y - ----- N), β(Z + ------ M)], _c_ _c_ _v_ _v_ [L′, M′, N′] = ψ (_v_) [L, β(M - ----- Z), β(N + ----- Y)], _c_ _c_ Then by reasoning similar to that followed in §(3), it can be shown that ψ(_v_) = 1. _v_ _v_ [X′, Y′, Z′] = [X, β(Y - ----- N), β(Z + ------ M)] _c_ _c_ _v_ _v_ [L′, M′, N′] = [L, β(M - ------ Z), β(N + ----- Y)], _c_ _c_