_d²x_ _d²y_ _d²z_ _m_ ----- = _e_X, _m_ ----- = _e_Y, _m_ ----- = _e_Z _dt²_ _dt²_ _dt²_ Where (_x_, _y_, _z_) are the co-ordinates of the electron, and _m_ is its mass. Let the electron possess the velocity _v_ at a certain epoch of time. Let us now investigate the laws according to which the electron will move in the ‘particle of time’ immediately following this epoch. Without influencing the generality of treatment, we can and we will assume that, at the moment we are considering, the electron is at the origin of co-ordinates, and moves with the velocity _v_ along the X-axis of the system. It is clear that at this moment (_t_ = 0) the electron is at rest relative to the system _k_, which moves parallel to the X-axis with the constant velocity _v_. From the suppositions made above, in combination with the principle of relativity, it is clear that regarded from the system _k_, the electron moves according to the equations _d²_ξ _d²_η _d²_ζ _m_ ----- = _e_X′, _m_ ----- = _e_Y′, _m_ ----- = _e_Z′ , _d_τ² _d_τ² _d_τ² in the time immediately following the moment, where the symbols (ξ, η, ζ, τ, X’, Y’, Z’) refer to the system _k_. If we now fix, that for _t_ = _v_ = _y_ = _z_ = 0, τ = ξ = η = ζ = 0, then the equations of transformation given in § 3 (and § 6) hold, and we have: _v_ τ = β(_t_ - ---- _x_), ξ = β(_x_ - _vt_), η = _y_, ζ = _z_, _c²_ _v_ _v_ X′ = X, Y′ = β(Y - --- N), Z′ = β(Z + --- M) _c_ _c_ With the aid of these equations, we can transform the above equations of motion from the system _k_ to the system K, and obtain:— (A) $$ \frac{d^2 x}{dt^2} = \frac{e}{m} \frac{1}{\beta} X $$ , $$ \frac{d^2 y}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Y - \frac{v}{c} N) $$ , $$ \frac{d^2 z}{dt^2} = \frac{e}{m} \frac{1}{\beta} (Z + \frac{v}{c} M) $$ Let us now consider, following the usual method of treatment, the longitudinal and transversal mass of a moving electron. We write the equations (A) in the form _d²x_ _m_β² ----- = _e_X = _e_X′ _dt²_ _d²y_ _v_ _m_β² ----- = _e_β (Y - --- N) = _e_Y′ _dt²_ _c_ _d²z_ _v_ _m_β² ----- = _e_β (Z - --- M) = _e_Z′ _dt²_ _c_