whence K_{_h_} = ∂S_{1 _h_}/∂_x₁_ + ∂S_{2 _h_}/∂_x₂_ + ∂S_{3 _h_}/∂_x₃_ + ∂S_{4 _h_}/∂_x₄_, (20) are components of the space-time vector 1st kind K = lor S, and X is a factor, which is to be determined from the relation _w__ẇ_ = - 1. By multiplying (19) by _w__{_h_}, and summing the four, we obtain X = K_ẇ_, and therefore clearly K + (K_ẇ_)_w_ will be a space-time vector of the 1st kind which is normal to _w_. Let us write out the components of this vector as X, Y, Z, ·_i_T Then we arrive at the following equation for the motion of matter, (21) ν _d_/_d_τ (_dx_/_d_τ) = X, ν _d_/_d_τ (_dy_/_d_τ) = Y, ν _d_/_d_τ (_dz_/_d_τ) = Z, ν _d_/_d_τ (_dx_/_d_τ) = T, and we have also (_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² > (_dt_/_d_τ)² = -1, and X _dx_/_d_τ + Y _dy_/_d_τ + Z _dz_/_d_τ = T _dt_/_d_τ. On the basis of this condition, the fourth of equations (21) is to be regarded as a direct consequence of the first three. From (21), we can deduce the law for the motion of a material point, _i.e._, the law for the career of an infinitely thin space-time filament. Let _x_, _y_, _z_, _t_, denote a point on a principal line chosen in any manner within the filament. We shall form the equations (21) for the points of the normal cross section of the filament through _x_, _y_, _z_, _t_, and integrate them, multiplying by the elementary contents of the cross section over the whole space of the normal section. If the integrals of the right side be R_{_x_} R_{_y_} R_{_z_} R_{_t_} and if _m_ be the constant mass of the filament, we obtain (22) _m_ _d_/_d_τ _dx_/_d_τ = R_{_x_}, _m_ _d_/_d_τ _dy_/_d_τ = R_{_y_}, _m_ _d_/_d_τ _dz_/_d_τ = R_{_z_}, _m_ _d_/_d_τ _dt_/_d_τ = R_{_t_} R is now a space-time vector of the 1st kind with the components (R_{_x_} R_{_y_} R_{_z_} R_{_t_}) which is normal to the space-time vector of the 1st kind _w_,—the velocity of the material point with the components _dx_/_d_τ, _dy_/_d_τ, _dz_/_d_τ, _i_ _dt_/_d_τ. We may call this vector R _the moving force of the material point_. If instead of integrating over the normal section, we integrate the equations over that cross section of the filament which is normal to the _t_ axis, and passes through (_x_, _y_, _z_, _t_), then [See (4)] the equations (22) are obtained, but are now multiplied by _d_τ/_dt_; in particular, the last equation comes out in the form, _m_ _d_/_dt_ (_dt_/_d_τ) = _w__{_x_} R_{_x_} _d_τ/_dt_ + _w__{_y_} R_{_y_} _d_τ/_dt_ + _w__{_z_} R_{_z_} _d_τ/_dt_. The right side is to be looked upon _as the amount of work done per unit of time_ at the material point. In this equation, we obtain the energy-law for the motion of the material point and the expression _m_ (_dt_/_d_τ - 1) = _m_ [1/√(1 - _w²_) - 1] = _m_ (½ |_w₁²_ + 3/8 |_w₁⁴_ + )