_f_ = | 0 _f₁₂_ _f₁₃_ _f₁₄_ | | _f₂₁_ 0 _f₂₃_ _f₂₄_ | | _f₃₁_ _f₃₂_ 0 _f₃₄_ | | _f₄₁_ _f₄₂_ _f₄₃_ 0 | F = | 0 F₁₂ F₁₃ F₁₄ | | F₂₁ 0 F₂₃ F₂₄ | | F₃₁ F₃₂ 0 F₃₄ | | F₄₁ F₄₂ F₄₃ 0 | are of much importance. Let us write, (70) _f_F =| S₁₁ - L S₁₂ S₁₃ S₁₄ | | S₂₁ S₂₂ - L S₂₃ S₂₄ | | S₃₁ S₃₂ S₃₃ - L S₃₄ | | S₄₁ S₄₂ S₄₃ S₄₄ - L | Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0. Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by (72) L = ½(_f₂₃_ F₂₃ + _f₃₁_F₃₁ + _f₁₂_ + F₁₂ + _f₁₄_ F₁₄ + _f₂₄_ F₂₄ + _f₃₄_ F₃₄) Then we shall have (73) S₁₁ = ½(_f₂₃_ F₂₃ + _f₃₄_ F₃₄ + _f₄₂_ F₄₂ - _f₁₂_ F₁₂ - _f₁₃_ F₁₃ _f₁₄_ F₁₄) S₁₂ = _f₁₃_ F₃₂ + _f₁₄_ F₄₂ etc.... In order to express in a real form, we write (74) S = | S₁₁ S₁₂ S₁₃ S₁₄ | | S₂₁ S₂₂ S₂₃ S₂₄ | | S₃₁ S₃₂ S₃₃ S₃₄ | | S₄₁ S₄₂ S₄₃ S₄₄ | = | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} | | X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} | | X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} | | -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_} | Now X_{_x_} = ½[_m__{_x_}M_{_x_} - _m__{_y_}M_{_y_} - _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}] so (75) X_{_y_} = _m__{_x_}M_{_y_} + _e__{_y_}E_{_x_}, Y_{_x_} = _m__{_y_}M_{_x_} + _e__{_x_}E_{_y_} etc. X_{_t_} = _e__{_y_}M_{_z_} - _e__{_z_}M_{_y_}, T_{_x_} = _m__{_x_}E_{_y_} - _m__{_y_}E_{_z_}, etc. T_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} + _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} + _e__{_y_}E_{_y_} + _e__{_z_}E_{_z_}] L_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} + _m__{_z_}M_{_z_} - _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}] These quantities[27] are all real. In the theory for bodies at rest, the combinations (X_{_x_}, X_{_y_}, X_{_z_}, Y_{_z_}, Y_{_y_}, Y_{_z_}, Z_{_x_}, Z_{_y_}, Z_{_z_}) are known as “Maxwell’s Stresses,” T_{_x_}, T_{_y_}, T_{_z_} are known as the Poynting’s Vector, T_{_t_} as the electromagnetic energy-density, and L as the Langrangian function. On the other hand, by multiplying the alternating matrices of _f_* and F*, we obtain (77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ | | -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ | | -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ | | -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L | and hence, we can put (78) _f_F = S - L, F*_f_* = -S - L,