(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,
where by L, we mean L-times the unit matrix, _i.e._ the matrix with
elements
| L_e__{_hk_} |, (_e__{_hh_} = 1, _e__{_hk_} = 0, _h_ ≠ _k_ _h_, _k_
= 1, 2, 3, 4).
Since here SL = LS, we deduce that,
F*_f_*_f_F = (-S - L)(S - L) = -SS + L²,
and find, since _f_*_f_ = Det^{½}_f_, F*F = Det^{½}F, we arrive at the
interesting
conclusion
(79) SS = L² - Det^{½}_f_ Det^{½}F
_i.e._ the product of the matrix S into itself can be expressed as the
multiple of a unit matrix—a matrix in which all the elements except
those in the principal diagonal are zero, the elements in the principal
diagonal are all equal and have the value given on the right-hand side
of (79). Therefore the general relations
(80) S_{_h_1} S_{1_k_} + S_{_h_2} S_{2_k_} + S_{_h_3} S_{3_k_} +
S_{_h_4} S_{4_k_} = 0,
_h_, _k_ being unequal indices in the series 1, 2, 3, 4, and
(81) S_{_h_1} S_{1_h_} + S_{_h_2} S_{2_h_} + S_{_h_3} S_{3_h_} +
S{_h_4} S_{4_h_} = L² -
Det^{½}_f_ Det^{½}F,
for _h_ = 1, 2, 3, 4.
Now if instead of F, and _f_ in the combinations (72) and (73), we
introduce the electrical rest-force Φ, the magnetic rest-force ψ, and
the rest-ray Ω [(55), (56) and (57)], we can pass over to the
expressions,—
(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],
(83) S_{_hk_} = - ½ ε Φ [=Φ] _e__{_hk_} - ½ μ ψ [=ψ] _e__{_hk_}
+ ε (Φ_{_h_} Φ_{_k_} - Φ ([=Φ]) ω_{_h_} Ω_{_k_}
+ μ (ψ_{_h_} ψ_{_k_} - Ψ [=ψ] Ω{_h_} ω_{_k_}) - ω_{_h_} ω_{_k_} - εμ
ω_{_h_} Ω_{_k_}
(_h₁_ _k_ = 1, 2, 3, 4).