(75) X_{y} = m_{x}M_{y} + e_{y}E_{x}, Y_{x} = m_{y}M_{x} + e_{x}E_{y} etc.[1]
X_{t} = e_{y}M_{z} - e_{z}M_{y}, T_{x} = m_{x}E_{y} - m_{y}E_{z}, etc.
T_{t} = 1/2[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} = 1/2[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 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 [function]* and F*, we obtain
(77) F*f* =| -S_{11} - L, -S_{12}, -S_{13}. -S_{14} |
| -S_{21}, -S_{22} - L, -S_{23}, -S_{24} |
| -S_{31} -S_{32}, -S_{33} - L, -S_{34} |
| -S_{41} -S_{42} -S_{43} -S_{44} - L |
and hence, we can put
(78) [function]F = S - L, F*[function]* = -S - L,
where by L, we mean L-times the unit matrix, i.e. the matrix with elements
| Le_{hk} |, (e_{hh} = 1, e_{hk} = 0, h [/=] k h, k = 1, 2, 3, 4).
Since here SL = LS, we deduce that,
F*[function]*[function]F = (-S - L)(S - L) = -SS + L^2,
and find, since [function]*[function] = Det^{1/2}[function], F*F = Det^{1/2}F, we arrive at the interesting conclusion
- ↑ Vide note 18.
