# Page:Grundgleichungen (Minkowski).djvu/41

 (75) $X_{x}=\frac{1}{2}(\mathfrak{m}_{x}\mathfrak{M}_{x}-\mathfrak{m}_{y}\mathfrak{M}_{y}-\mathfrak{m}_{z}\mathfrak{M}_{z}+\mathfrak{e}_{x}\mathfrak{E}_{x}-\mathfrak{e}_{y}\mathfrak{E}_{y}-\mathfrak{e}_{z}\mathfrak{E}_{z})$, $X_{y}=\mathfrak{m}_{x}\mathfrak{M}_{y}+\mathfrak{e}_{y}\mathfrak{E}_{x},\ Y_{x}=\mathfrak{m}_{y}\mathfrak{M}_{x}+\mathfrak{e}_{x}\mathfrak{E}_{y}$, u.s.f. $X_{t}=\mathfrak{e}_{y}\mathfrak{M}_{z}-\mathfrak{e}_{z}\mathfrak{M}_{y}$, $T_{x}=\mathfrak{m}_{z}\mathfrak{E}_{y}-\mathfrak{m}_{y}\mathfrak{E}_{z}$, u.s.f. $T_{t}=\frac{1}{2}(\mathfrak{m}_{x}\mathfrak{M}_{x}+\mathfrak{m}_{y}\mathfrak{M}_{y}+\mathfrak{m}_{z}\mathfrak{M}_{z}+\mathfrak{e}_{x}\mathfrak{E}_{x}+\mathfrak{e}_{y}\mathfrak{E}_{y}+\mathfrak{e}_{z}\mathfrak{E}_{z})$,
 (76) $L=\frac{1}{2}(\mathfrak{m}_{x}\mathfrak{M}_{x}+\mathfrak{m}_{y}\mathfrak{M}_{y}+\mathfrak{m}_{z}\mathfrak{M}_{z}-\mathfrak{e}_{x}\mathfrak{E}_{x}-\mathfrak{e}_{y}\mathfrak{E}_{y}-\mathfrak{e}_{z}\mathfrak{E}_{z})$,

These quantities are all real. In the theory for bodies at rest, the combinations ($X_{x},\ X_{y},\ X_{z},\ Y_{x},\ Y_{y},\ Y_{z},\ Z_{x},\ Z_{y},\ Z_{z}$ 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) $F^{*}f^{*}=\left|\begin{array}{llll} -S_{11}-L, & -S_{12}, & -S_{13}, & -S_{14}\\ -S_{21}, & -S_{22}-L, & -S_{23}, & -S_{23}\\ -S_{31}, & -S_{32}, & -S_{33}-L, & -S_{34}\\ -S_{41}, & -S_{42}, & -S_{43}, & -S_{44}-L\end{array}\right|$

and hence, we can put

 (78) $fF = S-L,\ F^{*}f^{*} = -S-L$,

where by L, we mean L-times the unit matrix, i.e. the matrix with elements

$\left|Le_{hk}\right|\ \left(\begin{array}{c} e_{hh}=1,\ e_{hk}=0,\ h\gtrless k\\ h,k=1,2,3,4\end{array}\right)$

Since here $SL = LS$, we deduce that,

$F^{*}f^{*}fF = (-S-L)(S-L) = -SS + L^{2}$,

and find, since $f^{*}f = Det^{\frac{1}{2}}f,\ F^{*}F = Det\frac{1}{2}F$, we arrive at the interesting conclusion

 (79) $SS = L^{2} -Det^{\frac{1}{2}}f Det^{\frac{1}{2}}F$,

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
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