Page:Grundgleichungen (Minkowski).djvu/19

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(2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves, i.e. for isotopic bodies; — they are comprised in the equations

 (V) $\mathfrak{e}=\epsilon\mathfrak{E},\ \mathfrak{M}=\mu\mathfrak{m},\ \mathfrak{s}=\sigma\mathfrak{E}$,

where $\epsilon$ = dielectric constant, $\mu$ = magnetic permeability, $\sigma$ = the conductivity of matter, all given as function of x, y, z, t. $\mathfrak{s}$ is here the conduction current.

By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work,

$x_{1} = x,\ x_{2} = y,\ x_{3} =z,\ x_{4} = it$

and write $s_{1},\ s_{2},\ s_{3},\ s_{4}$ for $\mathfrak{s}_{x},\ \mathfrak{s}_{y},\ \mathfrak{s}_{z},\ i\varrho$,

further $f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$

for $\mathfrak{m}_{x},\ \mathfrak{m}_{y},\ \mathfrak{m}_{z},\ -i\mathfrak{e}_{x},\ -i\mathfrak{e}_{y},\ -i\mathfrak{e}_{z}$,

and $F_{23},\ F_{31},\ F_{12},\ F_{14},\ F_{24},\ F_{34}$

for $\mathfrak{M}_{x},\ \mathfrak{M}_{y},\ \mathfrak{M}_{z},\ -i\mathfrak{E}_{x},\ i\mathfrak{E}_{y},\ i\mathfrak{E}_{z}$;

lastly we shall have the relation $f_{kh} = -f_{hk},\ F_{kh} = -F_{hk}$, (the letter f, F shall denote the field, s the (i.e. current).

Then the fundamental Equations can be written as

 (A) $\begin{array}{ccccccccc} & & \frac{\partial f_{12}}{\partial x_{2}} & + & \frac{\partial f_{13}}{\partial x_{3}} & + & \frac{\partial f_{14}}{\partial x_{4}} & = & s_{1},\\ \\\frac{\partial f_{21}}{\partial x_{1}} & & & + & \frac{\partial f_{23}}{\partial x_{3}} & + & \frac{\partial f_{24}}{\partial x_{4}} & = & s_{2},\\ \\\frac{\partial f_{31}}{\partial x_{1}} & + & \frac{\partial f_{32}}{\partial x_{2}} & & & + & \frac{\partial f_{34}}{\partial x_{4}} & = & s_{3},\\ \\\frac{\partial f_{41}}{\partial x_{1}} & + & \frac{\partial f_{42}}{\partial x_{2}} & + & \frac{\partial f_{43}}{\partial x_{3}} & & & = & s_{4}.\end{array}$