# Page:Grundgleichungen (Minkowski).djvu/19

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

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

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

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

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

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

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

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

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

Then the fundamental Equations can be written as

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