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}