Page:Grundgleichungen (Minkowski).djvu/5

if it is understood that of the symbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those which have not at all the suffix 4, or have it twice denote real quantities.

An individual system of values of x, y, z t, i. e., of $x_{1},\ x_{2},\ x_{3},\ x_{4}$ shall be called a space-time point.

Further let ${\mathfrak {w}}$ denote the velocity vector of matter, $\epsilon$ the dielectric constant, $\mu$, the magnetic permeability, $\sigma$ the conductivity of matter, while $\varrho$ denotes the density of electricity in space, and ${\mathfrak {s}}$ the vector of "Electric Current" which we shall come across in §7 and §8.

PART I. Consideration of the Limiting Case Æther.

§ 2. The Fundamental Equations for Æther.

By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electrodynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limiting case $\epsilon =1,\ \mu =1,\ \sigma =0$, they should constitute the laws for ponderable bodies. In this ideal limiting case $\epsilon =1,\ \mu =1,\ \sigma =0$, we shall have ${\mathfrak {E}}={\mathfrak {e}},{\mathfrak {M}}={\mathfrak {m}}$. At every space time point x, y, z, t we shall have the equations:

${\begin{array}{lcrl}(I)&\qquad &curl\ {\mathfrak {m}}-{\frac {\partial e}{\partial t}}&=\varrho {\mathfrak {w,}}\\\\(II)&&div\ {\mathfrak {e}}&={\mathfrak {\varrho ,}}\\\\(III)&&curl\ {\mathfrak {e}}+{\frac {\partial {\mathfrak {m}}}{\partial t}}&=0,\\\\(IV)&&div\ {\mathfrak {m}}&=0.\end{array}}$

I shall now write $x_{1},\ x_{2},\ x_{3},\ x_{4}$ for x, y, z, it $\left(i={\sqrt {-1}}\right)$ and

$\varrho _{1},\ \varrho _{2},\ \varrho _{3},\ \varrho _{4}$

for

$\varrho {\mathfrak {w}}_{x},\ \varrho {\mathfrak {w}}_{y},\ \varrho {\mathfrak {w}}_{z},\ i\varrho$