Page:Elektrische und Optische Erscheinungen (Lorentz) 033.jpg

We also want, after we have passed to the new coordinates, use the sign ${\tfrac {\partial }{\partial t}}$ instead of $\left({\tfrac {\partial }{\partial t}}\right)_{2}$ for a differentiation with respect to time at constant relative coordinates, so that

$\left({\frac {\partial }{\partial t}}\right)_{1}={\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}-{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}-{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}$

(18)

The derivative with respect to time, which occurs in the basic equations (I) - (V), are all of the kind indicated by $\left({\tfrac {\partial }{\partial t}}\right)_{1}$. We will maintain this sign as an abbreviation for the longer term (18).

In contrast, a point over a letter shall henceforth — such as $\partial /\partial t$ - indicate a differentiation with respect to time at constant relative coordinates. Thus the terms ${\dot {\mathfrak {d}}}$ and ${\dot {\mathfrak {H}}}$ in (4) and (IV) may not be left unaltered. By ${\mathfrak {d}}$, for example, we understood a vector with components

$\left({\frac {\partial {\mathfrak {d}}_{x}}{\partial t}}\right)_{1}{\text{, etc.,}}$

or

$\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}-{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}-{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}\right){\mathfrak {d}}_{x}{\text{, etc.}}$

We can suitably write this vector

$\left({\frac {\partial {\mathfrak {d}}}{\partial t}}\right)_{1}{,}$

while

${\dot {\mathfrak {d}}}$ or ${\frac {\partial {\mathfrak {d}}}{\partial t}}$

will mean the vector with components

${\frac {\partial {\mathfrak {d}}_{x}}{\partial t}}{\text{, etc.}}$

Based on the system of axes associated with ponderable matter, eventually the fundamental equations become

$Div\ {\mathfrak {d}}=\rho {,}$

(Ia)

${\mathfrak {S}}=\rho ({\mathfrak {p}}+{\mathfrak {v}})+\left({\frac {\partial {\mathfrak {d}}}{\partial t}}\right)_{1}{,}$

(4a)

$Div\ {\mathfrak {H}}=0{,}$

(IIa)