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

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