Page:Elektrische und Optische Erscheinungen (Lorentz) 034.jpg



§ 20. For some purposes, a different form of some equations is more appropriate.

The first of the three (IV) summarized relations is namely

where, by equation (IIa), we can write for the last three members

which is nothing else than the first component of

Accordingly, we obtain instead of (IVa)

Furthermore, the current $$\mathfrak{S}$$ can be entirely eliminated. The first of equations (IIIa) becomes, when we consider (4a) and (Ia),

$$\frac{\partial\mathfrak{H}_{z}}{\partial y}-\frac{\partial\mathfrak{H}_{y}}{\partial z}=4\pi\rho\left(\mathfrak{p}_{x}+\mathfrak{v}_{x}\right)+4\pi\left(\frac{\partial\mathfrak{d}_{x}}{\partial t}-\mathfrak{p}_{x}\frac{\partial\mathfrak{d}_{x}}{\partial x}-\mathfrak{p}_{y}\frac{\partial\mathfrak{d}_{x}}{\partial y}-\right.$$

$$\left.-\mathfrak{p}_{z}\frac{\partial\mathfrak{d}_{x}}{\partial z}\right)=4\pi\rho\mathfrak{v}_{x}+4\pi\left\{ \left(\mathfrak{p}_{x}\frac{\partial\mathfrak{d}_{y}}{\partial y}-\mathfrak{p}_{y}\frac{\partial\mathfrak{d}_{x}}{\partial y}\right)-\left(\mathfrak{p}_{z}\frac{\partial\mathfrak{d}_{x}}{\partial z}-\right.\right.$$

$$\left.\left.-\mathfrak{p}_{x}\frac{\partial\mathfrak{d}_{z}}{\partial z}\right)\right\}+4\pi\frac{\partial\mathfrak{d}_{x}}{\partial t}.$$

By that it follows, if we define a new vector $$\mathfrak{H}'$$ by means of the equation

thus

If we now introduce the sign $$\mathfrak{F}$$ for the electric force-action on stationary ions, we obtain the following set of formulas