Page:CunninghamExtension.djvu/6

82 current density in the two systems are,

these being vector equations.

Using equations (3) and (5) these give

$$\begin{array}{ll} \delta J_{x} & =\frac{\sum e\left(w'_{x}-w_{x}\right)}{\delta a\beta\left(1-\frac{vw_{x}}{c^{2}}\right)}=\frac{\delta j_{x}}{\beta\left(1-\frac{vw_{x}}{c^{2}}\right)}=\beta\left(1+\frac{vW_{x}}{c^{2}}\right)\delta j_{x},\\ \\\delta J_{y} & =\frac{\sum e}{\delta a}\left\{ \left(w'_{y}-w_{y}\right)-\frac{vw_{y}\left(w'_{x}-w_{x}\right)}{c^{2}\left(1-\frac{vw_{x}}{c^{2}}\right)}\right\} \\ \\ & =\delta j_{y}-\frac{\beta v}{c^{2}}W_{y}\delta j_{x},\\ \\\delta J_{z} & =\delta j_{z}-\frac{\beta v}{c^{2}}W_{z}\delta j_{x}.\end{array}$$

Summing for all electrons, we obtain Minkowski's equation,

{{MathForm2|(7)|$$\left.\begin{array}{l} J_{x}=\beta j_{x}\left(1+\frac{vW_{x}}{c^{2}}\right)\\ \\J_{y}=j_{y}-\frac{\beta vW_{y}}{c^{2}}j_{x}\\ \\J_{z}=j_{z}-\frac{\beta vW_{z}}{c^{2}}j_{x}\end{array}\right\} .$$}}

So far no distinction has been made between various types of motion of the electrons; but now they must be differentiated.

Consider first the so-called polarization electrons which move with the body, but have an electric moment. Taking &delta;a, &delta;A to be the volumes occupied by such a polarized element, its contributions to the polarization in the two systems are related as follows:

$$\delta P_{x}=\frac{\sum eX}{\delta A},$$

where X is measured relative to the centre of the element.