Page:BatemanElectrodynamical.djvu/39

 From this we may obtain the reciprocal invariant

$v_{x}dx+v_{y}dy+v_{z}dz+v_{t}dt,$

where

{{MathForm2|(3)|$$\left.\begin{array}{l} \sqrt{\Delta}v_{x}=\theta\left(Aw_{x}+Hw_{y}+Gw_{z}+U\right)\\ \sqrt{\Delta}v_{y}=\theta\left(Hw_{x}+Bw_{y}+Fw_{z}+V\right)\\ \sqrt{\Delta}v_{z}=\theta\left(Gw_{x}+Fw_{y}+Cw_{z}+W\right)\\ \sqrt{\Delta}v_{t}=\theta\left(Uw_{x}+Vw_{y}+Ww_{z}+D\right)\end{array}\right\} $$}}

Multiplying these and rejecting the invariant factor $$\sqrt{\Delta}dx\ dy\ dz\ dt$$, we obtain the invariant

Multiplying

$D_{x}dy\ dz+D_{y}dz\ dx+D_{z}dx\ dy+H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt$

by

$v_{x}dx+v_{y}dy+v_{z}dz+v_{t}dt,$

we obtain the invariant

We now assume that

$\begin{array}{l} s_{x}dy\ dz\ dt+s_{y}dz\ dx\ dt+s_{z}dx\ dy\ dz-\rho dx\ dy\ dz\\ \qquad-\theta\left[w_{x}dy\ dz\ dt+w_{y}dz\ dx\ dt+w_{z}dx\ dy\ dt-dx\ dy\ dz\right]\end{array}$

is an invariant multiple of this invariant. This gives the relation

$s_{x}v_{x}+s_{y}v_{y}+s_{z}v_{z}+\rho v_{t}-\theta\left[v_{x}w_{x}+v_{y}w_{y}+v_{z}w_{z}+v_{t}\right]=0,$

or

The constitutive relations are of the type

{{MathForm2|(7)|$$\left.\begin{array}{l} s_{x}-\theta w_{x}=\sigma\left(v_{t}D_{z}-v_{y}H_{z}+v_{z}H_{y}\right)\\ s_{y}-\theta w_{y}=\sigma\left(v_{t}D_{y}-v_{z}H_{x}+v_{x}H_{z}\right)\\ s_{z}-\theta w_{z}=\sigma\left(v_{t}D_{s}-v_{x}H_{y}+v_{y}H_{x}\right)\end{array}\right\},$$}}

and can be expressed in terms of E and B by means of the relations connecting these quantities with D and H.