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

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

(3)

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

${\begin{array}{ll}{\frac {\theta ^{2}}{\Delta }}\left[Aw_{x}^{2}+Bw_{y}^{2}+Cw_{z}^{2}+D+2Fw_{y}w_{z}\right.&+2Gw_{z}w_{x}+2Hw_{x}w_{y}\\&\left.+2Uw_{x}+2Vw_{y}+2Ww_{z}\right]=\Theta .\end{array}}$

(4)

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

${\begin{array}{l}\left(v_{t}D_{x}-v_{y}H_{z}+v_{z}H_{y}\right)dy\ dz\ dt+\left(v_{t}D_{y}-v_{z}H_{x}+v_{x}H_{z}\right)dx\ dy\ dt\\\qquad +\left(v_{t}D_{z}-v_{x}H_{y}+v_{y}H_{z}\right)dx\ dy\ dt+\left(v_{x}D_{x}+v_{y}D_{y}+v_{z}D_{z}\right)dx\ dy\ dz.\end{array}}$

(5)

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

$s_{x}v_{x}+s_{y}v_{y}+s_{z}v_{z}+\rho v_{t}=\Theta {\sqrt {\Delta }}.$

(6)

The constitutive relations are of the type

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

(7)

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