Page:BatemanElectrodynamical.djvu/38

is an invariant. When we put $dx=\delta x,\dots$ this implies that a certain quadratic form is an invariant.

Let us suppose that the constitutive relations connecting $B_{x},B_{y},B_{z},E_{x},E_{y},E_{z}$ with $H_{x},H_{y},H_{z},D_{x},D_{y},D_{z}$ are given by the circumstance that the invariant reciprocal to

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

is an invariant multiple of

$B_{x}dy\ dz+B_{y}dz\ dx+B_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt.$

This assumption preserves the analogy with the electron equations where the two fundamental integral invariants of the second order are reciprocals with regard to the quadratic form

$\lambda ^{2}\left[dx^{2}+dy^{2}+dz^{2}-dt^{2}\right].$

In the present case the relations between the two sets of vectors are of the type

$\left.{\begin{array}{rl}\kappa {\sqrt {\Delta }}B_{x}=&-\left(BC-F^{2}\right)H_{x}-(FG-CH)H_{y}-(HF-BG)H_{z}\\&+(HW-VG)D_{x}+(BW-VF)D_{y}+(FW-CV)D_{z}\\\\\kappa {\sqrt {\Delta }}E_{x}=&-\left(HW-VG\right)H_{x}-(GU-AW)H_{y}-(AV-HU)H_{z}\\&+(AD-U^{2})D_{x}+(HD-UV)D_{y}+(GD-UW)D_{z}\\\\{\frac {1}{\kappa }}{\sqrt {\Delta }}D_{x}=&\left(BC-F^{2}\right)E_{x}+(FG-CH)E_{y}+(HF-BG)E_{z}\\&+(HW-VG)B_{x}+(BW-VF)B_{y}+(FW-CV)B_{z}\\\\-{\frac {1}{\kappa }}{\sqrt {\Delta }}H_{x}=&\left(HW-VG\right)E_{x}+(GU-AW)E_{y}+(AV-HU)E_{z}\\&+(AD-U^{2})B_{x}+(HD-UV)B_{y}+(GD-UW)B_{z}\end{array}}\right\}$

(2)

where Δ denotes the determinant

$\left|{\begin{array}{cccc}A&H&G&U\\H&B&F&V\\G&F&C&W\\U&V&W&D\end{array}}\right|$

To obtain the other constitutive relations we start with the assumption that there is an integral invariant of the type^[1]

$\theta \left(w{}_{x}dy\ dz\ dt+w{}_{y}dz\ dx\ dt+w{}_{z}dx\ dy\ dt-dx\ dy\ dz\right)$

1. This assumption is justified by the remark made on p. 249.