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

{{MathForm2|(2)|$$\left.\begin{array}{rl} \kappa\sqrt{\Delta}B_{x}= & -\left(BC-F^{2}\right)H_{x}-(FG-CH)H_{y}-(HF-BG)H_{z}\\ & +(HW-VG)D_{x}+(BW-VF)D_{y}+(FW-CV)D_{z}\\ \\\kappa\sqrt{\Delta}E_{x}= & -\left(HW-VG\right)H_{x}-(GU-AW)H_{y}-(AV-HU)H_{z}\\ & +(AD-U^{2})D_{x}+(HD-UV)D_{y}+(GD-UW)D_{z}\\ \\\frac{1}{\kappa}\sqrt{\Delta}D_{x}= & \left(BC-F^{2}\right)E_{x}+(FG-CH)E_{y}+(HF-BG)E_{z}\\ & +(HW-VG)B_{x}+(BW-VF)B_{y}+(FW-CV)B_{z}\\ \\-\frac{1}{\kappa}\sqrt{\Delta}H_{x}= & \left(HW-VG\right)E_{x}+(GU-AW)E_{y}+(AV-HU)E_{z}\\ & +(AD-U^{2})B_{x}+(HD-UV)B_{y}+(GD-UW)B_{z}\end{array}\right\} $$}}

where &Delta; denotes the determinant

$\left

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

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