Page:BatemanElectrodynamical.djvu/36

is seen from (1), § 6, to be an invariant. If we multiply this by the invariants

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

we obtain the invariants

${\begin{array}{l}{\frac {\lambda }{\sqrt {1-w^{2}}}}\left[\left(B_{x}-w_{y}E_{z}+w_{z}E_{y}\right)dy\ dz\ dt+\left(B_{y}-w_{z}E_{x}+w_{x}E_{z}\right)dz\ dx\ dt\right.\\\\\qquad \left.+\left(B_{x}-w_{x}E_{y}+w_{y}E_{x}\right)dx\ dy\ dt-\left(w_{x}B_{x}+w_{y}B_{y}+w_{z}B_{z}\right)dx\ dy\ dz\right],\end{array}}$

(1)

${\begin{array}{l}{\frac {\lambda }{\sqrt {1-w^{2}}}}\left[\left(D_{x}+w_{y}E_{z}-w_{z}E_{y}\right)dy\ dz\ dt+\left(D_{y}+w_{z}H_{x}-w_{x}H_{z}\right)dz\ dx\ dt\right.\\\\\qquad \left.+\left(D_{x}+w_{x}H_{y}-w_{y}H_{x}\right)dx\ dy\ dt-\left(w_{x}D_{x}+w_{y}D_{y}+w_{z}D_{z}\right)dx\ dy\ dz\right],\end{array}}$

(2)

${\begin{array}{l}{\frac {\lambda }{\sqrt {1-w^{2}}}}\left[\left(H_{x}-w_{y}D_{z}+w_{z}D_{y}\right)dy\ dz\ dt+\left(H_{y}-w_{z}D_{x}+w_{x}D_{z}\right)dz\ dx\ dt\right.\\\\\qquad \left.+\left(H_{x}-w_{x}D_{y}+w_{y}D_{x}\right)dx\ dy\ dt-\left(w_{x}H_{x}+w_{y}H_{y}+w_{z}H_{z}\right)dx\ dy\ dz\right],\end{array}}$

(3)

${\begin{array}{l}{\frac {\lambda }{\sqrt {1-w^{2}}}}\left[\left(E_{x}+w_{y}B_{z}-w_{z}B_{y}\right)dy\ dz\ dt+\left(E_{y}+w_{z}B_{x}-w_{x}B_{z}\right)dz\ dx\ dt\right.\\\\\qquad \left.+\left(E_{x}+w_{x}B_{y}-w_{y}B_{x}\right)dx\ dy\ dt-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)dx\ dy\ dz\right],\end{array}}$

(4)

and the constitutive relations are obtained by making the first of these equal to μ times the third, and the second equal to ε times the fourth, where ε and μ are invariants. These, however, are not the only constitutive relations which remain invariant,^[1] for we may obtain the two integral invariants

${\begin{array}{l}\left({\frac {\partial E_{x}}{\partial t}}-{\frac {\partial B_{z}}{\partial y}}+{\frac {\partial B_{y}}{\partial z}}\right)dy\ dz\ dt+\left({\frac {\partial E_{y}}{\partial t}}-{\frac {\partial B_{x}}{\partial z}}+{\frac {\partial B_{z}}{\partial x}}\right)dz\ dx\ dt\\\\\qquad +\left({\frac {\partial E_{z}}{\partial t}}-{\frac {\partial B_{y}}{\partial x}}+{\frac {\partial B_{x}}{\partial y}}\right)dx\ dy\ dt+\left({\frac {\partial E_{x}}{\partial x}}+{\frac {\partial E_{y}}{\partial y}}+{\frac {\partial E_{z}}{\partial z}}\right)dx\ dy\ dz\end{array}}$

(5)

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

(6)

A set of constitutive relations given by two linear relations between

1. This was pointed out to me by Mr. Hassé,