Page:BatemanElectrodynamical.djvu/37

these six integral forms of the third order is invariant for the group of spherical wave transformations.

To obtain the other constitutive relations, we construct an integral form of the third order reciprocal to (X) and write it in the form

${\frac {\rho _{0}}{1-w^{2}}}\left[w_{x}dy\ dz\ dt+w_{y}dz\ dx\ dt+w_{z}dx\ dy\ dt-dx\ dy\ dz\right]$

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where $\rho _{0}$ is a quantity which will be determined presently.

Subtracting this from the invariant

$s_{x}dy\ dz\ dt+s_{y}dz\ dx\ dt+s_{z}dx\ dy\ dt-\rho dx\ dy\ dz$

(7)

we may obtain a set of constitutive relations by identifying the resulting invariant with

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

If we multiply the first coefficient in this invariant by $w_{x}$, the second by $w_{y}$, the third by $w_{z}$, the fourth by 1, and add, the result is zero. The same must hold in the case of the invariant to which it is equated; therefore

$s_{x}w_{x}+s_{y}w_{y}+s_{z}w_{z}-\rho -{\frac {\rho _{0}}{1-w^{2}}}\left[w_{x}^{2}+w_{y}^{2}+w_{z}^{2}-1\right]=0$

or

$\rho =\rho _{0}+(sw).$

Hence we have the constitutive relations

$s-{\frac {\rho -(sw)}{1-w^{2}}}w={\frac {\sigma \lambda }{\sqrt {1-w^{2}}}}[E+[wB]\},$

which agree with those obtained by Minkowski, Einstein and Laub, in the case when λ = 1.

We shall now show that it is possible to construct a set of constitutive relations which are invariant for a much wider class of transformations.

Let the transformation be biuniform within a certain domain of values of (x, y, z, t), and such that the bilinear form

${\begin{array}{ll}A\ dx\ \delta x&+B\ dy\ \delta y+C\ dz\ \delta z+D\ dt\ \delta t+F(dy\ \delta z+dz\ \delta y)\\\\&+G(dz\ \delta x+dz\ \delta x)+H(dx\ \delta y+dy\ \delta x)+U(dx\ \delta t+dx\ \delta t)\\\\&+V(dy\ \delta t+dy\ \delta t)+W(dz\ \delta t+dz\ \delta t),\end{array}}$

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