Page:BatemanElectrodynamical.djvu/30

Putting $dx=\delta x$ and extracting the square root, we obtain an invariant

${\sqrt {\Theta }}dx\ dy\ dz\ dt$

(18)

Now, let

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

be an invariant of the second order. Multiplying it by (12) and rejecting a factor ${\sqrt {\Theta }}dx\ dy\ dz\ dt$, we obtain an invariant

$D_{x}\delta y\ \delta z+D_{y}\delta z\ \delta x+D_{z}\delta x\ \delta y-H_{x}\delta x\ \delta t-H_{y}\delta y\ \delta t-H_{z}\delta z\ \delta t,$

where

${\begin{array}{c}{\sqrt {\Theta }}D_{x}=\kappa _{11}E_{x}+\kappa _{12}E_{y}+\kappa _{13}E_{z}+\kappa _{14}B_{x}+\kappa _{15}B_{y}+\kappa _{16}B_{z},\\\dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \\-{\sqrt {\Theta }}H_{x}=\kappa _{41}E_{x}+\kappa _{42}E_{y}+\kappa _{43}E_{z}+\kappa _{44}B_{x}+\kappa _{45}B_{y}+\kappa _{46}B_{z}.\end{array}}$

The relation between the two invariants will be a mutual one if the coefficients $\kappa _{11},\ \dots$. are the elements of an orthogonal matrix.

The Invariants of a Spherical Wave Transformation.

Starting from the fundamental invariants

$d\tau ^{2}=\lambda ^{2}\left[dt^{2}-dx^{2}-dy^{2}-dz^{2}\right],$

(1)

$\lambda ^{4}dx\ dy\ dz\ dt,$

(2)

$A_{x}dx+A_{y}dy+A_{z}dz-\Phi dt,$

(3)

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

(4)

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

(5)

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

(6)

we may obtain a number of others by the methods of multiplication and reciprocation. It will be sufficient to enumerate these if we mention the equations from which they are derived,

${\begin{array}{ccc}(1\ {\mathsf {and}}\ 6)&{\frac {1}{\lambda ^{2}}}\left[\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt\right],&(7)\\\\(5\ {\mathsf {and}}\ 4)&\left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt&(8)\\\\(4\ {\mathsf {and}}\ 4)&\left(E_{x}H_{x}+E_{y}H_{y}+E_{z}H_{z}\right)dx\ dy\ dz\ dt&(9)\\\\(3\ {\mathsf {and}}\ 6)&\rho \left[A_{x}w_{x}+A_{y}w_{y}+A_{z}w_{z}-\Phi \right]dx\ dy\ dz\ dt&(10)\\\\(6\ {\mathsf {and}}\ 7)&{\frac {\rho ^{2}}{\lambda ^{2}}}\left(1-w^{2}\right)dx\ dy\ dz\ dt,&(11)\\\\(5\ {\mathsf {and}}\ 7)&{\frac {\rho }{\lambda ^{2}}}\left[\left(E_{x}-w_{z}H_{y}+w_{y}H_{z}\right)dy\ dz\ dt+\left(E_{y}-w_{x}H_{z}+w_{z}H_{x}\right)dz\ dx\ dt\right.&(12)\\&\left.+\left(E_{z}-w_{y}H_{z}+w_{x}H_{y}\right)dx\ dy\ dt-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)dx\ dy\ dz\right],\\\\(12)&{\frac {\rho }{\lambda ^{4}}}\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.&(13)\\&\left.+\left(E_{z}+w_{z}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right],\\\\(13\ {\mathsf {and}}\ 2)&\rho \ dx\ dy\ dz\ dt\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.&(14)\\&\left.+\left(E_{z}+w_{x}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right].\end{array}}$