# Page:BatemanElectrodynamical.djvu/10

282
[March 11,
Mr. H. Bateman

limited by the condition that the total charge on a system of particles is invariant. This is expressed analytically by the equation

 ${\displaystyle {\begin{array}{l}\rho w_{x}dy\ dz\ dt+\rho w_{y}dz\ dx\ dt+\rho w_{z}dx\ dy\ dt-\rho dx\ dy\ dz\\\qquad =\rho 'w'_{x}dy'dz'dt'+\rho 'w'_{y}dz'dx'dt'+\rho 'w'_{z}dx'dy'dt'-\rho 'dx'dy'dz',\end{array}}}$ (1)

provided the axes form a right-handed system in each case.

If the transformation is such that the integral equations of the theory of electrons are invariant, we must have

 ${\displaystyle {\begin{array}{l}E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\\\qquad =E'_{x}dy'\ dz'+E'_{y}dz'\ dx'+E'_{z}dx'\ dy'-H'_{x}dx'\ dt'-H'_{y}dy'\ dt'-H'_{z}dz'\ dt',\end{array}}}$ ${\displaystyle {\begin{array}{l}H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\\\qquad =\theta \left[H'_{x}dy'\ dz'+H'_{y}dz'\ dx'+H'_{z}dx'\ dy'+E'_{x}dx'\ dt'+E'_{y}dy'\ dt'+E'_{z}dz'\ dt'\right],\end{array}}}$ (2)

where ${\displaystyle \theta }$ is a constant.

These relations give two sets of equations connecting the quantities ${\displaystyle E_{x},\dots ,H_{x},\dots }$ with ${\displaystyle E'_{x},\dots ,H'_{x},\dots }$ viz.,

 ${\displaystyle {\begin{array}{rl}E_{x}=E'_{x}{\frac {\partial (y',z')}{\partial (y,z)}}+E'_{y}{\frac {\partial (z',x')}{\partial (y,z)}}&+E'_{z}{\frac {\partial (x',y')}{\partial (y,z)}}-H'_{x}{\frac {\partial (x',t')}{\partial (y,z)}}\\\\&-H'_{y}{\frac {\partial (y',t')}{\partial (y,z)}}-H'_{z}{\frac {\partial (z',t')}{\partial (y,z)}}\\\\-H_{x}=E'_{x}{\frac {\partial (y',z')}{\partial (x,t)}}+E'_{y}{\frac {\partial (z',x')}{\partial (x,t)}}&+E'_{z}{\frac {\partial (x',y')}{\partial (x,t)}}-H'_{x}{\frac {\partial (x',t')}{\partial (x,t)}}\\\\&-H'_{y}{\frac {\partial (y',t')}{\partial (x,t)}}-H'_{z}{\frac {\partial (z',t')}{\partial (x,t)}}\end{array}}}$ ${\displaystyle {\mathsf {and}}\ {\begin{array}{rr}H_{x}=&\theta \left[H'_{x}{\frac {\partial (y',z')}{\partial (y,z)}}+H'_{y}{\frac {\partial (z',x')}{\partial (y,z)}}+H'_{z}{\frac {\partial (x',y')}{\partial (y,z)}}\right.\\\\&\left.+E'_{x}{\frac {\partial (x',t')}{\partial (y,z)}}+E'_{y}{\frac {\partial (y',t')}{\partial (y,z)}}+E'_{z}{\frac {\partial (z',t')}{\partial (y,z)}}\right],\\\\E_{x}=&\theta \left[H'_{x}{\frac {\partial (y',z')}{\partial (x,t)}}+H'_{y}{\frac {\partial (z',x')}{\partial (x,t)}}+H'_{z}{\frac {\partial (x',y')}{\partial (y,t)}}\right.\\\\&\left.+E'_{x}{\frac {\partial (x',t')}{\partial (x,t)}}+E'_{y}{\frac {\partial (y',t')}{\partial (x,t)}}+E'_{z}{\frac {\partial (z',t')}{\partial (x,t)}}\right].\end{array}}}$ (3)

In order that these equations may be equivalent to one another[1] we must

1. It is assumed here that the equations of transformation are independent of ${\displaystyle E_{x},E_{y,}E_{z},H_{x},H_{y},H_{z}}$