Page:BatemanElectrodynamical.djvu/17

These equations may be simplified by using the relations of type

${\frac {\partial (y',z',t')}{\partial (y,z,t)}}=\theta {\frac {\partial x'}{\partial t}}\lambda ^{2},$

which are proved in the same way as (A), the quantity λ² being defined by equation (C).

The new equations are

$\rho w_{x}=\lambda ^{2}\left[\rho 'w'_{x}{\frac {\partial x'}{\partial x}}+\rho 'w'_{y}{\frac {\partial y'}{\partial x}}+\rho 'w'_{z}{\frac {\partial z'}{\partial x}}-\rho '{\frac {\partial t'}{\partial x}}\right]$ $-\rho =\lambda ^{2}\left[\rho 'w'_{x}{\frac {\partial x'}{\partial t}}+\rho 'w'_{y}{\frac {\partial y'}{\partial t}}+\rho 'w'_{z}{\frac {\partial z'}{\partial t}}-\rho '{\frac {\partial t'}{\partial t}}\right],$

and these imply that

$\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt=\lambda ^{2}\left[\rho 'w'_{x}dx'+\rho 'w'_{y}dy'+\rho 'w'_{z}dz'-\rho 'dt'\right]$

The formulae connecting the electromagnetic potentials are obtained by putting

$A_{x}dx+A_{y}dy+A_{z}dz-\Phi dt=\theta \left[A'_{x}dx'+A'_{y}dy'+A'_{z}dz'-\Phi 'dt'\right].$

Since $\theta =+1$, they are

${\begin{array}{rl}A_{x}=&A'_{x}{\frac {\partial x'}{\partial x}}+A'_{y}{\frac {\partial y'}{\partial x}}+A'_{z}{\frac {\partial z'}{\partial x}}-\Phi '{\frac {\partial t'}{\partial x}},\\\\-\Phi =&A'_{x}{\frac {\partial x'}{\partial t}}+A'_{y}{\frac {\partial y'}{\partial t}}+A'_{z}{\frac {\partial z'}{\partial t}}-\Phi '{\frac {\partial t'}{\partial t}}.\end{array}}$

(ii) When the Jacobian is negative and $\partial t'/\partial t>0$, the axes are righthanded in the original system and left-handed in the system specified by the dashed letters. To obtain the correct formulae of transformation we must change the sign of H' in equations (2) and (8) and put $\theta =-1$. There is also a doubt about the sign in the equation

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

the sign depending upon whether positive electricity is transformed into negative electricity or positive electricity. If the negative sign be taken there must be a corresponding alteration in equation (3).