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 &lambda;² being defined by equation (C).

The new equations are

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).