Page:BatemanElectrodynamical.djvu/9

These equations are equivalent to the single integral equation

${\begin{array}{l}\int \left(A_{x}dx+A_{y}dy+A_{z}dz-\Phi dt\right)\\\qquad =\iint \left(H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{z}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\right),\end{array}}$

(IV)

provided a suitable interpretation is given to the integrals.

Take any closed curve C and a surface $\Sigma$ bounded by this curve. Let t be expressed in terms of (x, y, z) by an arbitrary known law; then the line integral may be understood to mean

$\int \left[\left(A_{x}-\Phi {\frac {\partial t}{\partial x}}\right)dx+\left(A_{y}-\Phi {\frac {\partial t}{\partial y}}\right)dy+\left(A_{z}-\Phi {\frac {\partial t}{\partial z}}\right)dz\right],$

and can be transformed into the surface integral by means of Stokes's theorem. Since

${\begin{array}{l}{\frac {d}{dy}}\left(A_{z}-\Phi {\frac {\partial t}{\partial z}}\right)-{\frac {d}{dz}}\left(A_{y}-\Phi {\frac {\partial t}{\partial y}}\right)\\\\\qquad ={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}+\left({\frac {\partial A_{z}}{\partial t}}+{\frac {\partial \Phi }{\partial z}}\right){\frac {\partial t}{\partial y}}-\left({\frac {\partial A_{y}}{\partial t}}+{\frac {\partial \Phi }{\partial y}}\right){\frac {\partial t}{\partial z}}\\\\\qquad =H_{x}-E_{z}{\frac {\partial t}{\partial y}}+E_{y}{\frac {\partial t}{\partial z}},\end{array}}$

the surface integral which is obtained is

${\begin{array}{ll}\iint \left[\left(H_{x}-E_{z}{\frac {\partial t}{\partial y}}+E_{y}{\frac {\partial t}{\partial z}}\right)dy\ dz\right.&+\left(H_{y}-E_{z}{\frac {\partial t}{\partial z}}+E_{z}{\frac {\partial t}{\partial x}}\right)dz\ dx\\\\&\left.+\left(H_{z}-E_{y}{\frac {\partial t}{\partial x}}+E_{x}{\frac {\partial t}{\partial y}}\right)dx\ dy\right],\end{array}}$

or

$\iint \left(H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\right).$

Hence equation (IV) is established.

3. The Group of Point Transformations for which the Integral Equations of the Theory of Electrons are Invariant.

Let us consider a transformation of coordinates from (x, y, z, t) to (x', y', z', t') which is biuniform within a certain domain of values of (x, y, z, t). We shall suppose that the choice of a transformation is