Page:BatemanElectrodynamical.djvu/9

1909.] These equations are equivalent to the single integral equation

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

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

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