Page:BatemanElectrodynamical.djvu/27

 and

$$\begin{array}{l} E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\\ \qquad=E'_{x}dy'dz'+E'_{y}dz'dx'+E'_{z}dx'dy'-H'_{x}dx'dt'-H'_{y}dy'dt'-H'_{z}dz'dt',\end{array}$$

we obtain, on multiplication,

$$\begin{array}{l} \left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt\\ \qquad=\left(E_{x}^{'2}+E_{y}^{'2}+E_{z}^{'2}-H_{x}^{'2}-H_{y}^{'2}-H_{z}^{'2}\right)dx'dy'dz'dt',\end{array}$$

while, if either form be multiplied by itself, we obtain

These equations indicate the invariance of the property that at a surface of discontinuity, or in a spherical wave, the electric force is equal in magnitude to the magnetic force and perpendicular to it.

A space time transformation from the variables (x, y, z, t) to (x', y, z, t') can be used to transform the whole motion in one dynamical system into a corresponding motion in another, as far as the kinematics is concerned, provided the velocities $$\left(w_{x},w_{y},w_{z}\right),\ \left(w'_{x},w'_{y},w'_{z}\right)$$ of corresponding points are such that the equations

are a consequence of the relations

This condition is satisfied, if

Multiplying these equations together by Grassmann's rule, we get

This shows that there is an integral invariant of the form

This fact has already been used in § 3 and will be required again in § 7.