Page:BatemanElectrodynamical.djvu/42

Then, if $a_{x},a_{y},a_{z},a,s$ are five functions of x, y, z, t satisfying the relations

$a_{x}s_{x}+a_{y}s_{y}+a_{z}s_{z}=as$ $s_{x}^{2}+s_{y}^{2}+s_{z}^{2}=s^{2},$

a transformation of coordinates for which

${\begin{array}{r}dx^{2}+dy^{2}+dz^{2}-dt^{2}+\left(a_{x}dx+a_{y}dy+a_{z}dz-a\ dt\right)\left(s_{x}dx+s_{y}dy+s_{z}dz-s\ dt\right)\\=\lambda \left(dx'^{2}+dy'^{2}+dz'^{2}-dt'^{2}\right),\end{array}}$

λ being a function of x', y', z', t', is in general suitable for the transformation of the problem in question. The above equation implies that, if the velocity of radiation is represented by unity in the first system of coordinates, it is also represented by unity in the second system.

[March 15th, 1910.—

The relation $a_{x}s_{x}+a_{y}s_{y}+a_{z}s_{z}=as$ can be omitted.]