Page:BatemanElectrodynamical.djvu/20

and is positive; hence t' increases as t increases, if (x, y, z) are kept constant.

The transformation which corresponds to a reflexion in the four-dimensional space is also of considerable interest. In the particular case when the reflecting space passes through the plane x = 0, s = 0, the reflexion may be replaced by a rotation round the plane x = 0, s = 0, and a reflexion in the space x = 0. The corresponding transformation is thus made up out of a transformation of Lorentz

$x'={\frac {x-vt}{\sqrt {1-v^{2}}}},\ y'=y,\ z'=z,\ t'={\frac {t-vx}{\sqrt {1-v^{2}}}}.$

and a change in the sign of x'. Putting

${\frac {v}{\sqrt {1-v^{2}}}}={\frac {2u}{1-u^{2}}},\ {\frac {1}{\sqrt {1-v^{2}}}}={\frac {1+u^{2}}{1-u^{2}}},$

and changing the sign of x', we get

${\begin{array}{l}x'=-{\frac {1+u^{2}}{1-u^{2}}}x+{\frac {2u}{1-u^{2}}}t,\\\\y'=y,\\\\z'=z\\\\t'={\frac {1+u^{2}}{1-u^{2}}}t-{\frac {2u}{1-u^{2}}}x.\end{array}}$

The quantity u is introduced because the angle of rotation in the four-dimensional space is twice the angle between the reflecting space and the space x = 0. Its geometrical meaning in the case of the spherical wave transformation is indicated by the equation

$x'-ut'=-(x-ut)$

which implies that a plane moving with the constant velocity u is transformed into itself. Further, when x = ut, we have

$x'=x,\ y'=y,\ z'=z,\ t'=t;$

hence every point of the plane is transformed into itself.^[1]

The formulae of transformation of the electromagnetic vectors are found from the first set of equations for a transformation with negative Jacobian.

1. Geometrically the transformation is equivalent to a reflexion in a moving plane mirror.