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

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

and a change in the sign of x'. Putting

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

and changing the sign of x', we get

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

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.

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