Page:BatemanElectrodynamical.djvu/21

 The general infinitesimal spherical wave transformation is

where the coefficients are all constants, and &epsilon; is a quantity whose square may be neglected. Since there are fifteen arbitrary constants the group is a fifteen parameter group.

We have

If (x, y, z) is kept constant as t varies, the corresponding point (x', y', z') moves along a parabola, but in one type of transformation it moves along a straight line with constant acceleration. This is the case, for example, when

$\begin{array}{lll} x'=x+\epsilon\left(y^{2}+z^{2}-x^{2}-t^{2}\right), & & y'=y(1-2\epsilon x),\\ \\z'=z(1-2\epsilon x), & & t'=t(1-2\epsilon x),\end{array}$

for, since quantities of order &epsilon;² may be neglected, the first equation may be written

$x'=x+\epsilon\left(y^{2}+z^{2}-x^{2}-t^{2}\right)$

Hence, if (x, y, z) are kept constant, the point (x', y', z') moves with constant acceleration &gamma; given by

$\gamma=-2\epsilon.$

Substituting for &gamma;, we get

$\begin{array}{lll} x'=x-\frac{1}{2}\gamma\left(y^{2}+z^{2}-x^{2}-t^{2}\right), & & y'=y(1+\gamma x),\\ \\z'=z(1+\gamma x), & & t'=t(1+\gamma x).\end{array}$

The last equation agrees with the one obtained by Einstein.

In the case of the general infinitesimal transformation, the expression

$(x'-x)^{2}+(y'-y)^{2}+(z'-z)^{2}-(t'-t)^{2},$