Page:BatemanElectrodynamical.djvu/32

 Thus

$\begin{array}{l} \frac{d^{2}w_{1}}{ds^{2}}-w_{1}\left\{ \left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2}\right\} \\ \\\qquad=\frac{\ddot{w}_{x}}{\left(1-w^{2}\right)^{\frac{1}{2}}}+\frac{3\dot{w}_{x}(w\dot{w})}{\left(1-w^{2}\right)^{\frac{1}{2}}}+\frac{w_{x}}{\left(1-w^{2}\right)^{\frac{1}{2}}}\left\{ (w\ddot{w})+\frac{3(w\dot{w})}{1-w^{2}}\right\} \end{array}$|undefined

Now Abraham has given a formula for the reaction of radiation upon a moving electron, the x component of the reaction being

$K_{x}=\frac{2e^{2}}{3\left(1-w^{2}\right)}\left[\ddot{w}_{x}+\dot{w}_{x}\frac{3(w\dot{w})}{1-w^{2}}+\frac{w_{x}}{1-w^{2}}\left\{ (w\ddot{w})+\frac{3(w\dot{w})^{2}}{1-w^{2}}\right\} \right];$|undefined

accordingly,

$\begin{array}{ll} K_{x}dt & =\frac{2e^{2}}{3c^{2}}\left[\frac{d^{2}w_{1}}{ds^{2}}-w_{1}\left\{ \left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2}\right\} \right]ds\\ \\ & =K_{1}ds.\end{array}$|undefined

To complete the symmetry of the result, we define quantities $$K_t$$ and $$K_4$$ by the equations

$\begin{array}{ll} K_{t}dt & =\frac{2e^{2}}{3c^{2}}\left[\frac{d^{2}w_{4}}{ds^{2}}-w_{4}\left\{ \left(\frac{dw_{1}}{ds}\right)^{2}+\left(\frac{dw_{2}}{ds}\right)^{2}+\left(\frac{dw_{3}}{ds}\right)^{2}-\left(\frac{dw_{4}}{ds}\right)^{2}\right\} \right]ds\\ \\ & =K_{4}ds.\end{array}$|undefined

we then have

Laue has shown recently that Abraham's formula may be derived by means of the principle of relativity. We shall complete this result by showing that

$\left(K_{x}\delta x+K_{y}\delta y+K_{z}\delta z-K\ \delta x\right)dt$

is an integral invariant for the whole group of spherical wave transformations. It will be sufficient to prove this for the case of the transformation

$x'=\frac{x}{r^{2}-t^{2}},\ y'=\frac{y}{r^{2}-t^{2}},\ z'=\frac{z}{r^{2}-t^{2}},\ t'=\frac{t}{r^{2}-t^{2}}.$|undefined