Page:BatemanElectrodynamical.djvu/31

 These invariants represent quantities which are of considerable importance in the theory of electrons. Another invariant which is of some importance is obtained in the following way.

Let

so that

$w_{1}=\frac{dx}{ds},\ w_{2}=\frac{dy}{ds},\ w_{3}=\frac{dz}{ds},\ w_{4}=\frac{dt}{ds},$

Then

Hence

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