Page:BatemanElectrodynamical.djvu/12

284 or all the quantities of type $$\beta_{1}\delta_{2}-\beta_{2}\delta_{1}$$ are zero, and that either

$(\beta\alpha)=(\beta\delta)=(\gamma\alpha)=(\gamma\delta)=0$

or all the quantities of type $$\alpha_{1}\delta_{2}-\alpha_{2}\delta_{1}$$ are zero. It follows from this that either (1) the six quantities (&alpha;&beta;) are zero or (2) that the thirty-six quantities $$\left(\alpha_{1}\delta_{2}-\alpha_{2}\delta_{1}\right),\ \left(\beta_{1}\delta_{2}-\beta_{2}\delta_{1}\right),\ \left(\gamma_{1}\delta_{2}-\gamma_{2}\delta_{1}\right),\ \dots$$ are all zero or (3) that there is a set of relations

It is easy to see, however, that in the latter case we also have

$(\gamma\delta)=(\alpha\beta)=0.$

Hence, in all cases,

$(\alpha\gamma)=(\alpha\beta)=(\alpha\delta)=(\beta\gamma)=(\beta\delta)=(\gamma\delta)=0.$

Again, we have

$\left

hence, since the last term is zero,

$\left

This gives

$\begin{array}{l} \left(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}\right)\left(\delta_{1}^{2}+\delta_{2}^{2}+\delta_{3}^{2}+\delta_{4}^{2}\right)=\end{array}\left

say, and there are similar equations in $$\alpha,\beta,\alpha\gamma,\beta\gamma,\beta\delta,\gamma\delta.$$. It follows that

$\begin{array}{cl} \alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2} & =\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+\beta_{4}^{2}\\ & =\gamma_{1}^{2}+\gamma_{2}^{2}+\gamma_{3}^{2}+\gamma_{4}^{2}=\delta_{1}^{2}+\delta_{2}^{2}+\delta_{3}^{2}+\delta_{4}^{2}=\pm\lambda^{2}\end{array}$

These conditions, combined with the previous set, imply that the sixteen quantities $$\alpha_{1},\beta_{2},\dots$$ are the elements of an orthogonal matrix.