Page:BatemanElectrodynamical.djvu/15

 we have

$\begin{array}{ll} \frac{\partial(x',y',z')}{\partial(x,y,z)}= & \theta\left[\frac{\partial t'}{\partial t}\left\{ \left(\frac{\partial x'}{\partial x}\right)^{2}+\left(\frac{\partial y'}{\partial x}\right)^{2}+\left(\frac{\partial z'}{\partial x}\right)^{2}-\left(\frac{\partial t'}{\partial x}\right)^{2}\right\} \right.\\ \\ & \qquad\left.-\frac{\partial x'}{\partial x}\left\{ \frac{\partial x'}{\partial x}\frac{\partial x'}{\partial t}+\frac{\partial y'}{\partial x}\frac{\partial y'}{\partial t}+\frac{\partial z'}{\partial x}\frac{\partial z'}{\partial t}-\frac{\partial t'}{\partial x}\frac{\partial t'}{\partial t}\right\} \right]\end{array}$

or

We also have

$\begin{array}{ll} \frac{\partial(x',y',z',t')}{\partial(x,y,z,t)} & =\frac{\partial(y',z')}{\partial(y,z)}\frac{\partial(x',t')}{\partial(x,t)}+\frac{\partial(z',x')}{\partial(y,z)}\frac{\partial(y',t')}{\partial(x,t)}+\dots\\ \\ & =\theta\left[\left\{ \frac{\partial(x',t')}{\partial(x,t)}\right\} ^{2}+\left\{ \frac{\partial(y',t')}{\partial(x,t)}\right\} ^{2}+\dots-\left\{ \frac{\partial(y',z')}{\partial(x,t)}\right\} ^{2}-\dots\right]\\ \\ & =-\theta\left[\left(\frac{\partial x'}{\partial x}\right)^{2}+\left(\frac{\partial y'}{\partial x}\right)^{2}+\left(\frac{\partial z'}{\partial x}\right)^{2}-\left(\frac{\partial t'}{\partial x}\right)^{2}\right]\\ \\ & \qquad\times\left\{ \left(\frac{\partial x'}{\partial t}\right)^{2}+\left(\frac{\partial y'}{\partial t}\right)^{2}+\left(\frac{\partial z'}{\partial t}\right)^{2}-\left(\frac{\partial t'}{\partial t}\right)^{2}\right\} \\ \\ & \qquad-\left\{ \frac{\partial x'}{\partial x}\frac{\partial x'}{\partial t}+\frac{\partial y'}{\partial x}\frac{\partial y'}{\partial t}+\frac{\partial z'}{\partial x}\frac{\partial z'}{\partial t}-\frac{\partial t'}{\partial x}\frac{\partial t'}{\partial t}\right\} \end{array}$

Therefore

$\begin{array}{ll} \frac{\partial(x',y',z',t')}{\partial(x,y,z,t)}= & -\theta\left[\left(\frac{\partial x'}{\partial x}\right)^{2}+\left(\frac{\partial y'}{\partial x}\right)^{2}+\left(\frac{\partial z'}{\partial x}\right)^{2}-\left(\frac{\partial t'}{\partial x}\right)^{2}\right]\\ \\ & \qquad\times\left[\left(\frac{\partial x'}{\partial t}\right)^{2}+\left(\frac{\partial y'}{\partial t}\right)^{2}+\left(\frac{\partial z'}{\partial t}\right)^{2}-\left(\frac{\partial t'}{\partial t}\right)^{2}\right]\end{array}$

This gives us the relation

which holds for any spherical wave transformation.

We shall now introduce the further restriction that the inequality

$(dx')^{2}+(dy')^{2}+(dz')^{2}<(dt')^{2}$

is a consequence of

$(dx)^{2}+(dy)^{2}+(dz)^{2}<(dt)^{2}$

This means that, if a particle is moving with a velocity less than that of light in one system of coordinates, it is also moving with a velocity less than that of light in the transformed system.