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

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

(A)

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^[1]

$-{\frac {\partial (x',y',z')}{\partial (x,y,z)}}\left[\left({\frac {\partial x'}{\partial t}}\right)^{2}+\left({\frac {\partial y'}{\partial t}}\right)^{2}+\left({\frac {\partial z'}{\partial t}}\right)^{2}-\left({\frac {\partial t'}{\partial t}}\right)^{2}\right]={\frac {\partial t'}{\partial t}}{\frac {\partial (x',y',z',t')}{\partial (x,y,z,t)}},$

(B)

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.

1. I am indebted to a referee for calling my attention to this relation and the necessity of distinguishing between the two types of transformation.