Page:BatemanElectrodynamical.djvu/13

If the signs of the α's are all changed, the relations take the form

$\beta _{2}\gamma _{3}-\beta _{3}\gamma _{2}=-\left(\alpha _{1}\delta _{4}-\alpha _{4}\delta _{1}\right),$

but the final result is unaltered; hence a set of relations of this type also imply that the quantities α are the elements of an orthogonal matrix.

Taking the quantities $\alpha ,\beta ,\dots$ to be the elements of the matrix

$\left|{\begin{array}{ccccccc}{\frac {\partial x'}{\partial x}}&&{\frac {\partial x'}{\partial y}}&&{\frac {\partial x'}{\partial z}}&&{\frac {\partial x'}{i\partial t}}\\\\{\frac {\partial y'}{\partial x}}&&{\frac {\partial y'}{\partial y}}&&{\frac {\partial y'}{\partial z}}&&{\frac {\partial y'}{i\partial t}}\\\\{\frac {\partial z'}{\partial x}}&&{\frac {\partial z'}{\partial y}}&&{\frac {\partial z'}{\partial z}}&&{\frac {\partial z'}{i\partial t}}\\\\i{\frac {\partial t'}{\partial x}}&&i{\frac {\partial t'}{\partial y}}&&i{\frac {\partial t'}{\partial z}}&&{\frac {\partial t'}{\partial t}}\end{array}}\right|$

we have the result that, if eighteen relations of the type

${\frac {\partial (y',z')}{\partial (x,y)}}={\frac {\partial (x',t')}{\partial (z,t)}},\ {\frac {\partial (y',z')}{\partial (x,t)}}=-{\frac {\partial (x',t')}{\partial (y,z)}}$

or of the type

${\frac {\partial (y',z')}{\partial (x,y)}}=-{\frac {\partial (x',t')}{\partial (z,t)}},\ {\frac {\partial (y',z')}{\partial (x,t)}}={\frac {\partial (x',t')}{\partial (y,z)}}$

are satisfied, then the matrix is an orthogonal one, and this implies that

$dx'^{2}+dy'^{2}+dz'^{2}-dt'^{2}=\lambda ^{2}\left[dx^{2}+dy^{2}+dz^{2}-dt^{2}\right];$

in other words, that the differential equation

$(dx)^{2}+(dy)^{2}+(dz)^{2}-(dt)^{2}=0$

is an invariant. The converse is also true, and may be easily verified.

This differential equation expresses the condition that two neighbouring particles should be in a position to act on one another; it also indicates that an elementary wave starting from a given point will have the form of a sphere. For this reason we shall call a transformation which leaves the differential equation invariant, a spherical wave transformation. Our analysis shows that a transformation which leaves the integral equations of the theory of electrons invariant is necessarily a spherical wave transformation. We must next inquire whether all spherical wave transformations are relevant for our purpose. If we call t' the local time, we must exclude transformations which make the local time run backwards as t increases. Hence, in order that a transformation may be relevant, the condition

${\frac {\partial t'}{\partial t}}>0$

must be satisfied.