Page:BatemanElectrodynamical.djvu/14

286
[March 11,
Mr. H. Bateman

Secondly, we must exclude transformations in which the axes are changed from a right-handed system to a left-handed system, for then equation (1) would imply a change from positive electricity to negative electricity. Such a transformation, however, becomes relevant when equation (1) is replaced by

 ${\displaystyle \rho w_{x}dy\ dz\ dt+\rho w_{y}dz\ dx\ dt+\rho w_{z}dx\ dy\ dt-\rho dx\ dy\ dz}$ ${\displaystyle =-\rho 'w'_{x}dy'dz'dt'-\rho 'w'_{y}dz'dx'dt'-\rho 'w'_{z}dx'dy'dt'+\rho 'dx'dy'dz',}$ (1)'

equation (2) by

 ${\displaystyle E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt}$ ${\displaystyle =-E'_{x}dy'dz'-E'_{y}dz'dx'-E'_{z}dx'\ dy'-H'_{x}dx'dt'-H'_{y}dy'dt'-H'_{z}dz'dt',}$ (2)'

and equation (3) by

 ${\displaystyle H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt}$ ${\displaystyle =\theta \left[-H'_{x}dy'dz'-H'_{y}dz'dx'-H'_{z}dx'dy'+E'_{x}dx'dt'+E'_{y}dy'dt'+E'_{z}dz'dt'\right].}$ (3)'

These relations also imply that the transformation is a spherical wave transformation, and so a transformation which changes a right-handed system of axes into a left-handed system is relevant, but the formulae of transformation of the components of the electric and magnetic force are not the same as before. The sign of θ will be determined later by the condition that a right-handed set of axes is transformed into a lefthanded set. If

${\displaystyle {\frac {\partial (x',y',z')}{\partial (x,y,z)}}>0,}$

corresponding sets of axes are both right-handed or both left-handed. If, on the other hand,

${\displaystyle {\frac {\partial (x',y',z')}{\partial (x,y,z)}}<0,}$

the axes are right-handed in one system and left-handed in the other.

It is easy to establish a relation between the two quantities

${\displaystyle {\frac {\partial t'}{\partial t}}}$ and ${\displaystyle {\frac {\partial (x',y',z')}{\partial (x,y,z)}}.}$

Since

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