limited by the condition that the total charge on a system of particles is invariant. This is expressed analytically by the equation
provided the axes form a right-handed system in each case.
If the transformation is such that the integral equations of the theory of electrons are invariant, we must have
H x d y d z + H y d z d x + H z d x d y + E x d x d t + E y d y d t + E z d z d t = θ [ H x ′ d y ′ d z ′ + H y ′ d z ′ d x ′ + H z ′ d x ′ d y ′ + E x ′ d x ′ d t ′ + E y ′ d y ′ d t ′ + E z ′ d z ′ d t ′ ] , {\displaystyle {\begin{array}{l}H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\\\qquad =\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],\end{array}}}
where θ {\displaystyle \theta } is a constant.
These relations give two sets of equations connecting the quantities E x , … , H x , … {\displaystyle E_{x},\dots ,H_{x},\dots } with E x ′ , … , H x ′ , … {\displaystyle E'_{x},\dots ,H'_{x},\dots } viz.,
a n d H x = θ [ H x ′ ∂ ( y ′ , z ′ ) ∂ ( y , z ) + H y ′ ∂ ( z ′ , x ′ ) ∂ ( y , z ) + H z ′ ∂ ( x ′ , y ′ ) ∂ ( y , z ) + E x ′ ∂ ( x ′ , t ′ ) ∂ ( y , z ) + E y ′ ∂ ( y ′ , t ′ ) ∂ ( y , z ) + E z ′ ∂ ( z ′ , t ′ ) ∂ ( y , z ) ] , E x = θ [ H x ′ ∂ ( y ′ , z ′ ) ∂ ( x , t ) + H y ′ ∂ ( z ′ , x ′ ) ∂ ( x , t ) + H z ′ ∂ ( x ′ , y ′ ) ∂ ( y , t ) + E x ′ ∂ ( x ′ , t ′ ) ∂ ( x , t ) + E y ′ ∂ ( y ′ , t ′ ) ∂ ( x , t ) + E z ′ ∂ ( z ′ , t ′ ) ∂ ( x , t ) ] . {\displaystyle {\mathsf {and}}\ {\begin{array}{rr}H_{x}=&\theta \left[H'_{x}{\frac {\partial (y',z')}{\partial (y,z)}}+H'_{y}{\frac {\partial (z',x')}{\partial (y,z)}}+H'_{z}{\frac {\partial (x',y')}{\partial (y,z)}}\right.\\\\&\left.+E'_{x}{\frac {\partial (x',t')}{\partial (y,z)}}+E'_{y}{\frac {\partial (y',t')}{\partial (y,z)}}+E'_{z}{\frac {\partial (z',t')}{\partial (y,z)}}\right],\\\\E_{x}=&\theta \left[H'_{x}{\frac {\partial (y',z')}{\partial (x,t)}}+H'_{y}{\frac {\partial (z',x')}{\partial (x,t)}}+H'_{z}{\frac {\partial (x',y')}{\partial (y,t)}}\right.\\\\&\left.+E'_{x}{\frac {\partial (x',t')}{\partial (x,t)}}+E'_{y}{\frac {\partial (y',t')}{\partial (x,t)}}+E'_{z}{\frac {\partial (z',t')}{\partial (x,t)}}\right].\end{array}}}
In order that these equations may be equivalent to one another[1] we must
![{\displaystyle {\mathsf {and}}\ {\begin{array}{rr}H_{x}=&\theta \left[H'_{x}{\frac {\partial (y',z')}{\partial (y,z)}}+H'_{y}{\frac {\partial (z',x')}{\partial (y,z)}}+H'_{z}{\frac {\partial (x',y')}{\partial (y,z)}}\right.\\\\&\left.+E'_{x}{\frac {\partial (x',t')}{\partial (y,z)}}+E'_{y}{\frac {\partial (y',t')}{\partial (y,z)}}+E'_{z}{\frac {\partial (z',t')}{\partial (y,z)}}\right],\\\\E_{x}=&\theta \left[H'_{x}{\frac {\partial (y',z')}{\partial (x,t)}}+H'_{y}{\frac {\partial (z',x')}{\partial (x,t)}}+H'_{z}{\frac {\partial (x',y')}{\partial (y,t)}}\right.\\\\&\left.+E'_{x}{\frac {\partial (x',t')}{\partial (x,t)}}+E'_{y}{\frac {\partial (y',t')}{\partial (x,t)}}+E'_{z}{\frac {\partial (z',t')}{\partial (x,t)}}\right].\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76883641949aeaa4f7ab472699bbb8b0cd2ac20c)
![{\displaystyle {\begin{array}{l}H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt\\\qquad =\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],\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac609d787a2c8f76fefc22b2e6fc266dbd6832be)
