Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/462

that is to say, at the place

$(\xi _{1}+\Delta \xi \cosh a,\qquad \eta _{1}+\Delta \eta ,\qquad \zeta _{1}+\Delta \zeta )$

at the instant {{Wikimath|(t₁ - sinh a. Δξ/c). Thus at the instant t₁, this charge will occupy the position

$(\xi _{1}+\Delta \xi \cosh a+\sinh a.\Delta \xi .v_{x_{1}}/c,\qquad \eta _{1}+\Delta \eta +\sinh a.\Delta \xi .v_{y_{1}}/c\qquad$

$\zeta _{1}+\Delta \zeta +\sinh a.\Delta \xi .v_{a_{1}}/c)$

The charges corresponding to those in the original system which were at the instant t contained in a volume ΔξΔηΔζ will therefore in the derived system at the instant t₁, occupy a volume

$\left|{\begin{matrix}\cosh a&+&\sinh a.v_{x_{1}}/c.&0&0\\\ &\ &\sinh a.v_{y_{1}}/c&1&0\\\ &\ &\sinh a.v_{z_{1}}/c&0&1\\\end{matrix}}\right|.\Delta \xi \Delta \eta \Delta \zeta$

or,

$(\cosh a+\sinh a.v_{x_{1}}/c)\Delta \xi \Delta \eta \Delta \zeta$.

Thus if ρ₁ denote the volume-density of electric charge in the transformed system, we shall have

$\rho _{1}(\cosh a+\sinh a.v_{x_{1}}/c)=\rho$;

this equation expresses the connexion between ρ₁ and ρ. We have moreover

${\begin{matrix}v_{x}&=&{\frac {{\frac {\partial x}{\partial x_{1}}}v_{x_{1}}+{\frac {\partial x}{\partial y_{1}}}v_{y_{1}}+{\frac {\partial x}{\partial z_{1}}}v_{z_{1}}+{\frac {\partial x}{\partial t_{1}}}}{{\frac {\partial t}{\partial x_{1}}}v_{x_{1}}+{\frac {\partial t}{\partial y_{1}}}v_{y_{1}}+{\frac {\partial t}{\partial z_{1}}}v_{z_{1}}+{\frac {\partial t}{\partial t_{1}}}}}\\\ &=&e\tanh a+{\frac {v_{x_{1}}{\text{sech }}a}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}\end{matrix}}$

and similarly

$v_{y}={\frac {v_{y_{1}}}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}$,

and

$v_{x}={\frac {v_{x_{1}}}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}$.