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

 that is to say, at the place

at the instant {{Wikimath|(t{{sub|1}} - sinh a. Δξ/c). Thus at the instant {{Wikimath|t{{sub|1}}}}, this charge will occupy the position The charges corresponding to those in the original system which were at the instant {{Wikimath|t}} contained in a volume {{Wikimath|ΔξΔηΔζ}} will therefore in the derived system at the instant {{Wikimath|t{{sub|1}}}}, occupy a volume {{c|$$\left|\begin{matrix} \cosh a &+& \sinh a.v_{x_1}/c. & 0 & 0 \\ \ &\ & \sinh a.v_{y_1}/c  & 1 & 0 \\ \ &\ &\sinh a.v_{z_1}/c  & 0 & 1 \\ \end{matrix}\right|.\Delta\xi\Delta\eta\Delta\zeta$$}} or, {{c|$$(\cosh a + \sinh a.v_{x_1}/c)\Delta\xi\Delta\eta\Delta\zeta$$.}} Thus if {{Wikimath|ρ{{sub|1}}}} denote the volume-density of electric charge in the transformed system, we shall have {{c|$$\rho_1(\cosh a + \sinh a.v_{x_1}/c) = \rho$$;}} this equation expresses the connexion between {{Wikimath|ρ{{sub|1}}}} and {{Wikimath|ρ}}. We have moreover {{c|$$\begin{matrix} v_x &=& \frac{ \frac{\partial x}{\partial x_1}v_{x_1} + \frac{\partial x}{\partial y_1}v_{y_1} + \frac{\partial x}{\partial z_1}v_{z_1} + \frac{\partial x}{\partial t_1} }{ \frac{\partial t}{\partial x_1}v_{x_1} + \frac{\partial t}{\partial y_1}v_{y_1} + \frac{\partial t}{\partial z_1}v_{z_1} + \frac{\partial t}{\partial t_1} } \\ \ &=& e \tanh a + \frac{v_{x_1}\text{sech }a}{\cosh a + v_{x_1}e^{-1}\sinh a} \end{matrix}$$}} and similarly {{c|$$v_y = \frac{v_{y_1}}{\cosh a + v_{x_1}e^{-1}\sinh a}$$,}} and {{c|$$v_x = \frac{v_{x_1}}{\cosh a + v_{x_1}e^{-1}\sinh a}$$.}}