Page:Lorentz Simplified1899.djvu/2

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§ 3. We shall apply these equations to a system of bodies, having a common velocity of translation ${\mathfrak {p}}$, of constant direction and magnitude, the aether remaining at rest, and we shall henceforth denote by ${\mathfrak {v}}$, not the whole velocity of a material element, but the velocity it may have in addition to ${\mathfrak {p}}$.

Now it is natural to use a system of axes of coordinates, which partakes of the translation ${\mathfrak {p}}$. If we give to the axis of x the direction of the translation, so that ${\mathfrak {p}}_{y}$ and ${\mathfrak {p}}_{z}$ are 0, the equations (Ia)— (Va) will have to be replaced by

$Div\ {\mathfrak {d}}=\varrho$,

(Ib)

$Div\ {\mathfrak {H}}=0$,

(IIb)

$\left.{\begin{array}{l}{\frac {\partial {\mathfrak {H}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}=4\pi \varrho ({\mathfrak {p}}_{x}+{\mathfrak {v}}_{x})+4\pi \left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{x,}\\\\{\frac {\partial {\mathfrak {H}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {H}}_{z}}{\partial x}}=4\pi \varrho {\mathfrak {v}}_{y}+4\pi \left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{y},\\\\{\frac {\partial {\mathfrak {H}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}=4\pi \varrho {\mathfrak {v}}_{z}+4\pi \left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{z},\end{array}}\right\}$

(IIIb)

$\left.{\begin{array}{l}4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {d}}_{y}}{\partial z}}\right)=-\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {H}}_{x},\\\\4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {d}}_{z}}{\partial x}}\right)=-\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {H}}_{y},\\\\4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {d}}_{x}}{\partial y}}\right)=-\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {H}}_{z},\end{array}}\right\}$

(IVb)

${\mathfrak {E}}=4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {p.H}}]+[{\mathfrak {v.H}}]$

(Vb)

In these formulae the sign Div, applied to a vector ${\mathfrak {A}}$, has still the meaning defined by

$Div\ {\mathit {\mathfrak {A}}}={\frac {\partial {\mathfrak {A}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {A}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {A}}_{z}}{\partial z}}$.

As has already been said, ${\mathfrak {v}}$ is the relative velocity with regard to the moving axes of coordinates. If ${\mathfrak {v}}=0$, we shall speak of a system at rest; this expression therefore means relative rest with regard to the moving axes.