Page:Lorentz Simplified1899.djvu/2

 § 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

{{MathForm2|(IIIb)|$$\left.\begin{array}{l} \frac{\partial\mathfrak{H}_{z}}{\partial y}-\frac{\partial\mathfrak{H}_{y}}{\partial z}=4\pi\varrho(\mathfrak{p}_{x}+\mathfrak{v}_{x})+4\pi\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{d}_{x,}\\ \\\frac{\partial\mathfrak{H}_{x}}{\partial z}-\frac{\partial\mathfrak{H}_{z}}{\partial x}=4\pi\varrho\mathfrak{v}_{y}+4\pi\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{d}_{y},\\ \\\frac{\partial\mathfrak{H}_{y}}{\partial x}-\frac{\partial\mathfrak{H}_{x}}{\partial y}=4\pi\varrho\mathfrak{v}_{z}+4\pi\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{d}_{z},\end{array}\right\} $$}}

{{MathForm2|(IV{{sub|b}})|$$\left.\begin{array}{l} 4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{z}}{\partial y}-\frac{\partial\mathfrak{d}_{y}}{\partial z}\right)=-\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{H}_{x},\\ \\4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{x}}{\partial z}-\frac{\partial\mathfrak{d}_{z}}{\partial x}\right)=-\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{H}_{y},\\ \\4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{y}}{\partial x}-\frac{\partial\mathfrak{d}_{x}}{\partial y}\right)=-\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{H}_{z},\end{array}\right\} $$}}

{{MathForm2|(V{{sub|b}})|$$\mathfrak{E}=4\pi V^{2}\mathfrak{d}+[\mathfrak{p.H}]+[\mathfrak{v.H}]$$}}

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.