Page:Lorentz Simplified1899.djvu/13

( 439 )

being the electric force. By both signs I shall now denote, not the whole vector, but the part that is due to the vibrations.

The equations may again be written in a form in which the velocity of translation does not explicitly appear. For this purpose, it is necessary to replace the variables x', y', z', t', ${\mathfrak {F}}',{\mathfrak {H}}',{\mathfrak {a}}$ and $\varrho _{0}$ by new ones, differing from the original quantities by certain constant factors.

For the sake of uniformity of notation all these new variables will be distinguished by double accents. Let $\varepsilon$ be an indeterminate coefficient, differing from unity by a quantity of the order ${\mathfrak {p}}_{x}^{2}/V^{2}$ and let us put

$x={\frac {\varepsilon }{k}}x'',\ y=\varepsilon y'',\ z=\varepsilon z''$,

(6)

${\mathfrak {a}}_{x}={\frac {\varepsilon }{k}}{\mathfrak {a}}''_{x},\ {\mathfrak {a}}_{y}=\varepsilon {\mathfrak {a}}''_{y},\ {\mathfrak {a}}_{z}=\varepsilon {\mathfrak {a}}''_{z}$,

(7)

$\varrho _{0}={\frac {k}{\varepsilon ^{3}}}\varrho _{0}''$,

(8)

${\begin{array}{lllll}{\mathfrak {F}}'_{x}={\frac {1}{\varepsilon ^{2}}}{\mathfrak {F}}''_{x},&&{\mathfrak {F}}'_{y}={\frac {1}{\varepsilon ^{2}}}{\mathfrak {F}}''_{y},&&{\mathfrak {F}}'_{z}={\frac {1}{\varepsilon ^{2}}}{\mathfrak {F}}''_{z},\\\\{\mathfrak {H}}'_{x}={\frac {k}{\varepsilon ^{2}}}{\mathfrak {H}}''_{x},&&{\mathfrak {H}}'_{y}={\frac {k}{\varepsilon ^{2}}}{\mathfrak {H}}''_{y},&&{\mathfrak {H}}'_{z}={\frac {k}{\varepsilon ^{2}}}{\mathfrak {H}}''_{z},\end{array}}$

$t'=k\varepsilon t''$,

(9)

so that t" is a modified local time; then we find

$Div''\ {\mathfrak {F}}''=4\pi V^{2}\left(-{\mathfrak {a}}''_{x}{\frac {\partial \varrho _{0}''}{\partial x''}}-{\mathfrak {a}}''_{y}{\frac {\partial \varrho _{0}''}{\partial y''}}-{\mathfrak {a}}''_{z}{\frac {\partial \varrho _{0}''}{\partial z''}}\right)$,

(Ie)

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

(IIe)

${\frac {\partial {\mathfrak {H}}''_{z}}{\partial y''}}-{\frac {\partial {\mathfrak {H}}''_{y}}{\partial z''}}=4\pi \varrho _{0}''{\frac {\partial \ {\mathfrak {a}}''_{x}}{\partial t''}}+{\frac {1}{V^{2}}}{\frac {\partial {\mathfrak {F}}''_{x}}{\partial t''}}$, etc.

(IIIe)

${\frac {\partial {\mathfrak {F}}''_{z}}{\partial y''}}-{\frac {\partial {\mathfrak {F}}''_{y}}{\partial z''}}=-{\frac {\partial {\mathfrak {H}}''_{x}}{\partial t''}}$, etc.

(IVe)

${\mathfrak {E}}_{x}={\frac {1}{\varepsilon ^{2}}}{\mathfrak {F}}''_{x},\ {\mathfrak {E}}_{y}={\frac {1}{k\varepsilon ^{2}}}{\mathfrak {F}}''_{y},\ {\mathfrak {E}}_{z}={\frac {1}{k\varepsilon ^{2}}}{\mathfrak {F}}''_{z}$.

(Ve)