Page:Electromagnetic phenomena.djvu/4

$div\ {\mathfrak {d}}=\varrho ,\quad div\ {\mathfrak {h}}=0$, ${\frac {\partial {\mathfrak {h}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {h}}_{y}}{\partial z}}={\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{x}+{\frac {1}{c}}\varrho \left(w+{\mathfrak {u}}_{x}\right)$, ${\frac {\partial {\mathfrak {h}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {h}}_{z}}{\partial x}}={\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{y}+{\frac {1}{c}}\varrho {\mathfrak {u}}_{y}$, ${\frac {\partial {\mathfrak {h}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {h}}_{x}}{\partial y}}={\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{z}+{\frac {1}{c}}\varrho {\mathfrak {u}}_{z}$, ${\frac {\partial {\mathfrak {d}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {d}}_{y}}{\partial z}}=-{\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {h}}_{x}$, ${\frac {\partial {\mathfrak {d}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {d}}_{z}}{\partial x}}=-{\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {h}}_{y}$, ${\frac {\partial {\mathfrak {d}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {d}}_{x}}{\partial y}}=-{\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {h}}_{z}$, ${\mathfrak {f}}_{x}={\mathfrak {d}}_{x}+{\frac {1}{c}}\left({\mathfrak {u}}_{y}{\mathfrak {h}}_{z}-{\mathfrak {u}}_{z}{\mathfrak {h}}_{y}\right)$, ${\mathfrak {f}}_{y}={\mathfrak {d}}_{y}-{\frac {1}{c}}w{\mathfrak {h}}_{z}+{\frac {1}{c}}\left({\mathfrak {u}}_{z}{\mathfrak {h}}_{x}-{\mathfrak {u}}_{x}{\mathfrak {h}}_{z}\right)$, ${\mathfrak {f}}_{z}={\mathfrak {d}}_{z}+{\frac {1}{c}}w{\mathfrak {h}}_{y}+{\frac {1}{c}}\left({\mathfrak {u}}_{x}{\mathfrak {h}}_{y}-{\mathfrak {u}}_{y}{\mathfrak {h}}_{x}\right)$.

§ 4. We shall further transform these formulae by a change of variables. Putting

${\frac {c^{2}}{c^{2}-w^{2}}}=k^{2}$,

(3)

and understanding by l another numerical quantity, to be determined further on, I take as new independent variables

$x'=klx,\quad y'=ly,\quad z'=lz$,

(4)

$t'={\frac {l}{k}}t-kl{\frac {w}{c^{2}}}x$,

(5)

and I define two new vectors ${\mathfrak {d}}'$ and ${\mathfrak {h}}'$ by the formulae

${\mathfrak {d}}_{x}^{'}={\frac {1}{l^{2}}}{\mathfrak {d}}_{x},\quad {\mathfrak {d}}_{y}^{'}={\frac {k}{l^{2}}}\left({\mathfrak {d}}_{y}-{\frac {w}{c}}{\mathfrak {h}}_{z}\right),\quad {\mathfrak {d}}_{z}^{'}={\frac {k}{l^{2}}}\left({\mathfrak {d}}_{z}+{\frac {w}{c}}{\mathfrak {h}}_{y}\right)$, ${\mathfrak {h}}_{x}^{'}={\frac {1}{l^{2}}}{\mathfrak {h}}_{x},\quad {\mathfrak {h}}_{y}^{'}={\frac {k}{l^{2}}}\left({\mathfrak {h}}_{y}+{\frac {w}{c}}{\mathfrak {d}}_{z}\right),\quad {\mathfrak {h}}_{z}^{'}={\frac {k}{l^{2}}}\left({\mathfrak {h}}_{z}-{\frac {w}{c}}{\mathfrak {d}}_{y}\right)$,

for which, on account of (3), we may also write

${\begin{cases}{\mathfrak {d}}_{x}=l^{2}{\mathfrak {d}}_{x}^{'},\quad {\mathfrak {d}}_{y}=kl^{2}\left({\mathfrak {d}}_{y}^{'}+{\frac {w}{c}}{\mathfrak {h}}_{z}^{'}\right),\quad {\mathfrak {d}}_{z}=kl^{2}\left({\mathfrak {d}}_{z}^{'}-{\frac {w}{c}}{\mathfrak {h}}_{y}^{\mathfrak {'}}\right),\\{\mathfrak {h}}_{x}=l^{2}{\mathfrak {h}}_{x}^{'},\quad {\mathfrak {h}}_{y}=kl^{2}\left({\mathfrak {h}}_{y}^{'}-{\frac {w}{c}}{\mathfrak {d}}_{z}^{'}\right),\quad {\mathfrak {h}}_{z}=kl^{2}\left({\mathfrak {h}}_{z}^{'}+{\frac {w}{c}}{\mathfrak {d}}_{y}^{\mathfrak {'}}\right),\end{cases}}$.

(6)