Page:Electromagnetic phenomena.djvu/5

As to the coefficient l, it is to be considered as a function of w, whose value is 1 for w=0, and which, for small values of w, differs from unity no more than by an amount of the second order.

The variable t' may be called the local time; indeed, for k=1, l=1 it becomes identical with what I have formerly understood by this name.

If, finally, we put

${\frac {1}{kl^{3}}}\varrho =\varrho '$,

(7)

$k^{2}{\mathfrak {u}}_{x}={\mathfrak {u}}_{x}^{'},\quad k{\mathfrak {u}}_{y}={\mathfrak {u}}_{y}^{'},\quad k{\mathfrak {u}}_{z}={\mathfrak {u}}_{z}^{'}$,

(8)

these latter quantities being considered as the components of a new vector ${\mathfrak {u}}'$, the equations take the following form:

$\left.{\begin{aligned}&div'\ {\mathfrak {d}}'=\left(1-{\frac {wu_{x}'}{c^{2}}}\right)\varrho ',\quad div'\ {\mathfrak {h}}'=0,\\&rot'\ {\mathfrak {h'}}={\frac {1}{c}}\left({\frac {\partial {\mathfrak {d}}'}{\partial t'}}+\varrho '{\mathfrak {u}}\right),\\&rot'\ {\mathfrak {d}}'=-{\frac {1}{c}}{\frac {\partial {\mathfrak {h}}'}{\partial t'}},\end{aligned}}\right\}$

(9)

$\left.{\begin{aligned}&{\mathfrak {f}}_{x}=l^{2}{\mathfrak {d}}_{x}^{'}+l^{2}{\frac {1}{c}}\left({\mathfrak {u}}_{y}^{'}{\mathfrak {h}}_{z}^{'}-{\mathfrak {u}}_{z}^{'}{\mathfrak {h}}_{y}^{'}\right)+l^{2}{\frac {w}{c^{2}}}\left({\mathfrak {u}}_{y}^{'}{\mathfrak {d}}_{y}^{'}-{\mathfrak {u}}_{z}^{'}{\mathfrak {d}}_{z}^{'}\right),\\&{\mathfrak {f}}_{y}={\frac {l}{k}}^{2}{\mathfrak {d}}_{y}^{'}+{\frac {l}{k}}^{2}{\frac {1}{c}}\left({\mathfrak {u}}_{z}^{'}{\mathfrak {h}}_{x}^{'}-{\mathfrak {u}}_{x}^{'}{\mathfrak {h}}_{z}^{'}\right)-{\frac {l}{k}}^{2}{\frac {w}{c^{2}}}{\mathfrak {u}}_{x}^{'}{\mathfrak {d}}_{y}^{'},\\&{\mathfrak {f}}_{z}={\frac {l}{k}}^{2}{\mathfrak {d}}_{z}^{'}+{\frac {l}{k}}^{2}{\frac {1}{c}}\left({\mathfrak {u}}_{x}^{'}{\mathfrak {h}}_{y}^{'}-{\mathfrak {u}}_{y}^{'}{\mathfrak {h}}_{x}^{'}\right)-{\frac {l}{k}}^{2}{\frac {w}{c^{2}}}{\mathfrak {u}}_{x}^{'}{\mathfrak {d}}_{z}^{'}.\end{aligned}}\right\}$

(10)

The meaning of the symbols div' and rot' in (9) is similar to that of div and rot in (2); only, the differentiations with respect to x, y, z are to be replaced by the corresponding ones with respect to x', y', z' .

§ 5. The equations (9) lead to the conclusion that the vectors ${\mathfrak {d}}'$ and ${\mathfrak {h}}'$ may be represented by means of a scalar potential φ and a vector potential ${\mathfrak {a}}'$. These potentials satisfy the equations ^[1]

$\triangle '\varphi '-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi '}{\partial t'^{2}}}=-\varrho '$,

(11)

$\triangle '{\mathfrak {a}}'-{\frac {1}{c^{2}}}{\frac {\partial ^{2}{\mathfrak {a}}'}{\partial t'^{2}}}=-{\frac {1}{c^{2}}}\varrho '{\mathfrak {u}}'$.

(12)

and in terms of them ${\mathfrak {d}}'$ and ${\mathfrak {h}}'$ are given by

1. M. E., §§ 4 and 10.