Page:Elektrische und Optische Erscheinungen (Lorentz) 053.jpg

and because we seek the value of ${\mathfrak {d}}$ outside the molecule, (III_b) is transformed into

$4\pi {\dot {\mathfrak {d}}}=Rot\ {\mathfrak {H}}'{,}$

or, due to (35), it is transformed into

$4\pi {\dot {\mathfrak {d}}}_{x}=\left({\frac {\partial {\mathfrak {H'}}_{z}}{\partial y}}\right)'-\left({\frac {\partial {\mathfrak {H'}}_{y}}{\partial z}}\right)'-{\frac {{\mathfrak {p}}_{y}}{V^{2}}}{\frac {\partial {\mathfrak {H'}}_{z}}{\partial t}}+{\frac {{\mathfrak {p}}_{z}}{V^{2}}}{\frac {\partial {\mathfrak {H'}}_{y}}{\partial t}}{\text{, etc.}}$

If we bring the last two terms on the left side, then we just obtain $\textstyle {{\frac {1}{V^{2}}}{\dot {\mathfrak {F}}}_{x}}$ or $\textstyle {{\frac {1}{V^{2}}}{\frac {\partial {\mathfrak {F}}_{x}}{\partial t'}}}$, as it can be seen by (V_b); since ${\mathfrak {H}}$ and ${\mathfrak {H}}'$ only differ by magnitudes of order ${\mathfrak {p}}$, we may replace the vector product (V_b) by $[{\mathfrak {p.H'}}]$.

From

${\frac {\partial {\mathfrak {F}}_{x}}{\partial t'}}=V^{2}\left[\left({\frac {\partial {\mathfrak {H'}}_{z}}{\partial y}}\right)'-\left({\frac {\partial {\mathfrak {H'}}_{y}}{\partial z}}\right)'\right]{\text{, etc.}}$

we obtain ${\mathfrak {F}}$ by integration; constants were omitted by us, since we are only dealing with vibrations.

We substitute the values (39) and put

$\left({\frac {\partial }{\partial x}}\right)'\left({\frac {{\mathfrak {m}}_{x}}{r_{0}}}\right)+\left({\frac {\partial }{\partial y}}\right)'\left({\frac {{\mathfrak {m}}_{y}}{r_{0}}}\right)+\left({\frac {\partial }{\partial z}}\right)'\left({\frac {{\mathfrak {m}}_{z}}{r_{0}}}\right)=S.$

It is then

${\mathfrak {F}}_{x}=V^{2}\left\{\left({\frac {\partial S}{\partial x}}\right)'-\Delta '\left({\frac {{\mathfrak {m}}_{x}}{r_{0}}}\right)\right\}{\text{, etc.,}}$

(40)

and namely, ${\mathfrak {m}}_{x}$, ${\mathfrak {m}}_{y}$, ${\mathfrak {m}}_{z}$ are still related to the instant given above.

As to how the other magnitudes occurring in (I_b)-(VII_b) can be determined, can immediately be seen.

§ 34. Just some words on the error committed in the above calculation. That in (38) the factor ${\tfrac {1}{r}}$ was replaced by ${\tfrac {1}{r_{0}}}$, needs surely no justification. But we also haven't taken the values of $\rho {\mathfrak {v}}$ for the function f at the the correct times. Once we have replaced $t'-{\tfrac {r}{V}}$ by $t'-{\tfrac {r_{0}}{V}}$ in (38), then in the time when l is one of the