Page:Elektrische und Optische Erscheinungen (Lorentz) 046.jpg

As to the second variation, it should be noted that each element ds describes an infinitely small parallelogram on the surface, and that the value of the surface integral $\textstyle {\int {\dot {\overline {{\mathfrak {H}}_{n}}}}\ d\ \sigma }$ of this parallelogram, by suitably chosen signs, goes into dP. This value is determined by the area of the parallelepiped, with $d\ s$, ${\mathfrak {H}}$ as its sides, and the distance ${\mathfrak {v}}\ d\ t$ in the direction of ${\mathfrak {v}}$. We will find for it

$-d\ t[{\mathfrak {v.{\bar {H}}}}]_{s}\ d\ s{,}$

and for the whole increase of (33)

$d\ P=d\ t\int {\dot {\overline {{\mathfrak {H}}_{n}}}}\ d\ \sigma -d\ t\int [{\mathfrak {v.{\bar {H}}}}]_{s}\ d\ s{,}$

or, if the relations (IV_b) and (V_b), as well as the theorem stated in (1) (§ 4, h), were considered,

$-d\ t\int \left\{4\pi V^{2}{\bar {{\mathfrak {d}}_{s}}}+[{\mathfrak {p.{\bar {H}}}}]_{s}\right\}\ d\ s-d\ t\int [{\mathfrak {v.{\bar {H}}}}]_{s}\ d\ s.$

Consequently, (32) transforms into

$i=-C\int d\ P=C\left(P_{1}+P_{2}\right){,}$

where $P_{1}$ and $P_{2}$ belong to the beginning and the end of the considered time.

The magnitude P depends on the different parts of ${\mathfrak {H}}$. Since an induced current neither exists at the beginning nor at the end of time T, we commit no mistake when we substitute into (33) for ${\mathfrak {H}}$ only the magnetic force generated by the primary current. The prime above the letter can be omitted here, and if the induced wire is very thin, we may calculate for all current-lines with the same P. Finally, if $C_{1}$ is the sum of all numbers C (i.e., the conductivity of the induced electrical circuit), then the integral-current which we wished to calculate, becomes

$I=C_{1}\left(P_{1}-P_{2}\right){,}$

which is consistent with a known theorem.

The motion of Earth was never overlooked during the given derivation; consequently the formula admits of a conclusion about the influence of this motion on the phenomena of induction. There, only magnitudes of second order