Page:Elektrische und Optische Erscheinungen (Lorentz) 114.jpg

Second, as regards the work, we don't have to care about the last member in equation (15) and the analogous formulas.^[1] Only the "tensions" come into account, and

$S\ d\sigma \ dt$

is the work of the tension with respect to $d\sigma$. The components of this tension are

$\left\{2\pi V^{2}(2{\mathfrak {d}}_{x}{\mathfrak {d}}-\alpha {\mathfrak {d}}^{2})+{\frac {1}{8\pi }}(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right\}d\sigma$, etc.,

from which it follows

$S=2\pi V^{2}\left\{2{\mathfrak {d}}_{n}({\mathfrak {p}}_{x}{\mathfrak {d}}_{x}+{\mathfrak {p}}_{y}{\mathfrak {d}}_{y}+{\mathfrak {p}}_{z}{\mathfrak {d}}_{z})-{\mathfrak {p}}_{n}{\mathfrak {d}}^{2}\right\}+$ $+{\frac {1}{8\pi }}\left\{2{\mathfrak {H}}_{n}({\mathfrak {p}}_{x}{\mathfrak {H}}_{x}+{\mathfrak {p}}_{y}{\mathfrak {H}}_{y}+{\mathfrak {p}}_{z}{\mathfrak {H}}_{z})-{\mathfrak {p}}_{n}{\mathfrak {H}}^{2}\right\}$,

Eventually, A means the sum of the two last members in (107).

The given values now actually satisfy the condition (109).

1. Namely, to calculate the work, we can multiply the path ${\mathfrak {p}}T$ with the average of the force acting in its direction. This average would be zero for the last member in (15), when the surface σ would be displace together with the body, from which it follows, that it is in reality of order ${\mathfrak {p}}$.