Page:Elektrische und Optische Erscheinungen (Lorentz) 101.jpg

the oscillation period of the light source, then this is also the absolute oscillation period in P.

From that if follows for the relative oscillation period related to the water

$\left(1\pm {\frac {\mathfrak {p}}{W'}}\right)T$,

(88)

where $W'$ is exactly the sought velocity of the waves, while (as also in the following formulas) the above or below sign is to be applied, depending on whether the considered light bundle is propagating in the direction of the water motion, or in the opposite direction.

We always neglect magnitudes of second order and thus we may put instead of (88)

$\left(1\pm {\frac {\mathfrak {p}}{W}}\right)T$

(89)

Under $W$ in equation (82) — and also in this expression (89) itself — we have now to understand the value, which belong to period (89) in the stationary body. The corresponding refraction exponent is

$n\pm {\frac {\mathfrak {p}}{W}}T{\frac {dn}{dT}}$,

in case we denote the refraction exponent for period T by n; consequently we have to substitute

$W={\frac {V}{n\pm {\frac {\mathfrak {p}}{W}}T{\frac {dn}{dT}}}}={\frac {V}{n}}\mp {\frac {\mathfrak {p}}{n^{2}}}{\frac {V}{W}}T{\frac {dn}{dT}}$,

or, when we replace W by ${\frac {V}{n}}$ in the last member,

$W={\frac {V}{n}}\mp {\frac {\mathfrak {p}}{n}}T{\frac {dn}{dT}}$.

Furthermore it is in (82)

${\mathfrak {p}}_{n}=\pm {\mathfrak {p}}$,

so that we find

$W'={\frac {V}{n}}\mp {\frac {\mathfrak {p}}{n^{2}}}\mp {\frac {\mathfrak {p}}{n}}T{\frac {dn}{dT}}$,

and for the relative velocity with respect to the aether, thus also with respect to the closing plates of the tubes,

$W''={\frac {V}{n}}\mp {\mathfrak {p}}\left(1-{\frac {1}{n^{2}}}\right)\mp {\frac {\mathfrak {p}}{n}}T{\frac {dn}{dT}}$.

(90)