Page:Elektrische und Optische Erscheinungen (Lorentz) 097.jpg

in the direction ($b'_{x},b'_{y},b'_{z}$) within the moving body.

From (81) we find

${\frac {1}{W^{'2}}}={\frac {1}{W^{2}}}+2{\frac {b_{x}{\mathfrak {p}}_{x}+b_{y}{\mathfrak {p}}_{y}+b_{z}{\mathfrak {p}}_{z}}{W\ V^{2}}}$,

and for that we can write, neglecting magnitudes of second order,

${\frac {1}{W^{'2}}}={\frac {1}{W^{2}}}+2{\frac {b'_{x}{\mathfrak {p}}_{x}+b'_{y}{\mathfrak {p}}_{y}+b'_{z}{\mathfrak {p}}_{z}}{W\ V^{2}}}={\frac {1}{W^{2}}}+2{\frac {p_{n}}{W\ V^{2}}}$.

Here, ${\mathfrak {p}}_{n}$ is the component of velocity into the direction of the wave normal, with which $W'$ is related. Eventually

$W'=W-{\mathfrak {p}}_{n}{\frac {W^{2}}{V^{2}}}$

(82)

§ 69. Up to now, the investigation was general. Now it shall be assumed, that the body be isotropic. The velocity W is thus independent of the direction of the waves, and also the ratio

${\frac {V}{W}}=N$,

the absolute refraction index of the stationary body, only depends on T.

When interpreting formula (82), which now passes to

$W'=W-{\frac {{\mathfrak {p}}_{n}}{N^{2}}}$,

(83)

we have to remember, that we have used a coordinate system for the description of the phenomena, that moves together with ponderable matter. Thus (83) is the velocity of the light waves relative to that matter. If we wish it know the relative velocity $W''$ with respect to the aether, we have to compose the velocity (83), which has the direction of the wave normal, with the component ${\mathfrak {p}}_{n}$ of the translation velocity (that exactly falls in that direction). By that we obtain

$W''=W+\left(1-{\frac {1}{N^{2}}}\right){\mathfrak {p}}_{n}$,

(84)

which is in agreement with the known assumption of Fresnel.