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In a similar manner we derive from the continuity of ${\mathfrak {H}}{}_{n},\ {\mathfrak {E}}_{h}$ and ${\mathfrak {E}}_{k}$, by means of the relation to be derived from ($VI_{c}$)

${\mathfrak {H}}'_{n}={\mathfrak {H}}_{n}-{\frac {1}{V^{2}}}[{\mathfrak {p.E}}]_{n}={\mathfrak {H}}_{n}-{\frac {1}{V^{2}}}[{\mathfrak {p}}_{h}{\mathfrak {E}}_{k}-{\mathfrak {p}}_{k}{\mathfrak {E}}_{h}]$,

the continuity of ${\mathfrak {H}}'_{n}$.

If we also notice the other equations ($VIII_{c}$), then it is clear, that all limiting conditions are contained in the formulas

${\mathfrak {D}}'_{n(1)}={\mathfrak {D}}'_{n(2)},\ {\mathfrak {E}}{}_{h(1)}={\mathfrak {E}}{}_{h(2)},\ {\mathfrak {H}}'_{(1)}={\mathfrak {H}}'_{(2)}$,

$(VIII_{d})$

in which h can be now any arbitrary direction in the border surface.

§ 59. The equations $(I_{d})-(V_{d})$ and ($VIII_{d}$) differ from the equations which apply to stationary bodies by § 52, only by the fact that

$t',\ {\mathfrak {D}}'$ und ${\mathfrak {H}}'$

has taken the place of

$t,\ {\mathfrak {D}}$ and ${\mathfrak {H}}$

This coincidence opens for as a way, to treat problems regarding the influence of Earth's motion on optical phenomena, in a very simple way.

Namely, if a state of motion for a system of stationary bodies is known, where

${\mathfrak {D}}_{x},{\mathfrak {D}}_{y},{\mathfrak {D}}_{z},\ {\mathfrak {E}}_{x},{\mathfrak {E}}_{y},{\mathfrak {E}}_{z},\ {\mathfrak {H}}_{x},{\mathfrak {H}}_{y},{\mathfrak {H}}_{z}$.

(69)

are certain functions of x, y, z and t, then in the same system, if it is displaced by the velocity ${\mathfrak {p}}$, there can exist a state of motion, where

${\mathfrak {D}}'_{x},{\mathfrak {D}}'_{y},{\mathfrak {D}}'_{z},\ {\mathfrak {E}}'_{x},{\mathfrak {E}}'_{y},{\mathfrak {E}}'_{z},\ {\mathfrak {H}}'_{x},{\mathfrak {H}}'_{y},{\mathfrak {H}}'_{z}$.

(70)

are exactly the same functions of x, y, z and t' [that is, $t-{\frac {1}{V^{2}}}\left({\mathfrak {p}}_{x}x+{\mathfrak {p}}_{y}y+{\mathfrak {p}}_{z}z\right)$].

Although we have given (in the previous consideration) to the coordinate axes the directions of the symmetry axis, the derived theorem applies to any right-angled coordinate system. We can easily recognize this, when we consider, that for local time $t'$ it can also be written

$t-{\frac {{\mathfrak {p}}_{r}r}{V^{2}}}$,

where r is the line drawn from the coordinate origin to the point (x, y, z), and $t'$ is independent of the direction of the coordinate axes.