§ 3. We shall apply these equations to a system of bodies, having a common velocity of translation
p
{\displaystyle {\mathfrak {p}}}
, of constant direction and magnitude, the aether remaining at rest, and we shall henceforth denote by
v
{\displaystyle {\mathfrak {v}}}
, not the whole velocity of a material element, but the velocity it may have in addition to
p
{\displaystyle {\mathfrak {p}}}
.
Now it is natural to use a system of axes of coordinates, which partakes of the translation
p
{\displaystyle {\mathfrak {p}}}
. If we give to the axis of x the direction of the translation, so that
p
y
{\displaystyle {\mathfrak {p}}_{y}}
and
p
z
{\displaystyle {\mathfrak {p}}_{z}}
are 0, the equations (Ia)— (Va) will have to be replaced by
D
i
v
d
=
ϱ
{\displaystyle Div\ {\mathfrak {d}}=\varrho }
,
(Ib )
D
i
v
H
=
0
{\displaystyle Div\ {\mathfrak {H}}=0}
,
(IIb )
∂
H
z
∂
y
−
∂
H
y
∂
z
=
4
π
ϱ
(
p
x
+
v
x
)
+
4
π
(
∂
∂
t
−
p
x
∂
∂
x
)
d
x
,
∂
H
x
∂
z
−
∂
H
z
∂
x
=
4
π
ϱ
v
y
+
4
π
(
∂
∂
t
−
p
x
∂
∂
x
)
d
y
,
∂
H
y
∂
x
−
∂
H
x
∂
y
=
4
π
ϱ
v
z
+
4
π
(
∂
∂
t
−
p
x
∂
∂
x
)
d
z
,
}
{\displaystyle \left.{\begin{array}{l}{\frac {\partial {\mathfrak {H}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}=4\pi \varrho ({\mathfrak {p}}_{x}+{\mathfrak {v}}_{x})+4\pi \left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{x,}\\\\{\frac {\partial {\mathfrak {H}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {H}}_{z}}{\partial x}}=4\pi \varrho {\mathfrak {v}}_{y}+4\pi \left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{y},\\\\{\frac {\partial {\mathfrak {H}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}=4\pi \varrho {\mathfrak {v}}_{z}+4\pi \left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{z},\end{array}}\right\}}
(IIIb )
4
π
V
2
(
∂
d
z
∂
y
−
∂
d
y
∂
z
)
=
−
(
∂
∂
t
−
p
x
∂
∂
x
)
H
x
,
4
π
V
2
(
∂
d
x
∂
z
−
∂
d
z
∂
x
)
=
−
(
∂
∂
t
−
p
x
∂
∂
x
)
H
y
,
4
π
V
2
(
∂
d
y
∂
x
−
∂
d
x
∂
y
)
=
−
(
∂
∂
t
−
p
x
∂
∂
x
)
H
z
,
}
{\displaystyle \left.{\begin{array}{l}4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {d}}_{y}}{\partial z}}\right)=-\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {H}}_{x},\\\\4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {d}}_{z}}{\partial x}}\right)=-\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {H}}_{y},\\\\4\pi V^{2}\left({\frac {\partial {\mathfrak {d}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {d}}_{x}}{\partial y}}\right)=-\left({\frac {\partial }{\partial t}}-{\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}\right){\mathfrak {H}}_{z},\end{array}}\right\}}
(IVb )
E
=
4
π
V
2
d
+
[
p
.
H
]
+
[
v
.
H
]
{\displaystyle {\mathfrak {E}}=4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {p.H}}]+[{\mathfrak {v.H}}]}
(Vb )
In these formulae the sign Div , applied to a vector
A
{\displaystyle {\mathfrak {A}}}
, has still the meaning defined by
D
i
v
A
=
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle Div\ {\mathit {\mathfrak {A}}}={\frac {\partial {\mathfrak {A}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {A}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {A}}_{z}}{\partial z}}}
.
As has already been said,
v
{\displaystyle {\mathfrak {v}}}
is the relative velocity with regard to the moving axes of coordinates. If
v
=
0
{\displaystyle {\mathfrak {v}}=0}
, we shall speak of a system at rest; this expression therefore means relative rest with regard to the moving axes.