D
i
v
d
=
ρ
,
{\displaystyle Div\ {\mathfrak {d}}=\rho {,}}
(Ib )
D
i
v
H
=
0
,
{\displaystyle Div\ {\mathfrak {H}}=0{,}}
(IIb )
R
o
t
H
′
=
4
π
ρ
v
+
4
π
d
˙
,
{\displaystyle Rot\ {\mathfrak {H}}'=4\pi \rho {\mathfrak {v}}+4\pi {\dot {\mathfrak {d}}}{,}}
(IIIb )
R
o
t
F
=
−
H
˙
,
{\displaystyle Rot\ {\mathfrak {F}}=-{\dot {\mathfrak {H}}}{,}}
(IVb )
F
=
4
π
V
2
d
+
[
p
.
H
]
,
{\displaystyle {\mathfrak {F}}=4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {p.H}}]{,}}
(Vb )
H
′
=
H
−
4
π
[
p
.
d
]
,
{\displaystyle {\mathfrak {H}}'={\mathfrak {H}}-4\pi [{\mathfrak {p.d}}]{,}}
(VIb )
E
=
F
+
[
v
.
H
]
.
{\displaystyle {\mathfrak {E}}={\mathfrak {F}}+[{\mathfrak {v.H}}].}
(VIIb )
§ 21. From equations (Ia )—(Va ) (§ 19) also some formulas can be derived, any of them contains only one of the magnitudes
d
x
{\displaystyle {\mathfrak {d}}_{x}}
,
d
y
{\displaystyle {\mathfrak {d}}_{y}}
,
d
z
{\displaystyle {\mathfrak {d}}_{z}}
,
H
x
{\displaystyle {\mathfrak {H}}_{x}}
,
H
y
{\displaystyle {\mathfrak {H}}_{y}}
,
H
z
{\displaystyle {\mathfrak {H}}_{z}}
.
At first, it follows from (IVa )
−
4
π
V
2
R
o
t
R
o
t
d
=
R
o
t
(
∂
H
∂
t
)
1
=
(
∂
R
o
t
H
∂
t
)
1
.
{\displaystyle -4\pi V^{2}Rot\ Rot\ {\mathfrak {d}}=Rot\left({\frac {\partial {\mathfrak {H}}}{\partial t}}\right)_{1}=\left({\frac {\partial Rot\ {\mathfrak {H}}}{\partial t}}\right)_{1}.}
If we consider here what has been said in § 4, h , as well as the relations (Ia ), (IIIa ) and (4a ), we arrive at the three formulas
V
2
Δ
d
x
−
(
∂
2
d
x
∂
t
2
)
1
=
V
2
∂
ρ
∂
x
+
(
∂
∂
t
)
1
{
ρ
(
p
x
+
v
x
)
}
, etc.
{\displaystyle V^{2}\Delta {\mathfrak {d}}_{x}-\left({\frac {\partial ^{2}{\mathfrak {d}}_{x}}{\partial t^{2}}}\right)_{1}=V^{2}{\frac {\partial \rho }{\partial x}}+\left({\frac {\partial }{\partial t}}\right)_{1}\left\{\rho \left({\mathfrak {p}}_{x}+{\mathfrak {v}}_{x}\right)\right\}{\text{, etc.}}}
(A)
Similarly, we find
V
2
Δ
H
x
−
(
∂
2
H
x
∂
t
2
)
1
=
4
π
V
2
[
∂
∂
z
{
ρ
(
p
y
+
v
y
)
}
−
{\displaystyle V^{2}\Delta {\mathfrak {H}}_{x}-\left({\frac {\partial ^{2}{\mathfrak {H}}_{x}}{\partial t^{2}}}\right)_{1}=4\pi V^{2}\left[{\frac {\partial }{\partial z}}\left\{\rho \left({\mathfrak {p}}_{y}+{\mathfrak {v}}_{y}\right)\right\}-\right.}
−
∂
∂
y
{
ρ
(
p
z
+
v
z
)
}
]
, etc.
{\displaystyle \left.-{\frac {\partial }{\partial y}}\left\{\rho \left({\mathfrak {p}}_{z}+{\mathfrak {v}}_{z}\right)\right\}\right]{\text{, etc.}}}
(B)
The last members of these six equations are completely known once we know how the ions are moving.
Application to electrostatics.
§ 22. We want to calculate by which forces the ions act on one another, when all of them are at rest with respect to ponderable matter. In this case a state occurs, where at every point
d
{\displaystyle {\mathfrak {d}}}
and
H
{\displaystyle {\mathfrak {H}}}
are independent of time. We have
(
∂
∂
t
)
1
=
−
(
p
x
∂
∂
x
+
p
y
∂
∂
y
+
p
z
∂
∂
z
)
,
{\displaystyle \left({\frac {\partial }{\partial t}}\right)_{1}=-\left({\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}+{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}+{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}\right){,}}
(19)
and equations (A) and (B) will be reduced, when for brevity's sake the operation
Δ
−
1
V
2
(
p
x
∂
∂
x
+
p
y
∂
∂
y
+
p
z
∂
∂
z
)
2
{\displaystyle \Delta -{\frac {1}{V^{2}}}\left({\mathfrak {p}}_{x}{\frac {\partial }{\partial x}}+{\mathfrak {p}}_{y}{\frac {\partial }{\partial y}}+{\mathfrak {p}}_{z}{\frac {\partial }{\partial z}}\right)^{2}}