Using (4) and (5) this becomes
δ
P
x
=
∑
e
x
δ
a
=
δ
p
x
,
…
S
i
m
i
l
a
r
l
y
,
δ
P
x
=
∑
e
Y
δ
A
=
∑
e
δ
a
{
β
(
1
−
v
w
x
c
2
)
y
+
β
v
w
y
x
c
2
}
=
β
(
1
−
v
w
x
c
2
)
δ
p
y
+
β
v
w
y
c
2
δ
p
x
,
…
a
n
d
δ
P
z
=
β
(
1
−
v
w
x
c
2
)
δ
p
z
+
β
v
w
z
c
2
δ
p
x
,
…
}
.
{\displaystyle \left.{\begin{array}{cll}&\delta P_{x}&={\frac {\sum ex}{\delta a}}=\delta p_{x},\dots \\\\{\mathsf {Similarly,}}&\delta P_{x}&={\frac {\sum eY}{\delta A}}\\\\&&={\frac {\sum e}{\delta a}}\left\{\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)y+{\frac {\beta vw_{y}x}{c^{2}}}\right\}\\\\&&=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{y}+{\frac {\beta vw_{y}}{c^{2}}}\delta p_{x},\dots \\\\{\mathsf {and}}&\delta P_{z}&=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{z}+{\frac {\beta vw_{z}}{c^{2}}}\delta p_{x},\dots \end{array}}\right\}.}
(a)
Since the polarization electrons have no velocity relative to the body, they contribute nothing to the magnetization in either system.
Consider next the magnetization electrons , that is, systems of electrons spinning round centres fixed in the body and having a magnetic moment; possibly also an electric moment in a moving system.
If these have an electric moment, consider the contribution to the polarization
δ
P
X
=
∑
e
X
δ
A
=
∑
e
δ
a
β
(
1
−
v
w
x
′
c
2
)
X
=
∑
e
x
δ
a
,
{\displaystyle \delta P_{X}={\frac {\sum eX}{\delta A}}={\frac {\sum e}{\delta a}}\beta \left(1-{\frac {vw'_{x}}{c^{2}}}\right)X={\frac {\sum ex}{\delta a}},}
where we consider in the first place only those electrons having a definite value of
w
x
′
{\displaystyle w'_{x}}
Similarly
δ
P
Y
{\displaystyle \delta P_{Y}}
i.e. ,
∑
e
Y
/
δ
A
{\displaystyle \sum eY/\delta A}
β
(
1
−
v
w
x
c
2
)
δ
p
y
+
β
v
w
y
c
2
δ
p
x
+
β
v
c
δ
m
z
,
{\displaystyle \beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{y}+{\frac {\beta vw_{y}}{c^{2}}}\delta p_{x}+{\frac {\beta v}{c}}\delta m_{z},}
and likewise
δ
P
z
=
β
(
1
−
v
w
x
c
2
)
δ
p
z
+
β
v
w
z
c
2
δ
p
x
−
β
v
c
δ
m
y
.
{\displaystyle \delta P_{z}=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)\delta p_{z}+{\frac {\beta vw_{z}}{c^{2}}}\delta p_{x}-{\frac {\beta v}{c}}\delta m_{y}.}
In the same manner for the same group of electrons
δ
M
X
=
1
2
c
∑
e
{
Y
(
W
Z
′
−
W
Z
)
−
Z
(
W
Y
′
−
W
Y
)
}
/
δ
A
,
{\displaystyle \delta M_{X}={\frac {1}{2c}}\sum e\left\{Y\left(W'_{Z}-W_{Z}\right)-Z\left(W'_{Y}-W_{Y}\right)\right\}/\delta A,}
which, on reduction, becomes
δ
m
x
−
(
β
v
W
Y
δ
m
y
+
β
v
W
Z
δ
m
z
)
/
c
2
,
{\displaystyle \delta m_{x}-\left(\beta vW_{Y}\delta m_{y}+\beta vW_{Z}\delta m_{z}\right)/c^{2},}
