Taking into account the symmetry conditions:
Y
z
=
Z
y
,
Z
x
=
X
z
,
X
y
=
Y
x
,
{\displaystyle Y_{z}=Z_{y},\ Z_{x}=X_{z},\ X_{y}=Y_{x},}
we have
Y
z
′
−
Z
y
′
=
q
z
f
y
−
q
y
f
z
,
Z
x
′
−
X
z
′
=
q
x
f
z
−
q
z
f
x
,
X
y
′
−
Y
x
′
=
q
y
f
x
−
q
x
f
y
,
{\displaystyle {\begin{array}{c}Y'_{z}-Z'_{y}={\mathfrak {q}}_{z}{\mathfrak {f}}_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{z},\\Z'_{x}-X'_{z}={\mathfrak {q}}_{x}{\mathfrak {f}}_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{x},\\X'_{y}-Y'_{x}={\mathfrak {q}}_{y}{\mathfrak {f}}_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{y},\end{array}}}
relationships that are, as I demonstrated in the first paper, satisfied by the expressions given for relative pressures. Only to prove that, we set the function
2
Φ
′
=
X
x
′
x
2
+
Y
y
′
y
2
+
Z
z
′
z
2
+
(
Y
z
′
+
Z
y
′
)
y
z
+
(
Z
x
′
+
X
z
′
)
z
x
+
(
X
y
′
+
Y
x
′
)
x
y
{\displaystyle 2\Phi '=X'_{x}x^{2}+Y'_{y}y^{2}+Z'_{z}z^{2}+\left(Y'_{z}+Z'_{y}\right)yz+\left(Z'_{x}+X'_{z}\right)zx+\left(X'_{y}+Y'_{x}\right)xy}
equal to
2
Φ
′
=
2
Φ
+
x
2
q
x
f
x
+
y
2
q
y
f
y
+
z
2
q
z
f
z
+
(
q
y
f
z
+
q
z
f
y
)
y
z
+
(
q
z
f
x
+
q
x
f
z
)
z
x
+
(
q
x
f
y
+
q
y
f
x
)
x
y
,
{\displaystyle {\begin{array}{ll}2\Phi '&=2\Phi +x^{2}{\mathfrak {q}}_{x}{\mathfrak {f}}_{x}+y^{2}{\mathfrak {q}}_{y}{\mathfrak {f}}_{y}+z^{2}{\mathfrak {q}}_{z}{\mathfrak {f}}_{z}\\&+\left({\mathfrak {q}}_{y}{\mathfrak {f}}_{z}+{\mathfrak {q}}_{z}{\mathfrak {f}}_{y}\right)yz+\left({\mathfrak {q}}_{z}{\mathfrak {f}}_{x}+{\mathfrak {q}}_{x}{\mathfrak {f}}_{z}\right)zx+\left({\mathfrak {q}}_{x}{\mathfrak {f}}_{y}+{\mathfrak {q}}_{y}{\mathfrak {f}}_{x}\right)xy,\end{array}}}
namely
(27)
2
Φ
′
=
2
Φ
+
(
r
q
)
(
r
f
)
{\displaystyle 2\Phi '=2\Phi +({\mathfrak {rq}})({\mathfrak {rf}})}
and introducing the value (24a) of
2
Φ
{\displaystyle 2\Phi }
, which is an expression identical to the one resulting from the fundamental formulas (
V
a
{\displaystyle V_{a}}
) of the first paper. These finally give
(27a)
2
Φ
′
=
(
r
E
′
)
(
r
D
)
−
1
2
r
2
(
E
′
D
)
+
(
r
H
′
)
(
r
B
)
−
1
2
r
2
(
H
′
B
)
{\displaystyle 2\Phi '=({\mathfrak {rE}}')({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {E'D}})+({\mathfrak {rH}}')({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {H'B}})}
The identity of the values (27) and (27a) will be demonstrated, by proving that the relationship is satisfied:
(28)
{
(
r
q
)
(
r
f
)
+
(
r
q
)
(
r
W
)
=
(
r
,
E
′
−
E
)
(
r
D
)
+
(
r
,
H
′
−
H
)
(
r
B
)
−
1
2
r
2
{
(
E
′
−
E
,
D
)
+
(
H
′
−
H
,
B
)
}
{\displaystyle \left\{{\begin{array}{c}({\mathfrak {rq}})({\mathfrak {rf}})+({\mathfrak {rq}})({\mathfrak {rW}})\\\\=({\mathfrak {r,\ E'-E}})({\mathfrak {rD}})+({\mathfrak {r,\ H'-H}})({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}\left\{({\mathfrak {E'-E,D}})+({\mathfrak {H'-H,B}})\right\}\end{array}}\right.}
Taking account of (26c) and (25), we can write
(28a)
{
(
r
q
)
(
r
[
D
B
]
)
=
(
r
[
q
B
]
)
(
r
D
)
−
(
r
[
q
D
]
)
(
r
B
)
−
1
2
r
2
{
(
D
[
q
B
]
)
−
B
[
q
D
]
}
=
r
[
q
,
B
(
r
D
)
−
D
(
r
B
)
]
+
r
2
(
q
[
D
B
]
)
{\displaystyle {\begin{cases}({\mathfrak {rq}})({\mathfrak {r[DB]}})&=\left({\mathfrak {r[qB]}}\right)({\mathfrak {rD}})-\left({\mathfrak {r[qD]}}\right)({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}\left\{\left({\mathfrak {D[qB]}}\right)-{\mathfrak {B[qD]}}\right\}\\&={\mathfrak {r[q,\ B(rD)-D(rB)]+r^{2}(q[DB])}}\end{cases}}}
Now, it is identically
[
q
,
B
(
r
D
)
−
D
(
r
B
)
]
=
−
[
q
[
r
[
D
B
]
]
]
=
−
r
(
q
[
D
B
]
)
+
(
r
q
)
[
D
B
]
,
{\displaystyle {\mathfrak {[q,\ B(rD)-D(rB)]=-\left[q[r[DB]]\right]=-r(q[DB])+(rq)[DB],}}}
and the second part of equation (28a) gives in fact:
(
r
q
)
(
r
[
D
B
]
)
{\displaystyle ({\mathfrak {rq}})({\mathfrak {r[DB]}})}
so that the relationship (28) is identically satisfied. So, by formula (27a) which is postulated from our system of electrodynamics, the values of the pressures of Maxwell follow for the special case of the theory of Minkowski , which obeys the principle of relativity in agreement with relation (24a).