This is by (26), (14) and (13):
∂
H
′
∂
x
˙
=
∂
∂
x
˙
(
H
′
c
2
−
q
′
2
c
2
−
q
2
)
=
c
c
2
−
v
2
c
2
−
v
x
˙
∂
H
∂
x
˙
+
v
c
c
2
−
v
2
(
c
2
−
v
x
˙
)
2
H
{\displaystyle {\frac {\partial H'}{\partial {\dot {x}}}}={\frac {\partial }{\partial {\dot {x}}}}\left(H'{\sqrt {\frac {c^{2}-q'^{2}}{c^{2}-q{}^{2}}}}\right)={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial {\dot {x}}}}+{\frac {vc{\sqrt {c^{2}-v^{2}}}}{(c^{2}-v{\dot {x}})^{2}}}H}
∂
H
′
∂
y
˙
=
c
c
2
−
v
2
c
2
−
v
x
˙
∂
H
∂
y
˙
,
∂
H
′
∂
z
˙
=
c
c
2
−
v
2
c
2
−
v
x
˙
∂
H
∂
z
˙
{\displaystyle {\frac {\partial H'}{\partial {\dot {y}}}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial {\dot {y}}}},\ {\frac {\partial H'}{\partial {\dot {z}}}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial {\dot {z}}}}}
∂
H
′
∂
V
=
c
c
2
−
v
2
c
2
−
v
x
˙
∂
H
∂
V
,
∂
H
′
∂
T
=
c
c
2
−
v
2
c
2
−
v
x
˙
∂
H
∂
T
{\displaystyle {\frac {\partial H'}{\partial V}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial V}},\ {\frac {\partial H'}{\partial T}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}{\frac {\partial H}{\partial T}}}
∂
x
˙
∂
x
˙
′
=
(
c
2
−
v
x
˙
)
2
c
2
(
c
2
−
v
2
)
,
∂
y
˙
∂
x
˙
′
=
−
v
y
˙
(
c
2
−
v
x
˙
)
c
2
(
c
2
−
v
2
)
,
∂
x
˙
∂
x
˙
′
=
−
v
z
˙
(
c
2
−
v
x
˙
)
c
2
(
c
2
−
v
2
)
{\displaystyle {\frac {\partial {\dot {x}}}{\partial {\dot {x}}'}}={\frac {(c^{2}-v{\dot {x}})^{2}}{c^{2}(c^{2}-v^{2})}},\ {\frac {\partial {\dot {y}}}{\partial {\dot {x}}'}}=-{\frac {v{\dot {y}}(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}},\ {\frac {\partial {\dot {x}}}{\partial {\dot {x}}'}}=-{\frac {v{\dot {z}}(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}}}
,
∂
V
∂
x
˙
′
=
−
v
(
c
2
−
v
x
˙
)
c
2
(
c
2
−
v
2
)
V
,
∂
T
∂
x
˙
′
=
−
v
(
c
2
−
v
x
˙
)
c
2
(
c
2
−
v
2
)
T
{\displaystyle {\frac {\partial V}{\partial {\dot {x}}'}}=-{\frac {v(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}}V,\ {\frac {\partial T}{\partial {\dot {x}}'}}=-{\frac {v(c^{2}-v{\dot {x}})}{c^{2}(c^{2}-v^{2})}}T}
.
This is given by substitution with respect to (8) and (7):
G
x
′
′
=
1
c
c
2
−
v
2
{
(
c
2
−
v
x
˙
)
G
x
+
v
H
−
v
y
˙
G
y
−
v
z
˙
G
z
−
v
p
V
−
v
T
S
}
{\displaystyle {\mathfrak {G}}'_{x'}={\frac {1}{c{\sqrt {c^{2}-v^{2}}}}}\{(c^{2}-v{\dot {x}}){\mathfrak {G}}_{x}+vH-v{\dot {y}}{\mathfrak {G}}_{y}-v{\dot {z}}{\mathfrak {G}}_{z}-vpV-vTS\}}
or from the introduction of the energy E (10):
G
x
′
′
=
c
c
2
−
v
2
(
G
x
−
v
(
E
+
p
V
)
c
2
)
{\displaystyle {\mathfrak {G}}'_{x'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {G}}_{x}-{\frac {v(E+pV)}{c^{2}}}\right)}
.
(29)
If we introduce instead of the energy E the "thermal function at constant pressure" R by Gibbs:
R
=
E
+
p
V
,
{\displaystyle R=E+pV,}
(30)
whose variation in isobaric processes describes the supplied heat, then the last relation is simply given by:
G
x
′
′
=
c
c
2
−
v
2
(
G
x
−
v
c
2
R
)
{\displaystyle {\mathfrak {G}}'_{x'}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left({\mathfrak {G}}_{x}-{\frac {v}{c^{2}}}R\right)}
.
(31)
By differentiating the equation (29) by time t :
d
G
x
′
′
d
t
=
d
G
x
′
′
d
t
′
⋅
d
t
d
t
=
c
c
2
−
v
2
{
d
G
x
d
t
−
v
c
2
(
d
E
d
t
+
p
p
V
d
t
+
V
d
p
d
t
)
}
{\displaystyle {\frac {d{\mathfrak {G}}'_{x'}}{dt}}={\frac {d{\mathfrak {G}}'_{x'}}{dt'}}\cdot {\frac {dt}{dt}}={\frac {c}{\sqrt {c^{2}-v^{2}}}}\left\{{\frac {d{\mathfrak {G}}{}_{x}}{dt}}-{\frac {v}{c^{2}}}\left({\frac {dE}{dt}}+p{\frac {pV}{dt}}+V{\frac {dp}{dt}}\right)\right\}}
,
the relation between the x-components of the force
F
{\displaystyle {\mathfrak {F}}}
follows with consideration of (27), (20), (14) and (11), namely:
F
x
′
′
=
F
x
−
v
c
2
−
v
x
˙
(
F
y
y
˙
+
F
z
z
˙
+
V
p
˙
+
T
S
˙
)
{\displaystyle {\mathfrak {F}}'_{x'}={\mathfrak {F}}_{x}-{\frac {v}{c^{2}-v{\dot {x}}}}({\mathfrak {F}}_{y}{\dot {y}}+{\mathfrak {F}}_{z}{\dot {z}}+V{\dot {p}}+T{\dot {S}})}
.
(32)
Comparing this relation with the one found above (21), it follows that those have no general meaning, but only apply if
p
˙
=
0
{\displaystyle {\dot {p}}=0}
and
S
˙
=
0
{\displaystyle {\dot {S}}=0}
, that is, when the process runs isobaric and adiabatic. In fact, this property is