This is why Abraham gave
d
D
d
V
{\displaystyle {\tfrac {dD}{dV}}}
the name longitudinal mass and
D
V
{\displaystyle {\tfrac {D}{V}}}
the name transverse mass ; recall that
D
=
d
H
d
V
.
{\displaystyle D={\tfrac {dH}{dV}}.}
In the hypothesis of Lorentz , we have:
D
=
−
d
H
d
V
=
−
∂
H
∂
V
,
{\displaystyle D=-{\frac {dH}{dV}}=-{\frac {\partial H}{\partial V}},}
∂
H
∂
V
{\displaystyle {\tfrac {\partial H}{\partial V}}}
represent the derivative with respect to V, after r and θ were replaced by their values as functions of V from the first two equations (1); we will also have, after the substitution,
H
=
+
A
1
−
V
2
.
{\displaystyle H=+A{\sqrt {1-V^{2}}}.}
We choose units so that the constant factor A is equal to 1, and I pose
1
−
V
2
=
h
{\displaystyle {\sqrt {1-V^{2}}}=h}
, hence:
H
=
+
h
,
D
=
V
h
,
d
D
d
V
=
h
−
3
,
d
D
d
V
1
V
2
−
D
V
3
=
h
−
3
.
{\displaystyle H=+h,\quad D={\frac {V}{h}},\quad {\frac {dD}{dV}}=h^{-3},\quad {\frac {dD}{dV}}{\frac {1}{V^{2}}}-{\frac {D}{V^{3}}}=h^{-3}.}
We will pose again:
M
=
V
d
V
d
t
=
∑
ξ
d
ξ
d
t
,
X
1
=
∫
X
d
τ
{\displaystyle M=V{\frac {dV}{dt}}=\sum \xi {\frac {d\xi }{dt}},\quad X_{1}=\int Xd\tau }
and we find the equation for quasi-stationary motion:
(5)
h
−
1
d
ξ
d
t
+
h
−
3
ξ
M
=
X
1
.
{\displaystyle h^{-1}{\frac {d\xi }{dt}}+h^{-3}\xi M=X_{1}.}
Let's see what happens to these equations by the Lorentz transformation. We will pose:
1
+
ξ
ϵ
=
μ
{\displaystyle 1+\xi \epsilon =\mu }
, and we have first:
μ
ξ
′
=
ξ
+
ϵ
,
μ
η
′
=
η
k
,
μ
ζ
′
=
ζ
k
{\displaystyle \mu \xi ^{\prime }=\xi +\epsilon ,\quad \mu \eta ^{\prime }={\frac {\eta }{k}},\quad \mu \zeta ^{\prime }={\frac {\zeta }{k}}}
from which we derive easily
μ
h
′
=
h
k
.
{\displaystyle \mu h^{\prime }={\frac {h}{k}}.}
We also have
d
t
′
=
k
μ
d
t
{\displaystyle dt'=k\ \mu \ dt}
where:
d
ξ
′
d
t
′
=
d
ξ
d
t
1
k
3
μ
3
,
d
η
′
d
t
′
=
d
η
d
t
1
k
2
μ
2
−
d
ξ
d
t
η
ϵ
k
2
μ
3
,
d
ζ
′
d
t
′
=
d
ζ
d
t
1
k
2
μ
2
−
d
ξ
d
t
ζ
ϵ
k
2
μ
3
{\displaystyle {\frac {d\xi ^{\prime }}{dt^{\prime }}}={\frac {d\xi }{dt}}{\frac {1}{k^{3}\mu ^{3}}},\quad {\frac {d\eta ^{\prime }}{dt^{\prime }}}={\frac {d\eta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\eta \epsilon }{k^{2}\mu ^{3}}},\quad {\frac {d\zeta ^{\prime }}{dt^{\prime }}}={\frac {d\zeta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\zeta \epsilon }{k^{2}\mu ^{3}}}}
where again:
M
′
=
d
ξ
d
t
ϵ
h
2
k
3
μ
4
+
M
k
3
μ
3
{\displaystyle M^{\prime }={\frac {d\xi }{dt}}{\frac {\epsilon h^{2}}{k^{3}\mu ^{4}}}+{\frac {M}{k^{3}\mu ^{3}}}}
and
(6)
h
′
−
1
d
ξ
′
d
t
′
+
h
′
−
3
ξ
′
M
′
=
[
h
−
1
d
ξ
d
t
+
h
−
3
(
ξ
+
ϵ
)
M
]
μ
−
1
,
{\displaystyle h^{\prime -1}{\frac {d\xi ^{\prime }}{dt^{\prime }}}+h^{\prime -3}\xi ^{\prime }M^{\prime }=\left[h^{-1}{\frac {d\xi }{dt}}+h^{-3}(\xi +\epsilon )M\right]\mu ^{-1},}