Note that
ξ
′
=
ξ
+
ϵ
1
+
ξ
ϵ
,
η
′
=
η
k
(
1
+
ξ
ϵ
)
{\displaystyle \xi ^{\prime }={\frac {\xi +\epsilon }{1+\xi \epsilon }},\quad \eta ^{\prime }={\frac {\eta }{k\left(1+\xi \epsilon \right)}}}
It follows, by replacing δt' by its value
k
l
(
1
+
ξ
ϵ
)
δ
U
=
δ
U
′
(
1
+
ξ
ϵ
)
+
(
ξ
+
ϵ
)
k
l
ϵ
δ
U
,
{\displaystyle kl\left(1+\xi \epsilon \right)\delta U=\delta U^{\prime }\left(1+\xi \epsilon \right)+\left(\xi +\epsilon \right)kl\epsilon \delta U,}
l
(
1
+
ξ
ϵ
)
δ
V
=
δ
V
′
(
1
+
ξ
ϵ
)
+
η
l
ϵ
δ
U
.
{\displaystyle l\left(1+\xi \epsilon \right)\delta V=\delta V^{\prime }\left(1+\xi \epsilon \right)+\eta l\epsilon \delta U.}
If we recall the definition of k , we draw from this:
δ
U
=
k
l
δ
U
′
+
k
ϵ
l
ξ
δ
U
′
,
{\displaystyle \delta U={\frac {k}{l}}\delta U^{\prime }+{\frac {k\epsilon }{l}}\xi \delta U^{\prime },}
δ
V
=
1
l
δ
V
′
+
k
ϵ
l
η
δ
U
′
,
{\displaystyle \delta V={\frac {1}{l}}\delta V^{\prime }+{\frac {k\epsilon }{l}}\eta \delta U^{\prime },}
and also
δ
W
=
1
l
δ
W
′
+
k
ϵ
l
ζ
δ
U
′
;
{\displaystyle \delta W={\frac {1}{l}}\delta W^{\prime }+{\frac {k\epsilon }{l}}\zeta \delta U^{\prime };}
hence
(3)
∑
X
δ
U
=
1
l
(
k
X
δ
U
′
+
Y
δ
V
′
+
Z
δ
W
′
)
+
k
ϵ
l
δ
U
′
∑
X
ξ
.
{\displaystyle \sum X\delta U={\frac {1}{l}}\left(kX\delta U^{\prime }+Y\delta V^{\prime }+Z\delta W^{\prime }\right)+{\frac {k\epsilon }{l}}\delta U^{\prime }\sum X\xi .}
Now, in virtue of equations (2) we must have:
∫
∑
X
′
δ
U
′
d
t
′
d
τ
′
=
∫
∑
X
δ
U
d
t
d
τ
=
1
l
4
∑
X
δ
U
d
t
′
d
τ
′
.
{\displaystyle \int \sum X^{\prime }\delta U^{\prime }dt^{\prime }\ d\tau ^{\prime }=\int \sum X\delta U\ dt\ d\tau ={\frac {1}{l^{4}}}\sum X\delta U\ dt^{\prime }\ d\tau ^{\prime }.}
By replacing ΣXδU by its value (3) and by identifying, it follows:
X
′
=
k
l
5
X
+
k
ϵ
l
5
∑
X
ξ
,
Y
′
=
1
l
5
Y
,
Z
′
=
1
l
5
Z
.
{\displaystyle X^{\prime }={\frac {k}{l^{5}}}X+{\frac {k\epsilon }{l^{5}}}\sum X\xi ,\quad Y^{\prime }={\frac {1}{l^{5}}}Y,\quad Z^{\prime }={\frac {1}{l^{5}}}Z.}
These are the equations (11) of § 1. The principle of least action leads us to the same result as the analysis of § 1.
If we turn to formulas (1), we see that Σf² - Σα² is not affected by the Lorentz transformation, except one constant factor; it is not the case with expression Σf² + Σα² which represents the energy. If we confine ourselves to the case where ε is sufficiently small, so that the square can be neglected so that k = 1, and if we also assume l = 1, we find:
∑
f
′
2
=
∑
f
2
+
2
ϵ
(
g
γ
−
h
β
)
,
{\displaystyle \sum f^{\prime 2}=\sum f^{2}+2\epsilon \left(g\gamma -h\beta \right),}
∑
α
′
2
=
∑
α
2
+
2
ϵ
(
g
γ
−
h
β
)
,
{\displaystyle \sum \alpha ^{\prime 2}=\sum \alpha ^{2}+2\epsilon \left(g\gamma -h\beta \right),}
or by addition
∑
f
′
2
+
∑
α
′
2
=
∑
f
2
+
∑
α
2
+
4
ϵ
(
g
γ
−
h
β
)
.
{\displaystyle \sum f^{\prime 2}+\sum \alpha ^{\prime 2}=\sum f^{2}+\sum \alpha ^{2}+4\epsilon \left(g\gamma -h\beta \right).}
§ 4. — The Lorentz group
It is important to note that the Lorentz transformations form a group.
Indeed, if we set:
x
′
=
k
l
(
x
+
ϵ
t
)
,
y
′
=
l
y
,
z
′
=
l
z
,
t
′
=
k
l
(
t
+
ϵ
x
)
,
{\displaystyle x^{\prime }=kl(x+\epsilon t),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=kl(t+\epsilon x),}