we obtain Lorentz's transformation
x
′
=
x
+
v
t
1
−
v
2
,
y
′
=
y
,
z
′
=
z
,
t
′
=
t
+
v
x
1
−
v
2
.
{\displaystyle x'={\frac {x+vt}{\sqrt {1-v^{2}}}},\ y'=y,\ z'=z,\ t'={\frac {t+vx}{\sqrt {1-v^{2}}}}.}
The formulae of transformation of the electric and magnetic vectors are obtained at once from the general formulae; they are
A
x
=
β
(
A
x
′
−
v
Φ
′
)
,
−
Φ
=
β
(
v
A
x
′
−
Φ
′
)
,
A
y
=
A
y
′
,
A
z
=
A
z
′
,
E
x
=
E
x
′
,
H
x
=
H
x
′
,
E
y
=
β
(
E
y
′
−
v
H
z
′
)
,
H
y
=
β
(
H
y
′
−
v
E
z
′
)
,
E
z
=
β
(
E
z
′
−
v
H
y
′
)
,
H
z
=
β
(
H
z
′
−
v
E
y
′
)
,
{\displaystyle {\begin{array}{rlcrl}A_{x}=&\beta (A'_{x}-v\Phi '),&&-\Phi =&\beta (vA'_{x}-\Phi '),\\\\A_{y}=&A'_{y},&&A_{z}=&A'_{z},\\\\E_{x}=&E'_{x},&&H_{x}=&H'_{x},\\\\E_{y}=&\beta \left(E'_{y}-vH'_{z}\right),&&H_{y}=&\beta \left(H'_{y}-vE'_{z}\right),\\\\E_{z}=&\beta \left(E'_{z}-vH'_{y}\right),&&H_{z}=&\beta \left(H'_{z}-vE'_{y}\right),\end{array}}}
ρ
w
x
=
β
(
ρ
′
w
′
−
v
ρ
′
)
,
p
w
y
=
ρ
′
w
y
′
,
p
w
z
=
ρ
′
w
z
′
,
{\displaystyle \rho w_{x}=\beta (\rho 'w'-v\rho '),\ pw_{y}=\rho 'w'_{y},\ pw_{z}=\rho 'w'_{z},}
−
ρ
=
β
(
v
ρ
′
w
x
′
−
ρ
′
)
{\displaystyle -\rho =\beta (v\rho 'w'_{x}-\rho ')}
where
β
=
1
1
−
v
2
{\displaystyle \beta ={\frac {1}{\sqrt {1-v^{2}}}}}
These agree with the formulae of Einstein, Lorentz's formulae for the convection currents being slightly different.
In the case of an inversion with regard to a hypersphere whose centre is at the origin, and whose radius is a real quantity k , the formulas of transformation are
x
′
=
k
2
x
x
2
+
y
2
+
z
2
+
s
2
,
y
′
=
k
2
y
x
2
+
y
2
+
z
2
+
s
2
,
z
′
=
k
2
z
x
2
+
y
2
+
z
2
+
s
2
{\displaystyle x'={\frac {k^{2}x}{x^{2}+y^{2}+z^{2}+s^{2}}},\ y'={\frac {k^{2}y}{x^{2}+y^{2}+z^{2}+s^{2}}},\ z'={\frac {k^{2}z}{x^{2}+y^{2}+z^{2}+s^{2}}}}
s
′
=
k
2
s
x
2
+
y
2
+
z
2
+
s
2
{\displaystyle s'={\frac {k^{2}s}{x^{2}+y^{2}+z^{2}+s^{2}}}}
Putting
s
=
i
t
,
x
2
+
y
2
+
z
2
=
r
2
{\displaystyle s=it,\ x^{2}+y^{2}+z^{2}=r^{2}}
x
′
=
k
2
x
r
2
−
t
2
,
y
′
=
k
2
y
r
2
−
t
2
,
z
′
=
k
2
z
r
2
−
t
2
,
t
′
=
k
2
t
r
2
−
t
2
.
{\displaystyle x'={\frac {k^{2}x}{r^{2}-t^{2}}},\ y'={\frac {k^{2}y}{r^{2}-t^{2}}},\ z'={\frac {k^{2}z}{r^{2}-t^{2}}},\ t'={\frac {k^{2}t}{r^{2}-t^{2}}}.}
This transformation has a negative Jacobian; the formulae for transforming the components of the electric and magnetic force have been obtained by E. Cunningham.[1]
It should be noticed that
d
t
′
d
t
=
r
2
+
t
2
(
r
2
−
t
2
)
2
,
{\displaystyle {\frac {dt'}{dt}}={\frac {r^{2}+t^{2}}{\left(r^{2}-t^{2}\right)^{2}}},}
↑ Proc. London Math. Soc. , Ser, 2, Vol. 8, p. 77. 
