Putting
(
d
x
′
)
2
+
(
d
y
′
)
2
+
(
d
z
′
)
2
−
(
d
t
′
)
2
=
λ
2
[
(
d
x
)
2
+
(
d
y
)
2
+
(
d
z
)
2
−
(
d
t
)
2
]
{\displaystyle (dx')^{2}+(dy')^{2}+(dz')^{2}-(dt')^{2}=\lambda ^{2}\left[(dx)^{2}+(dy)^{2}+(dz)^{2}-(dt)^{2}\right]}
we see that λ² is positive, and therefore
(
∂
x
′
∂
t
)
2
+
(
∂
y
′
∂
t
)
2
+
(
∂
z
′
∂
t
)
2
−
(
∂
t
′
∂
t
)
2
{\displaystyle \left({\frac {\partial x'}{\partial t}}\right)^{2}+\left({\frac {\partial y'}{\partial t}}\right)^{2}+\left({\frac {\partial z'}{\partial t}}\right)^{2}-\left({\frac {\partial t'}{\partial t}}\right)^{2}}
is negative. This shows that
∂
(
x
′
,
y
′
,
z
′
)
∂
(
x
,
y
,
z
)
{\displaystyle {\frac {\partial (x',y',z')}{\partial (x,y,z)}}}
∂
t
′
∂
t
∂
(
x
′
,
y
′
,
z
′
,
t
′
)
∂
(
x
,
y
,
z
,
t
)
{\displaystyle {\frac {\partial t'}{\partial t}}{\frac {\partial (x',y',z',t')}{\partial (x,y,z,t)}}}
∂
t
′
∂
t
{\displaystyle {\frac {\partial t'}{\partial t}}}
∂
(
x
′
,
y
′
,
z
′
)
∂
(
x
,
y
,
z
)
{\displaystyle {\frac {\partial (x',y',z')}{\partial (x,y,z)}}}
The sign of θ may now be determined from equation (A). Since
(
∂
x
′
∂
x
)
2
+
(
∂
y
′
∂
x
)
2
+
(
∂
z
′
∂
x
)
2
−
(
∂
t
′
∂
x
)
2
=
λ
2
{\displaystyle \left({\frac {\partial x'}{\partial x}}\right)^{2}+\left({\frac {\partial y'}{\partial x}}\right)^{2}+\left({\frac {\partial z'}{\partial x}}\right)^{2}-\left({\frac {\partial t'}{\partial x}}\right)^{2}=\lambda ^{2}}
it is necessarily positive. Consequently θ must have the same sign as
∂
(
x
′
,
y
′
,
z
′
)
∂
(
x
,
y
,
z
)
,
{\displaystyle {\frac {\partial (x',y',z')}{\partial (x,y,z)}},}
We can now obtain the formulae of transformation in the two possible cases.
(i) When the Jacobian is positive,
θ
=
+
1
{\displaystyle \theta =+1}
E
x
=
E
x
′
∂
(
y
′
,
z
′
)
∂
(
y
,
z
)
+
E
y
′
∂
(
z
′
,
x
′
)
∂
(
y
,
z
)
+
E
z
′
∂
(
x
′
,
y
′
)
∂
(
y
,
z
)
−
H
x
′
∂
(
x
′
,
t
′
)
∂
(
y
,
z
)
−
H
y
′
∂
(
y
′
,
t
′
)
∂
(
y
,
z
)
−
H
z
′
∂
(
z
′
,
t
′
)
∂
(
y
,
z
)
−
H
x
=
E
x
′
∂
(
y
′
,
z
′
)
∂
(
x
,
t
)
+
E
y
′
∂
(
z
′
,
x
′
)
∂
(
x
,
t
)
+
E
z
′
∂
(
x
′
,
y
′
)
∂
(
x
,
t
)
−
H
x
′
∂
(
x
′
,
t
′
)
∂
(
x
,
t
)
−
H
y
′
∂
(
y
′
,
t
′
)
∂
(
x
,
t
)
−
H
z
′
∂
(
z
′
,
t
′
)
∂
(
x
,
t
)
ρ
w
x
=
ρ
′
w
x
′
∂
(
y
′
,
z
′
,
t
′
)
∂
(
y
,
z
,
t
)
+
ρ
′
w
y
′
∂
(
z
′
,
x
′
,
t
′
)
∂
(
y
,
z
,
t
)
+
ρ
′
w
z
′
∂
(
x
′
,
y
′
,
t
′
)
∂
(
y
,
z
,
t
)
−
ρ
′
∂
(
x
′
,
y
′
,
z
′
)
∂
(
y
,
z
,
t
)
,
−
ρ
=
ρ
′
w
x
′
∂
(
y
′
,
z
′
,
t
′
)
∂
(
x
,
y
,
z
)
+
ρ
′
w
y
′
∂
(
z
′
,
x
′
,
t
′
)
∂
(
x
,
y
,
z
)
+
ρ
′
w
z
′
∂
(
x
′
,
y
′
,
t
′
)
∂
(
x
,
y
,
z
)
−
ρ
′
∂
(
x
′
,
y
′
,
z
′
)
∂
(
x
,
y
,
z
)
.
{\displaystyle {\begin{array}{rl}E_{x}=&E'_{x}{\frac {\partial (y',z')}{\partial (y,z)}}+E'_{y}{\frac {\partial (z',x')}{\partial (y,z)}}+E'_{z}{\frac {\partial (x',y')}{\partial (y,z)}}\\\\&\qquad -H'_{x}{\frac {\partial (x',t')}{\partial (y,z)}}-H'_{y}{\frac {\partial (y',t')}{\partial (y,z)}}-H'_{z}{\frac {\partial (z',t')}{\partial (y,z)}}\\\\-H_{x}=&E'_{x}{\frac {\partial (y',z')}{\partial (x,t)}}+E'_{y}{\frac {\partial (z',x')}{\partial (x,t)}}+E'_{z}{\frac {\partial (x',y')}{\partial (x,t)}}\\\\&\qquad -H'_{x}{\frac {\partial (x',t')}{\partial (x,t)}}-H'_{y}{\frac {\partial (y',t')}{\partial (x,t)}}-H'_{z}{\frac {\partial (z',t')}{\partial (x,t)}}\\\\\rho w_{x}=&\rho 'w'_{x}{\frac {\partial (y',z',t')}{\partial (y,z,t)}}+\rho 'w'_{y}{\frac {\partial (z',x',t')}{\partial (y,z,t)}}+\rho 'w'_{z}{\frac {\partial (x',y',t')}{\partial (y,z,t)}}-\rho '{\frac {\partial (x',y',z')}{\partial (y,z,t)}},\\\\-\rho =&\rho 'w'_{x}{\frac {\partial (y',z',t')}{\partial (x,y,z)}}+\rho 'w'_{y}{\frac {\partial (z',x',t')}{\partial (x,y,z)}}+\rho 'w'_{z}{\frac {\partial (x',y',t')}{\partial (x,y,z)}}-\rho '{\frac {\partial (x',y',z')}{\partial (x,y,z)}}.\end{array}}}
![{\displaystyle (dx')^{2}+(dy')^{2}+(dz')^{2}-(dt')^{2}=\lambda ^{2}\left[(dx)^{2}+(dy)^{2}+(dz)^{2}-(dt)^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b38bc8460f99ad9ab0a0d2b796668a12b68477d)
