Putting
d
x
=
δ
x
{\displaystyle dx=\delta x}
and extracting the square root, we obtain an invariant
Θ
d
x
d
y
d
z
d
t
{\displaystyle {\sqrt {\Theta }}dx\ dy\ dz\ dt}
(18)
Now, let
B
x
d
y
d
z
+
B
y
d
z
d
x
+
B
z
d
x
d
y
+
E
x
d
x
d
t
+
E
y
d
y
d
t
+
E
z
d
y
d
t
,
{\displaystyle B_{x}dy\ dz+B_{y}dz\ dx+B_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dy\ dt,}
be an invariant of the second order. Multiplying it by (12) and rejecting a factor
Θ
d
x
d
y
d
z
d
t
{\displaystyle {\sqrt {\Theta }}dx\ dy\ dz\ dt}
, we obtain an invariant
D
x
δ
y
δ
z
+
D
y
δ
z
δ
x
+
D
z
δ
x
δ
y
−
H
x
δ
x
δ
t
−
H
y
δ
y
δ
t
−
H
z
δ
z
δ
t
,
{\displaystyle D_{x}\delta y\ \delta z+D_{y}\delta z\ \delta x+D_{z}\delta x\ \delta y-H_{x}\delta x\ \delta t-H_{y}\delta y\ \delta t-H_{z}\delta z\ \delta t,}
where
Θ
D
x
=
κ
11
E
x
+
κ
12
E
y
+
κ
13
E
z
+
κ
14
B
x
+
κ
15
B
y
+
κ
16
B
z
,
…
…
…
…
…
…
…
−
Θ
H
x
=
κ
41
E
x
+
κ
42
E
y
+
κ
43
E
z
+
κ
44
B
x
+
κ
45
B
y
+
κ
46
B
z
.
{\displaystyle {\begin{array}{c}{\sqrt {\Theta }}D_{x}=\kappa _{11}E_{x}+\kappa _{12}E_{y}+\kappa _{13}E_{z}+\kappa _{14}B_{x}+\kappa _{15}B_{y}+\kappa _{16}B_{z},\\\dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \dots \qquad \\-{\sqrt {\Theta }}H_{x}=\kappa _{41}E_{x}+\kappa _{42}E_{y}+\kappa _{43}E_{z}+\kappa _{44}B_{x}+\kappa _{45}B_{y}+\kappa _{46}B_{z}.\end{array}}}
The relation between the two invariants will be a mutual one if the coefficients
κ
11
,
…
{\displaystyle \kappa _{11},\ \dots }
. are the elements of an orthogonal matrix.
The Invariants of a Spherical Wave Transformation.
Starting from the fundamental invariants
d
τ
2
=
λ
2
[
d
t
2
−
d
x
2
−
d
y
2
−
d
z
2
]
,
{\displaystyle d\tau ^{2}=\lambda ^{2}\left[dt^{2}-dx^{2}-dy^{2}-dz^{2}\right],}
(1)
λ
4
d
x
d
y
d
z
d
t
,
{\displaystyle \lambda ^{4}dx\ dy\ dz\ dt,}
(2)
A
x
d
x
+
A
y
d
y
+
A
z
d
z
−
Φ
d
t
,
{\displaystyle A_{x}dx+A_{y}dy+A_{z}dz-\Phi dt,}
(3)
H
x
d
y
d
z
+
H
y
d
z
d
x
+
H
z
d
x
d
y
+
E
x
d
x
d
t
+
E
y
d
y
d
t
+
E
z
d
z
d
t
{\displaystyle H_{x}dy\ dz+H_{y}dz\ dx+H_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dz\ dt}
(4)
E
x
d
y
d
z
+
E
y
d
z
d
x
+
E
z
d
x
d
y
−
H
x
d
x
d
t
−
H
y
d
y
d
t
−
H
z
d
z
d
t
{\displaystyle E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt}
(5)
ρ
w
x
d
y
d
z
d
t
+
ρ
w
y
d
z
d
x
d
t
+
ρ
w
z
d
x
d
y
d
t
−
ρ
d
x
d
y
d
z
{\displaystyle \rho w_{x}dy\ dz\ dt+\rho w_{y}dz\ dx\ dt+\rho w_{z}dx\ dy\ dt-\rho dx\ dy\ dz}
(6)
we may obtain a number of others by the methods of multiplication and reciprocation. It will be sufficient to enumerate these if we mention the equations from which they are derived,
(
1
a
n
d
6
)
1
λ
2
[
ρ
w
x
d
x
+
ρ
w
y
d
y
+
ρ
w
z
d
z
−
ρ
d
t
]
,
(
7
)
(
5
a
n
d
4
)
(
E
x
2
+
E
y
2
+
E
z
2
−
H
x
2
−
H
y
2
−
H
z
2
)
d
x
d
y
d
z
d
t
(
8
)
(
4
a
n
d
4
)
(
E
x
H
x
+
E
y
H
y
+
E
z
H
z
)
d
x
d
y
d
z
d
t
(
9
)
(
3
a
n
d
6
)
ρ
[
A
x
w
x
+
A
y
w
y
+
A
z
w
z
−
Φ
]
d
x
d
y
d
z
d
t
(
10
)
(
6
a
n
d
7
)
ρ
2
λ
2
(
1
−
w
2
)
d
x
d
y
d
z
d
t
,
(
11
)
(
5
a
n
d
7
)
ρ
λ
2
[
(
E
x
−
w
z
H
y
+
w
y
H
z
)
d
y
d
z
d
t
+
(
E
y
−
w
x
H
z
+
w
z
H
x
)
d
z
d
x
d
t
(
12
)
+
(
E
z
−
w
y
H
z
+
w
x
H
y
)
d
x
d
y
d
t
−
(
w
x
E
x
+
w
y
E
y
+
w
z
E
z
)
d
x
d
y
d
z
]
,
(
12
)
ρ
λ
4
[
(
E
x
+
w
y
H
z
−
w
z
H
y
)
δ
x
+
(
E
y
+
w
z
H
x
−
w
x
H
z
)
δ
y
(
13
)
+
(
E
z
+
w
z
H
y
−
w
y
H
x
)
δ
z
−
(
w
x
E
x
+
w
y
E
y
+
w
z
E
z
)
δ
t
]
,
(
13
a
n
d
2
)
ρ
d
x
d
y
d
z
d
t
[
(
E
x
+
w
y
H
z
−
w
z
H
y
)
δ
x
+
(
E
y
+
w
z
H
x
−
w
x
H
z
)
δ
y
(
14
)
+
(
E
z
+
w
x
H
y
−
w
y
H
x
)
δ
z
−
(
w
x
E
x
+
w
y
E
y
+
w
z
E
z
)
δ
t
]
.
{\displaystyle {\begin{array}{ccc}(1\ {\mathsf {and}}\ 6)&{\frac {1}{\lambda ^{2}}}\left[\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt\right],&(7)\\\\(5\ {\mathsf {and}}\ 4)&\left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt&(8)\\\\(4\ {\mathsf {and}}\ 4)&\left(E_{x}H_{x}+E_{y}H_{y}+E_{z}H_{z}\right)dx\ dy\ dz\ dt&(9)\\\\(3\ {\mathsf {and}}\ 6)&\rho \left[A_{x}w_{x}+A_{y}w_{y}+A_{z}w_{z}-\Phi \right]dx\ dy\ dz\ dt&(10)\\\\(6\ {\mathsf {and}}\ 7)&{\frac {\rho ^{2}}{\lambda ^{2}}}\left(1-w^{2}\right)dx\ dy\ dz\ dt,&(11)\\\\(5\ {\mathsf {and}}\ 7)&{\frac {\rho }{\lambda ^{2}}}\left[\left(E_{x}-w_{z}H_{y}+w_{y}H_{z}\right)dy\ dz\ dt+\left(E_{y}-w_{x}H_{z}+w_{z}H_{x}\right)dz\ dx\ dt\right.&(12)\\&\left.+\left(E_{z}-w_{y}H_{z}+w_{x}H_{y}\right)dx\ dy\ dt-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)dx\ dy\ dz\right],\\\\(12)&{\frac {\rho }{\lambda ^{4}}}\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.&(13)\\&\left.+\left(E_{z}+w_{z}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right],\\\\(13\ {\mathsf {and}}\ 2)&\rho \ dx\ dy\ dz\ dt\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right.&(14)\\&\left.+\left(E_{z}+w_{x}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right].\end{array}}}