and
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
=
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 {\begin{array}{l}E_{x}dy\ dz+E_{y}dz\ dx+E_{z}dx\ dy-H_{x}dx\ dt-H_{y}dy\ dt-H_{z}dz\ dt\\\qquad =E'_{x}dy'dz'+E'_{y}dz'dx'+E'_{z}dx'dy'-H'_{x}dx'dt'-H'_{y}dy'dt'-H'_{z}dz'dt',\end{array}}}
we obtain, on multiplication,
(
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
=
(
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
′
,
{\displaystyle {\begin{array}{l}\left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt\\\qquad =\left(E_{x}^{'2}+E_{y}^{'2}+E_{z}^{'2}-H_{x}^{'2}-H_{y}^{'2}-H_{z}^{'2}\right)dx'dy'dz'dt',\end{array}}}
while, if either form be multiplied by itself, we obtain
(
E
x
H
x
+
E
y
H
y
+
E
z
H
z
)
d
x
d
y
d
z
d
t
=
(
E
x
′
H
x
′
+
E
y
′
H
y
′
+
E
z
′
H
z
′
)
d
x
′
d
y
′
d
z
′
d
t
′
.
{\displaystyle \left(E_{x}H_{x}+E_{y}H_{y}+E_{z}H_{z}\right)dx\ dy\ dz\ dt=\left(E'_{x}H'_{x}+E'_{y}H'_{y}+E'_{z}H'_{z}\right)dx'dy'dz'dt'.}
These equations indicate the invariance of the property that at a surface of discontinuity, or in a spherical wave, the electric force is equal in magnitude to the magnetic force and perpendicular to it.
A space time transformation from the variables (x, y, z, t ) to (x', y, z, t') can be used to transform the whole motion in one dynamical system into a corresponding motion in another, as far as the kinematics is concerned, provided the velocities
(
w
x
,
w
y
,
w
z
)
,
(
w
x
′
,
w
y
′
,
w
z
′
)
{\displaystyle \left(w_{x},w_{y},w_{z}\right),\ \left(w'_{x},w'_{y},w'_{z}\right)}
d
x
′
=
w
x
′
d
t
′
,
d
y
′
=
w
y
′
d
t
′
,
d
z
′
=
w
z
′
d
t
′
,
{\displaystyle dx'=w'_{x}dt',\ dy'=w'_{y}dt',\ dz'=w'_{z}dt',}
are a consequence of the relations
d
x
=
w
x
d
t
,
d
y
=
w
y
d
t
,
d
z
=
w
z
d
t
.
{\displaystyle dx=w_{x}dt,\ dy=w_{y}dt,\ dz=w_{z}dt.}
This condition is satisfied, if
d
x
′
−
w
x
′
d
t
′
=
ν
11
(
d
x
−
w
x
d
t
)
+
ν
12
(
d
y
−
w
y
d
t
)
+
ν
13
(
d
z
−
w
z
d
t
)
,
{\displaystyle dx'-w'_{x}dt'=\nu _{11}\left(dx-w_{x}dt\right)+\nu _{12}\left(dy-w_{y}dt\right)+\nu _{13}\left(dz-w_{z}dt\right),}
d
y
′
−
w
y
′
d
t
′
=
ν
21
(
d
x
−
w
x
d
t
)
+
ν
22
(
d
y
−
w
y
d
t
)
+
ν
23
(
d
z
−
w
z
d
t
)
,
{\displaystyle dy'-w'_{y}dt'=\nu _{21}\left(dx-w_{x}dt\right)+\nu _{22}\left(dy-w_{y}dt\right)+\nu _{23}\left(dz-w_{z}dt\right),}
d
z
′
−
w
z
′
d
t
′
=
ν
31
(
d
x
−
w
x
d
t
)
+
ν
32
(
d
y
−
w
y
d
t
)
+
ν
33
(
d
z
−
w
z
d
t
)
.
{\displaystyle dz'-w'_{z}dt'=\nu _{31}\left(dx-w_{x}dt\right)+\nu _{32}\left(dy-w_{y}dt\right)+\nu _{33}\left(dz-w_{z}dt\right).}
Multiplying these equations together by Grassmann's rule, we get
d
x
′
d
y
′
d
z
′
−
w
x
′
d
y
′
d
z
′
d
t
′
−
w
y
′
d
z
′
d
x
′
d
t
′
−
w
z
′
d
x
′
d
y
′
d
t
′
{\displaystyle dx'dy'dz'-w'_{x}dy'dz'dt'-w'_{y}dz'dx'dt'-w'_{z}dx'dy'dt'}
=
|
ν
11
ν
12
ν
13
ν
21
ν
22
ν
23
ν
31
ν
32
ν
33
|
(
d
x
d
y
d
z
−
w
x
d
y
d
z
d
t
−
w
y
d
z
d
x
d
t
−
w
z
d
x
d
y
d
t
)
.
{\displaystyle {\begin{array}{c}=\end{array}}\left|{\begin{array}{ccc}\nu _{11}&\nu _{12}&\nu _{13}\\\nu _{21}&\nu _{22}&\nu _{23}\\\nu _{31}&\nu _{32}&\nu _{33}\end{array}}\right|{\begin{array}{c}\left(dx\ dy\ dz-w{}_{x}dy\ dz\ dt-w{}_{y}dz\ dx\ dt-w{}_{z}dx\ dy\ dt\right).\end{array}}}
This shows that there is an integral invariant of the form
θ
(
w
x
d
y
d
z
d
t
+
w
y
d
z
d
x
d
t
+
w
z
d
x
d
y
d
t
−
d
x
d
y
d
z
)
{\displaystyle \theta \left(w{}_{x}dy\ dz\ dt+w{}_{y}dz\ dx\ dt+w{}_{z}dx\ dy\ dt-dx\ dy\ dz\right)}
This fact has already been used in § 3 and will be required again in § 7.
