DOC.
24 173
Let
there
be
a
body
at rest
in
the
system
(x,
y,
z),
whose
energy,
referred
to
the
system
(x,
y,
z),
is
E0.
The energy
of the
body
with respect
to
the
system
(£,
n, c),
which
is
moving
with
velocity
v
as
above, shall
be
H0.
Let
this
body
simultaneously
emit
plane
waves
of light of
energy
L/2
(measured
relative
to (x,
y, z))
in
a
direction
forming
an
angle
(p
with the
xaxis
and
an
equal amount
of light
in
the
opposite
direction. All the while,
the
body
shall
stay at rest
with
respect to
the
system (x,
y,
z).
This
process
must
satisfy
the
energy
principle,
and
this
must
be true
(according
to
the
principle
of relativity) with
respect to
both coordinate
systems.
If E1
and
H1
denote the
energy
of the
body
after
the emission of light,
as
measured
relative
to
the
system (x,
y, z)
and
(£,n,c), respectively,
we
obtain,
using
the relation indicated
above,
E0 
Et
+
L
+
L
2
+
2
*0 =
H1
+
1
2
1

v v
jCOS
if
£
1
+
yCOS
If
1

V
7
7
+
2
1

V
V
=
«l
+
1

V
V
7
Subtracting,
we
get
from
these
equations
(H0

E0

(H1
 E1]
=
L
1

Li
V
V

1
The two
differences
of
the
form
H

E
occurring
in
this
expression have
a
simple
physical
meaning. H
and
E
are
the
energy
values of
the
same body,
referred
to two
coordinate
systems
in relative
motion, the
body
being at rest
in
one
of the
systems
(system (x,
y,
z)). Hence
it is
clear that the differ
ence
H

E
can
differ
from the
body's
kinetic
energy
K
with
respect
to
the
other
system
(system
(£,*,)) solely
by an
additive
constant
C,
which
depends
on
the
choice of
the
arbitrary
additive
constants
of the
energies
H
and
E.
We
can
therefore
put