DOC.
45
241
the
body's
electrical
charges.1 The
energy AE
transferred
from
the field of
force
to
the
body
between
the times
t0
and
t1 is
given
by
the
expression
11
=
I
dlJvIfjdxdydz
`Ito
where
the
space integral
is
to be
extended
over
the
body
and
we
have
put
"
_
dX
A
dY
A
dZ
p
=
dx
+
dy
+
dz
.
Using
the transformation
equations given
in the
paper
cited
above2,
we
transform this
expression to
the
space-time
system
(E,
n,
c,
r),
which
corresponds
to
a
coordinate
system
that is
at rest
with
respect to
the
body
and
whose
axes
are
parallel
to
(x,y,z).
One
thus
obtains after
a
simple
calculation,
in
a
notation that
corresponds
exactly
to
that
used
in the
paper
quoted,
=
if
f3vP
ds~diy1Cdr
where,
as
there,
ß
denotes the
expression
1
1-(u/v)2.
Note
that
according
to
our
assumptions
the forces
X'
cannot be
arbitrary.
Rather,
at
all times
they
must
be such
that the
body
under consideration
does
not experience
any
acceleration.
The
necessary
(but
not
sufficient)
condition
for this,
according to
a
theorem
of
statics,
is
that,
observed
from
a
coordi-
nate
system
that
moves
together with the
body,
the
sum
of
the
X-components
of
the forces
acting
upon
the
body always
vanishes.
One
thus
has
for
each
r
J
X'pdtdr)d(
=
0
.
1We
introduce this
assumption
in
order
to be
able
to
assume
that the
acting
forces
are
not subjected
to
any
restricting
conditions
due to
the
way
they
are
produced.
[6]
2A.
Einstein,
Ann.
d.
Phys.
17
(1905),
§§
3
and
6.