100
DOC.
3
STATICS OF GRAVITATIONAL FIELD
the first
equation
the
two
equations5
x=A1c,
[14]
x=2a1c-ac.
Eliminating
A1,
one obtains from these two equations
cx-2cx=-ac2,
or the equation
d(fl
a
dt(~c2)
c2
In
an
analogous way
one
obtains
for
the other
two
components
d
dt
y.
C2
d
dt
z.
C2
)
These three
equations hold,
first of
all,
for the time
t
=
0.
However,
they
hold in
general,
because this time is
not
distinguished
from other times
except
for the
circumstance that
we
have
made it the
starting point
of
our
series
expansion.
The
equations
thus found
are
the
equations
of motion that
we were seeking
for the
freely
moving point
in
a
constant
acceleration field.
If
we
take into
account
that
a
=
dc/dx,
and that
(dc/dy)
=
(dc/dz)
=
0,
then
we can
also write these
equations
in the form
[15]
(6)
d(i
i&c
dt(~c2
cax'
d(y iac
dt(~c2
c&y
_ __
ldc
caz
With the
equations
written in this
form,
the x-direction is
no
longer distinguished;
both sides have
a
vectorial character. For this
reason,
these
equations
are
also
to
be
viewed
as equations
of motion of
a
material
point
in
a
static
gravitational
field if the
point
is
subjected solely
to
the action
of
gravitation.
From
(6)
we can
first of
all
find
out
the
relationship
between the
constant k
appearing
in
(5b)
and the
gravitational
constant K
in the usual
sense.
In the
case
of
5The terms omitted in
(2)
do
not
show
up
in the result when
one
does this twofold
differentiation and
subsequently
sets t
equal
to
zero.