DOC.
23
153
x
=
vi.
7 1
T
-
t
1
-
V
=
t
~
1 1
J
V
t,
J
We
thus have
which
shows
that the clock
(observed
in the
system
at
rest)
is retarded
each
second
by (1
-
1
-
v/V2)
sec
or,
with
quantities
of the fourth
and
higher
orders
neglected,
by
1/2(v/V)2
sec.
This
yields
the
following
peculiar
consequence:
If
at
the points
A
and
B
of
K
there
are
located clocks
at rest which,
observed
in
a
system
at
rest,
are
synchronized,
and
if the clock in
A
is
transported
to
B
along
the
connecting
line with
velocity
v,
then
upon
arrival of this clock
at
B
the
two
clocks will
no
longer
be
synchronized;
instead, the clock that
has
been transported from
A
to
B
will
lag
1/2tv2/V2
sec (up
to quantities
of
the fourth and
higher orders)
behind the
clock that has
been
in
B
from
the
outset,
if
t
is the time
needed
by
the
clock
to
travel
from
A
to
B.
We see
at
once
that this
result holds
even
when
the clock
moves
from
A
to
B
along
any
arbitrary
polygonal
line,
and
even
when
the points
A
and
B
coincide.
If
we assume
that
the
result
proved
for
a
polygonal
line holds also for
a
continuously curved
line, then
we
arrive
at
the
following
proposition:
If
there
are
two
synchronous
clocks in
A,
and
one
of them
is
moved
along
a
closed
curve
with
constant
velocity
until it
has returned
to
A,
which
takes,
say,
t
sec,
then this clock will
lag
on
its arrival
at
A
it(v/V)2
sec
behind the clock
that
has not been
moved. From
this
we
conclude that
a
balance-wheel clock that is located
at
the Earth's
equator
must be
very
slightly slower than
an
absolutely
identical
clock,
subjected to
otherwise
identical conditions, that is located
at
one
of
the
Earth's
poles.
§5. The
addition
theorem
of
velocities
[18]
[19]
In
the
system
k
moving
with
velocity
v
along
the X-axis of the
system K
let there
be
a
point
moving
according
to
the
equation