296 DOC. 71 PRINCETON LECTURES
SPECIAL RELATIVITY
the
cylinder,
and
can
thus, from
the
above,
be
only
an
even
function
of
v.
If
we
introduce
a
third
system
of
co-ordinates, K",
which
moves
relatively to
K' with
velocity
v
in
the direc-
tion of the
negative
x-axis
of
K,
we
obtain,
by
apply-
ing (29) twice,
x'\
=
X(v)X(-w)xi
[38]
/"
=
X(v)X(-v)l.
Now,
since
X(v)
must
be
equal to
A(-v),
and
since
we
assume
that
we use
the
same
measuring
rods in all
the
systems,
it follows
that the transformation
of
K"
to
K
must
be
the identical transformation
(since
the
possibility
X
=
-1
does
not
need
to
be considered).
It
is
essential
for these
considerations
to
assume
that the behaviour of
the
measuring
rods
does
not depend upon
the
history
of
[39]
their
previous
motion.
Moving
Measuring
Rods
and
Clocks.
At
the definite K
time,
l
=
0,
the
position
of
the
points
given by
the
integers
x'1
=
n,
is
with
respect
to
K,
given
by
x1
=
n
V1
-
v2;
this follows from
the
first of
equations
(29)
and
expresses
the Lorentz contraction.
A clock
at rest at
the
origin
x1
=
0 of
K,
whose
beats
are
characterized
by
l
=
n,
will,
when observed
from K',
have beats characterized
by
/'
=
B
• Vi
-
v2,
this follows from the second of
equations (29)
and
shows
that
the
clock
goes
slower
than
if it
were
at rest relatively
to
K'.
These
two
consequences,
which
hold,
mutatis
mutandis,
for
[36]