260
THE
RELATIVITY
PRINCIPLE
t
I
_
where
p(v)-ß
v
"
t
-
x1
=
tp(v)
•/?•
fa;
-
vt)
y'
=
tp(v)-y
(p(v)'Z
9
ß
=
1
V
J
7^
Now we
will
determine the function
of
v,
which has not
yet
been
determined.
If
we
introduce
a
third
system,
S", which
is
equivalent to
S
and
S',
is
moving
with the velocity
-v
relative
to
S',
and
is oriented
relative
to S'
in the
same
way
S'
is oriented relative
to
S,
we
obtain,
by
twofold application of the
equations
we
have
just
found,
t"
u
x
y
z
II
_
II
_
p(v)-p{
p(v)-p{
ip{v)-p(
p(v) 'p(
v)'t
v) 'X
v)-y
v)
-
z
Since
the
coordinate
origins of
S and S"
coincide
permanently, the
axes
have
identical directions
and
the
systems
are
"equivalent," this
substitution is the identity,1
so
that
(p(v) *p(-v)
=
1
Further, since the
relation
between
y
and
y'
cannot depend
on
the
sign
of
v,
we
have
p{v)
=
ip(~v)
Thus,2
p(v)
=
1,
and
the transformation
equations
read
1This
conclusion is
based
on
the
physical
assumption
that the
length
of
a
measuring
rod
or
the
rate
of
a
clock
do
not
undergo any
permanent
changes
if these objects
are
set
in
motion and
then
brought
to
rest again.
2(f(v)
=
-1
is
obviously
out of
the
question.