132
DOC. 2
RELATIVITY AND ITS CONSEQUENCES
ß
=
1
1-
v
N
and
p(v)
is
a
function of
v
that
is
to
be determined. We
can easily
find
p(v)
by
introducing
a
third coordinate
system
S",
which
is equivalent
to
the
first
two
systems,
is
moving relatively
to
S'
with
a
uniform
velocity -v,
and
is
oriented
with
respect
to
S'
as
S'
is
oriented
with
respect
to
S.
Then,
applying
equation
(5)
twice,
we
obtain
t"
=
x"
=
y"
= z"
=
q(v).
p(-v).
t
p(v).
p(-v).x
p(v).
(p(-v).y
q(v).
(p(-v).z
Since
the
origins
of
S and
S"
are
permanently coincident,
the
axes
have
the
same
orientation,
and
the
systems
are
equivalent,
we
must
necessarily
have
p(v).p(-v)
=
1.
Since,
moreover,
the relation between
y
and
y'
(as
also
that between
z
and z')
does not
depend
on
the
sign
of
v,
we
have
p(v)
=
p(-v).
From
this it follows
that
p(v) = 1
(p(v)
=
-1
is
here
inappropriate),
and that the transformation
equations
are
(I)
f
=
ß
where
t
-
-X
c2
=
ß(*
-
vi)
,
/
-y
z'
=
z
ß
-
1
1- V
N