DOC.
47
259
we
choose
the
position
of the x'-axis in
such
a
way
that it
has,
with
reference
to S,
the
same
direction
as
the translational
motion of S' has
with reference
to S,
then
it follows for
reasons
of
symmetry
that the
S-referred coordinate
planes
of
S'
must
be mutually
perpendicular.
We
can
and
will
choose
the positions of the
two
coordinate
systems
in
such
a
way
that
the x-axis
of
S
and
the
x'-axis
of S'
coincide
at
all
times,
and
that
the S-referred y'-axis of
S' be
parallel
to
the
y-axis
of
S.
Further,
we
shall
choose
the instant at which
the coordinate
origins
coincide
as
the
starting
time in both
systems;
the linear transformation
equations
sought
are
then
homogeneous.
From
the
now
known
position of
the coordinate
planes
of S'
relative
to
S,
we
immediately
conclude
that
the
following
pairs of
equations
are
equiva-
lent:
x'
=
0
and
x
-
vt
=
0
y'
=
0
and
y
=
0
z' =
0
and
z
=
0
Three of
the transformation
equations
sought
thus
have
the
form:
xx
-
(ax
-
vt)
y'
=
by
z'
=
cz
.
Since the
propagation
velocity
of light in
empty
space
is
c
with
respect
to
both reference
systems,
the
two equations
x2 +
y2
+
z2
=
c2H2
and
x'2
+
yX2
+ z]2
=
c2V2
must
be
equivalent.
From
this
and
the
expressions
for x', y', z' just
found
we
conclude after
a
simple
calculation that the transformation
equations
must
be
of the
form