150
ELECTRODYNAMICS
OF
MOVING
BODIES
The
transformation
equations
we
have
derived
also
contain
an
unknown
function
p
of
v,
which
we now
wish
to
determine.
To
this
end
we
introduce
a
third coordinate
system
K', which relative
to
the
system k
is in
parallel-translational
motion
parallel
to
the axis
5
such
that its
origin
moves
along
the 5-axis with velocity
-v.
Let
all three
coordinate
origins
coincide
at
time
t
=
0,
and
let the time t'
of
the
system
K' be
zero
at
t
=
x
=
y
=
z
=
0.
We
denote the coordinates
measured
in
the
system
K'
by
x',y'
,z1
and,
by
twofold
application of
our
transformation
equations,
we
get
V
=
(ß(-v)ß(-v){T
+
yi
=
v(v)ip{-v)t*
=
p{-v)ß{-v)-{{ + vt}
=
p(v)p(-v)x9
y] =
ip(-v)r)
=
p(v)p(-v)y,
Z1
=
p{-v)(
=
ip(v)(fi(-v)z.
Since the relations between
x1 ,y1,z1
and
x,y,z
do
not
contain
the
time
t,
the
systems
K
and
K'
are
at rest
relative
to each
other,
and
it
is clear that the transformation
from
K
to
K' must
be
the
identity
transformation.
Hence,
ip{v)p(-v)
=
1.
Let
us now
explore the
meaning
of
p(v).
We
shall focus
on
that portion
of
the H-axis
of
the
system
k
that
lies
between (
=
0,
n =
0,
(
=
0, and
E
=
0,
n
=
l,
C
=
0.
This portion
of
the H-axis is
a
rod
that
moves
perpendicular
to
its axis with
a
velocity
v
relative
to
the
system
K
and
whose
ends
possess
in
K
the coordinates
and
=
vt,
y1
=
Zj
=
0
x2
=
vt,
yi
-
0,