DOC.
71
PRINCETON LECTURES 351
THE GENERAL THEORY
we
have
the
possibility
of
getting
the relation between
naturally
measured
lengths
and
times,
on
the
one
hand,
and the
corresponding
differences of
co-ordinates,
on
the
other
hand.
As
the
division
into
space
and time
is
in
[107]
agreement
with
respect
to
the
two
systems
of
co-ordinates,
so
when
we
equate
the
two
expressions
for
ds2
we get
two
relations.
If,
by (101a),
we
put
ds2=-(1+k/4x
/
adVo/t)
(dx12
+
dx22
+
dx23)
+
(1
-
k/4x
/
adv0/r)
dl2
we
obtain,
to
a sufficiently
close
approximation,
(106)
IdX12dt=(1-k/8xfadV0)dl.
+
dX22
+
dX32
=
(
1
+
k/8x
ƒ
adv0)
+
dx22
+dx32
The unit
measuring
rod
has
therefore the coordinate
length’
1_k/8x_
8x
]fadV0r
in
respect to
the
system
of
co-ordinates
we
have
selected.
The
particular
system
of
co-ordinates
we
have
selected
[108]
insures
that
this
length
shall
depend only upon
the
place,
and
not upon
the direction. If
we
had
chosen
a
different
system
of
co-ordinates
this would
not
be
so.
But
however
we
may
choose
a
system
of
co-ordinates,
the
laws of
con-
figuration
of
rigid
rods do
not agree
with
those of
Euclidean
geometry;
in
other
words,
we
cannot choose
any
system
of
co-ordinates
so
that the co-ordinate
differences,
Ax1,
[91]