168
DOC. 30 FOUNDATION
OF GENERAL RELATIVITY
and
»•
-
56
fe
Sf
^+
s(S)
-
;••
we
obtain
from
(20a),
after
a
partial
integration,
*2
where
[14]
KvhxvdX
=
0,
Ai
d/flW
1_
dxv
9nM
d\\w
d\)
210
ix. d\
'
(20b)
Since
the
values of
dxo
are
arbitrary,
it
follows
from this
that
Ko
=
0
....
(20c)
are
the
equations
of
the
geodetic
line.
If
ds does
not vanish
along
the
geodetic
line
we
may
choose
the
"length
of
the
arc"
s,
measured
along
the
geodetic
line,
for the
parameter
A.
Then
w
=
1,
and in
place
of
(20c)
we
obtain
[15]
a
j
dX(f
dXfi
^
1
^)c/nV
dx^
dxv
^
ds2
'bxa
ds ds
2
dxo
ds ds
=
or,
by
a
mere change
of
notation,
+
fay
,]0
...
(20d)
where,
following Christoffel,
we
have
written
']
=
Kfe+£
-
fe)
...
(21)
Finally,
if
we
multiply
(20d) by gor
(outer
multiplication
with
respect
to
r,
inner with
respect
to
o),
we
obtain the
equations
of
the
geodetic
line
in the form
d^Xf
«
\dXu
dxy
A
w
+
^Tldf
di-0'
...
(22)__.
where,
following
Christoffel,
we
have
set
{/*",
r}
=
gTa\jiv,
a]
...
(23)