DOC.
13
GENERALIZED THEORY OF RELATIVITY
179
(29)
rs
_
dg
"äT
=
~?YrpYi7~ät.'pa pa
then the three middle
terms
under the summation
sign
in
equation
(28)
cancel
one
another
out,
and there
remains,
along
with the first
term,
£
~Y
dg
rs
'yh^k
yuT,
3
lOg
yfg
rs dx dx
rskl
t
kl
t
so
that
one
obtains for the
divergence
of the
covariant
four-vector9
(30)
£ 3
[JgyrsTr\
rs
fg
rs
dxs
(c) X
=
2
Let the
starting
tensor
be
a
contravariant
tensor
of second rank
©rs,
the extension
of
which, according
to
formula
(20),
is
(31)
/
//:]
ik
©rst
=
E
Ytf
d©
re
+
©fc
+
,
h
dX:
ik
\
/
This
yields
for the
divergence
of the contravariant
tensor
©rs
either the
row
divergence
(32)
©r
=
Esst
©rst
E
d©
re
+
sk\
+
sk
5*
ks rk
st sk
r
/
or
the column
divergence
____
ag,,
g~1
Ykl
=
-Elk!
ax,
I
ax,
where
t is
one
of
the numbers
1,2,
...
n.
For
any
given
k
one
thus obtains
n equations
(i
=
1,2,
...
n)
with
n
unknowns
ü
(l
= 1,
2,
...
n),
the solution
of
which
yields
dxt
the formula in the
text.
9The
same
rresult
is
obtained
by
Kottler
(l.c.,
p.
21),
who
starts out
from
a
special
third–
rank
tensor
(cf. §3
of this
paper)
and
applies
the
theory
of
integral
forms.
[62]