DOC.
13
GENERALIZED THEORY OF RELATIVITY
187
0a /
dy"v
dy
U
=
A
-
B
+
Y
V^'Y
3r
3x
aß
/xv
*^Aa
OXn
/3
ox ox
or
if
we
apply
formula
(29)
of
§2
in
the second summand and
integrate by parts
in
the third summand
U=
A
Yl
dgik
^i,/-
dg"v drfiv
dgik
•
y"lidxa
dxß
'
2 Vildxa
Zj
V9'
dxa ' dx, 'y«tYß*
dx"«
a
ßtivilc
P
"
a
ß/uvik
ß
+2
ZgMv
dy.
Zi
SJ
dyfiv
dxa
dxfi'
d
{/Hr.ß
dx
aß(.i v
13
dxa
dx
a
p
fiv
The first
two
sums
have the form of
terms
such
as we place on
the left side of
our
identity.
We denote them
by
(50)
F
^
E
dglk
\fg-ynRy
r-
dguv
/XV
.
dy/XVI
dx"
aßUk
dx..
dx
aßfiv
ik
ß
(51)
w
=
y
^8ik
-Jc-y
y""
dg^v-dy.
dx_ dx.. dxn
aßfiv
ik
a
^a
"*ß
The third of
the
sums appearing on
the
right
has
the form of
a
sum
of differential
dy
quotients;
if
we
eliminate from it with the
help
of
the
above formula
(29),
this
dx_
a
sum proves
to be
the
quantity A
that has
already
been introduced.
Finally, we replace
dx
^
in the last
sum
in
accord with the
same
formula. In this
way we
find
/
U-V+W
=
2A-B
+
Y
Y^-Yv*
d8ik
d
sfgyaß
dg^
a/5/xv ik
^ v
dxa
dx
ß
V
dx
a
/
or
/
\
U
-
V
+
W
=
2A
-
B
+
Y
dgik
d
\/g
VaßvmVvk
dg
/XV
aßfiM
ik
dx
q
dXß
dx
V
a
dgik
dgßV
d
aß^ik
E
dxo dxa
&•Y
aß
dxß
(Y"iYvi)
x'IAl
By
virtue of
(29),
i.e., by
virtue of
E
Y,Yvt
dg
ßV
-
dy,
ik
/xv
dx
a
d*«
'
the first of these
sums
becomes
dgik
d
dju
JiSfc
dxa
dxß
JgyM
d*a
j
=
-u