182 DOC. 30 FOUNDATION OF
GENERAL RELATIVITY
quantity
TauB,
which
is
symmetrical
with
respect
to
the
in-
dices
u
and
ß.
Thus there remains
only
the
first term in
round brackets
to be
considered,
so
that,
taking
(31)
into
ac-
count,
we
obtain
m
=
-
rßßrßJr" +
r;ßsg:ß.
Thus
SH
=
-
r*
r*va
*9
(iv
M0
2H
...
(48)
=
r
ö9
Ii v
1
*»
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Carrying
out the
variation
in
(47a),
we
get
in the
first
place
7)
/7H
=
o,.
.
.
(47b)
which,
on
account of
(48),
agrees
with
(47),
as was
to be
proved.
If
we
multiply
(47b)
by
guva,
then
because
*9
fiv s9afxy
T
7)x,
üx
and,
consequently,
)H
"dga
(IV
"
^Xa\i,ga
*x,
(«r=)
*9
I»
tXn
9
we
obtain the
equation
ÖH
m
~bxa
*9aHV
^bXcr
=
0
or*
K
7x*
=
0
...
(49)
-
2*r
=
_
s:H
*9atkv
where,
on
account
of
(48),
the
second
equation
of
(47),
and
(34)
=
fx
-
r"Kßr?r
. .
(50)
*The
reason
for the introduction
of
the factor
-
2k
will
be apparent
later.
[24]