378
DOC.
382
SEPTEMBER
1917
To
obtain
expression
(15)
for
ds2
we
must
set[23]
p
=
1
(corresponding
to
v
=
r
for
Droste)
u2
=
W 2
=
1--.
-i
a
j
_
a
1
r
(17)
Hence
this
holds for
a
very specific
coordinate
system.
In
a
point
of
space
where
x1
=
r, x2
=
x3
=
0,
you
obtain
911 =
~
922 =
93’
=
~
9n
=
9W
=
0,
dgn
1
dg22 0533
dp
dx2
-~(u2-p2),
r
dxi -
-2pp'
- dx\
dr
dgn dgu
=
2
ww'
dx3
dx\
(18)
Now
U1
must be calculated
at
a
point
where
x2
=
x3
=
0.
Thus
ai
=
^S(*V-9V‘)(tf
+
(19)
You
will
notice,
first
of
all,
that the
terms with
a
=
i
contribute
zero.
To
obtain
a
term
differing
from
zero, moreover,
either i
or a
must have
the
value
1,
and
the
remaining
two
of
the
three
indices
a,
i,
l
must
be
equal.
i
=
1, a
=
l # 1
contributes
^ gll\g22^
Í
9.922
+
9^
î
dg33
+
gAi
dga
dx\ dxi dx\
and
a
=
1,
i
=
l
#
1
gives
"44
dg44
U1
consists of these two
terms. Introducing
the
expressions
(18), one
obtains
twpp 2
/
p2\ v2w'
2ti
=
-4--1-
-uw
1--t
-
2--.
u r
\
uz
u
(20)
For
the
special
coordinate
system,
where
equations
(17)
apply,
one
finds
we4y we4ty34 34
(21)