74 DOC.
62
MARCH
1915
In
order to arrive at
a
limit,
one
would
only
have
to
set
I
do
not want
to
go
into
the
issue of
the
existence
or
nonexistence
of
this limit
at
all,
as
this
question
is,
in
my view,
of
no
importance;
rather, I
shall
attempt
to
prove
that the
conclusion
drawn from formula
(71)
of
the
paper
is correct.[3] We
can
then
refer
our
later
observations
to
this
proof.
Assumption:
61= [
dr'E^g*'.
. .
J
E
(1)
is
invariant,
where the
ôguv's
vanish at
the
boundaries
of
E,
but
are
otherwise
freely
selectable.
Posit:
£v
is
a
covariant tensor.
Proof:
As
thus
^
=
• •
^
dxß
dxu
y
dx1 f)xr
«íT
=
...
oXfj,
dx"
(2)
(2a)
I multiply
(2a) by
/g’dr'
=
ygdr
and
integrate
over
E. Then
I
obtain
qwtt43
2t
t23tg
2T 3T4
3TT
43T234T
(3)
Since
integration
region
E
is
supposed
to
be
infinitely small,
I
can
substitute the
factors
dx'pdxudx'o/dxv
with
the
(constant)
values that these factors obtain for
any
one
place
in
the
integration
space.
In this
way
I
disregard only
a
relatively infinitely
small amount
on
the
righthand
side of
(3).
Therefore,
if
I set
as a
shortcut
it
is:
A“"
=
f
V^gSg^dr
. . . J
s
Fix'

y'
P
^
dx^
dx"
(4)
(3a)
Thus it
is proven
first
of
all
that
Auv
is
a
contravariant
tensor
and,
in
particular,
one
with
components
that
can
be selected
independently
from
one
another.[4]