82 DOCS.
68,
69
MARCH-APRIL
1915
good
if Mr. de
Marval
repeated
the
lecture in
many
German cities.
His
evidence
would
not
be
suspect,
as one
of
his brothers
is
an
officer
in
the
German
Army.
69. To
Tullio Levi-Civita
[Berlin,] 2 April
1915
Highly
esteemed
Colleague,
You
letter
of the 28th of March
was
extraordinarily
interesting
for
me.[1] I
had
to ponder
for
one
and
a
half
days
without
interruption
before it became clear
to
me
how
to
reconcile
your example
with
my proof.
I enclose
your
letter
so
that
I
can
refer to
it without
any
inconvenience to
you.
Your
deduction
is
entirely
correct.
1-gEuv
does not have
a
tensor
character
with the infinitesimal
transformation
envisaged
by
you, even
though
the
trans-
formation
follows
from
a
justified
coordinate
system. Oddly enough, my
proof
is
not refuted
by
this for
the
following
reason:
My proof
fails in
precisely
that
special
case
you
have addressed.
In
order for
the
tensor
character
of
1-gEuv
to follow
from
the
assumption[2]
Bu
=
0
ABu
=
0,
•••
(1)
as
well
as
from
the
conclusion drawn from it
J
dr''£2 8
g1*" (Ep,,
=
invariant
for
the
infinitesimal
substitution
under
examination,
it
is
necessary
that
the
ôguv's
be
freely
selectable,
the
fulfillment of
the
region’s
boundary
conditions
notwith-
standing.
To be
more
precise,
this does
not
even
have
to
apply
to
the
6guv's,
but
only
to
the
integrals
ƒ
y/gôg^dr,
which
as
proven earlier,
of
course,
have
a
tensor
character if the chosen
integration
area
is
infinitely
small.[3]
Of
equal
importance
is
the
requirement
that
for
a
given
infinitesimal
integration
region, one
must be able
to choose
the
integrals freely
ƒ
ÔgTdr.
If
this
is
not
the
case,
then
the
tensorial
character
of
Euv/g
cannot
be concluded
from
(1).