180 DOC.
183
JANUARY
1916
in
your
third
letter.[3] I
could do
this
myself,
of
course,
to
the
extent
that
it
is
all clear
to
me.
However,
nature has
unfortunately
deprived
me
of the talent
of
being
able to communicate in
writing,
so
that
what
I
write
may
well
be
correct
but
is quite
indigestible.
Judging
from
your
second letter,
it
seems
to have
escaped you
that
I
already
indicated the
generally
covariant form of
the
field
equations.
It
is
provided
in
equations
(2a)
and
(1)
of
the
paper,
“The Field
Equations of
Gravitation.”[4]
I
am
of the conviction
that the
depiction
of the
theory
would
gain
much
clarity
if Hamilton’s formulation
were
taken
as
a
point
of
departure,
as
you
have done
in
your
fine
paper published by
the
Amsterd[am]
Acad.[5]
The
path
toward this
seems
to be
the
following.
The
V-scalar[6]
G
=
y/=g
(ik,
lm)gHgim
iklm
is
a
function of
the
quantities
guv,
dguv/dxo,
d2guv/dxodxr.
The
integral
fGdr
is
hence
invariant.
Through
partial integration
(in
only
one
way)
the
same can
be
presented
in
the
form
ƒLdr
+
surface
integral,
whereby
L is
now
just dependent
on guv
and
dguv/dxo.
In this
way
I
find
(calculating
only
once!)
L
-
V-g
+
dig
(
dg™
_
o^dlg
gxß].
If,
therefore,
the
generally
covariant
field
equations
can
be
presented
in
the
form of Hamilton’s
principle,
of which
I
am
quite certain,
then this
must be
the
Hamiltonian
function
to
use.
The second
term
in brackets
can
also be written in
the
form
ZgaB{aBa}{opp}=-ZgaBTaaBTpop
The
computation
of
dl/dguv
and
dL/dgouv
is
quite
arduous, though,
at
least
with
my
limited
proficiency
in
calculation.[7]