DOC. 10 RESEARCH NOTES
205
Quadratisch
82cp 86cp
-2 -2-2-2
etc.
wird
notwendig
4.
Ordnung.
dritten Grades
in 9
wird
2.
Ordnung,
wie
es
sein
muss.
82(p
829
34cp
8x2
8x2
8x38x4
[8]A
result
in
the
theory
of differential forms
due to
Beltrami
is
that
all
differential invari-
ants
of
a
scalar
(p
can
be
produced
by
such combination of
(p
with the
interrelated
Beltrami
operators
A,
0,
and
A2
(see
Wright
1908,
pp.
56-57).
Einstein's
grad
is Beltrami's first
oper-
ator A
and
Einstein's
A
is
Beltrami's second
operator
A2
in
Cartesian
coordinates. For
further
discussion,
see
the editorial
note,
"Einstein's Research Notes
on a
Generalized
Theory
of Rel-
ativity," sec.
III.
[9]On
the
assumption
that
G^v
is
transformationally equivalent
to
the second derivative of
9
([eq.
7]),
Einstein seeks
to construct
differential
invariants of
G^v
from the differential
in-
variants of
(p.
He
considers
terms
linear,
quadratic, and
of
third
order
in
9
with
"etc.,"
sug-
gesting
a sum
of
terms
with coordinates
permuted
so
that
all
coordinates
enter symmetrically.
The
highest
order derivative
in
the
quadratic expression
is the sixth-order derivative
369
------,
which
corresponds
to
a
fourth-order derivative of the
quantity equivalent to
G.
dx2 3*3 3x4
Similarly,
the
expression
of third order
in
9 corresponds to
a
second-order derivative of
G.
The
term
linear
in 9 is
an
eighth-order
derivative of
9 and
corresponds
to
a
sixth-order deriv-
ative of
G.
[p.
4]
812
£34
[10]
813
£24
814
&23
32p
dx3dx4
+
32cp
32(p
32p 32cp
[11]
3jCj5jc3
dx2dx4 dx1dx4 dx2dx
[eq. 8]
anxJ
+2a
l2xy
+
2
_
,
[12]
a
33z
1
[eq. 9]
g12
g34 g12
£34
823\
841
dx^dx2
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