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