692

DOC.

650

NOVEMBER

1918

650.

From

Felix

Klein

Göttingen, 10

November

1918

Esteemed

Colleague,

Your

postcard

of

8

November has

just arrived.[1]

Meanwhile,

with Miss Noe-

ther’s

help,

I

understand

that the

proof

for

the

vector

character

of

Ea

from

“higher

principles”

as

I

had

sought

was

already

given

by

Hilbert

on

pp.

6,

7

of his first

note,[2]

although

in

a

version

that

does not draw attention

to

the

essential

point.

The

specific

structure

of

K

is

not

used

there,[3]

rather K

can

be

any

invariant

formed

from

the

arguments coming

into consideration. In

the

following

manner:[4]

1)

At

the

start,

we

cut

away

from

E°

or

rf,

respectively,

all

components

that

recognizably

have

the

desired vector character. There remains

(q=(

9pcr

for

investigation

of its nature.

2)

In

any case,

dc/dw2

is

an

invariant.[5] The

highest

term

(a

term

with the

highest

differential quotients

for

Puvp)

to

appear

in it

is:

-dk/dguvpuvpo.

3)

Now

we

have

the

obvious statement:

puv

transforms

(under

arbitrary

trans-

formation of

w) as a

tensor,

if

the

terms with

lower

differential quotients

are

disregarded.

This

is

enough

to

conclude

that-dk/dguv

that-dk/dguv

is

a

tensor,

which

now,

as

in

my

previous

letter, will

be called

Kpouv.[6]

4)

Once

we

had

made this

finding,

we

set,

again

as

in

my

earlier

letter,

in

making

use

of

the

process

of covariant differentiation

P¡T

=

A?

+

T^p™

+

TuTppTp,

where

Auvp

is

now a

tensor.

5)

Accordingly,

-KpouvAuvg

is

a

regular

vector,

which

we

eliminate.

There

remains:

©a

=

(

)puv

to

be

analyzed,

which

we

immediately

want

to abbreviate,

=

Bouvpuv.

6)

Then

it’s

the

same

game anew as

in

(1),

(2)

....

:

a)

In

any event,

do/dw

is

an

invariant.[7]

b)

The

highest

term to

occur

in it

is

B°uvpuvo.