78

DOC.

66

MARCH

1915

First of

all,

once

again

to

the

objection

relating

to

the

special

case

that with

an

appropriate choice of coordinates

the

guv's

are

constant.[3]

You state

that

in

this

case

the

Euv’s

do not vanish

if

the

adapted

system

is selected

in

such

a

way

that the

guv's

are

not constant. But

you

did

not support

this

statement,

and

I

am

considering

it

incorrect

as

long

as

you

have not

provided

an

example

or a

general proof

for this.

Now

to

the

second

objection. Although you

concede

that

T

_(tßu A^

+

£

=

invariant,

(A^

=

ƒ

SgV'y/^dr),

where

e

is

an

infinitesimal amount of

higher

order,

you

contest,

however,

that

from

this

it

can

be concluded

that

Euv/-g

is

a

tensor.

Well,

I must

admit

that

the

basis of this

objection

is not clear

to

me.

For,

since

I

have

proven

that the

Auv's

transform

contravariantly,[4]

in

my opinion

it

is quite

irrelevant to

the

proof

that

these

Auv’s

are

infinitesimal

quantities.

Nevertheless,

an

addition

must

be made

to

the

proof

that instead

of

an

infinitesimal

tensor

Auv,

a

finite

one

is

introduced

through

the

formation of

a

limit.

I

shall not

go

into

this,

however,

but

am

convinced

that

you

will

acknowledge

the

following

derivation

as

sound:

Up

to

a

relatively

oo

small

quantity,

E

• • •

(1)

for

any justified

substitution,

as

you yourself

concede.

Up

to

a

relatively

oo

small quantity-as

you

likewise

concede-and

as

I

have

proven

in

my

last

letter,[5]

dx1 dx1Ud/T

j^l(JT

_

UJy(T

J^LV ^

dxn

dxv

(2)

From

(1)

and

(2)

it

follows,

however,

that

up

to

a

relatively

oo

small

quantity,

the

equation

V""'

&CTT

dxa

dxT

V7?

dx"

dxu

thus

also

applies.

Since

the

Auv’s

can

be chosen

independently

of

one

another,

hence

_

y dx'a

dx'T

è'aT

V-9

y

dxn

dxv