606

DOC.

581

JULY

1918

581. From

Felix Klein

Göttingen,

5

July 1918

Esteemed

Colleague,

I

still must write

you

about

my Monday

lecture.[1] I

have

the

proof you

had

intended:[2]

that the

Jo's form

a

contragradient

vector,

carried out

under the

restriction that

the world

tube

of

the closed

system

can

be

contained within

a

cylinder of finite

transverse

dimensions.[3]

The

accompanying figure

probably

does not need

any

explanation

(I

choose

the

co-

ordinate

system

in such

a way

that the

x4

axis

is

parallel

to the

cylinder’s

orientation;

a,

a'

are

any

two

points

linked

together

by

another

parallel;

they

define

the

vector

0, 0, 0,

Ax4).-

One

obtains

for

the

corresponding

integrals[4]

x4 =

C'

x4

=

C'

A"

=

J

ƒ

ƒ

ƒ

Uvodx1dx2dx3dx4

the

precise

scheme:[5]

0 0 0 Ax4

.

J1

0 0 0

Ax4

.

J2

0 0 0 Ax4

.

J3

0

0

0 Ax4

.

J4

Now I

introduce,

through

some

Lorentz

transformation,

instead

of

x,

new

x

coordinates

and

define

the

new

Avo's

by

means

of

a

cylinder

section whose

edges

x4

=

C,

and

=

C',

resp., again

go

through

the

points

a,

a1.

The

integra-

tion

domain

is

thereby visibly

different

from

the

original

one,

but

this

difference

bears

less

weight against

the

overall

ex-

tension,

the

larger Ax4

is

relative to

the

cylinder’s

cross

section.

We

conclude that

for

lim

Ax4

=

oo,

the

Avo’s

must

be

related

exactly

to the

Ava’s

as

the

Uvo’s are

to

the

Uvo's.[6]

Simultaneously,

Avo

=

Ja.Ax4.[7]

From

this then

follows

explicitly

the

desired correlation between Ja and

Jo,[8]

first