474

DOC.

463

FEBRUARY

1918

only one

unknown each of

the 4th

order.). Unfortunately,

they

then

also

contain

some

nasty

terms of

the

2nd

order

that

are

quadratic

to

the other

unknowns.

On

this

occasion,

I also discovered

some

differential relations between

the

Buv's, or

Bao,

or

BaB,ro.

Those between

the

Bao’s

lead to

your energy-momentum

law

and

read

(without

the

constraining

conditions

\/-g

= 1):[9]

dB/dx-1/2dB/dxa=Bao.TBaB-BaBTBao.

These

relations

are

the direct

consequence

of

more

general,

apparently

still

entirely

unknown

relations:[10]

dBik,lm/dXa

+

dBik,mn/l

+

dBik,nl/m

=

TaklBai,mn

+

TakmBai,nl

+

TaknBai,lm- (3

analogous

terms

with

i and

k

exchanged.).

I

am

probably

going

to take

the

opportunity

to

publish

this,

after

all.

My

calculations showed

me now

that the

field

equations

are

inadequate

for

determining

the

guv’s,

that

is,

not

even

to

the

extent

of

allowing general

arbi-

trariness for

the

coordinates. Yet

the

space-time

geometry

in the

vicinity

of

a

point

should be determined

entirely by

the

energy

state-etc.-prevailing

within

it. Hence conditions must

necessarily be

added

to

your

field

equations.

For

this,

it

seems

to

me, my

condition

of potential is

very suitable,

which states

that

a

four

vector

Au must

exist such

that

gua-dAo/v

+

gua-dAo/u

+

Aadguv/a

=

0 (:or,

as

far

as

I

am

concerned, equal

to

another

2nd-order

tensor:).[11]

For it

seems

to

me

that,

on

the

basis of

this

assumption,

the

guv's

are

definitely

determined

essentially

by

the

energy

tensor and

the

electromagnet,

field (:if

the

electromagnet,

field is

derived from

Au

through

the

formulas

pa

= guo Au;

Fpa

=

dpp/o-dpo/p,

as

I

have

done

in

my

last

letter:).

All these reflections

are

terribly

drawn

out

and

tricky,

and

I

am

almost inclined

to

give

up hope

of

arriving

at

a

positive

result here with

the

current state of

mathematics,

with

my

lack

of

access

to

the literature

and

with

my

limited time.

I

cannot

spend

my 17-day

vacation

on

a

thorough study of

mathematics

either,

disregarding

that

nothing

would

come

out of

it

in such

a

short

time.

I

therefore

would like to

ask

you

to

direct

your

attention

to these

problems yourself or,

if

your

time

also is

taxed

too much

otherwise,

to refer

some

mathematicians

to

your

equations. My

esteemed teacher

Hilbert[12] will

certainly

find

some

results here.

The author

of

the

Encyclopedia

article cited

above,

Mr. Ed.

von

Weber,[13]

might

possibly

be in

the

position

to

shake

some

theorems

effortlessly

out of

his

rich store

of

special

mathematical

skills,

which

might

illuminate

the

applied

boundary

value

problem.