DOC.

69

APRIL

1915 83

In

the

special

case

examined

by you, however,

the

ƒSguvdT

quantities

are

not

only not freely choosable,

but

they

even

vanish

altogether.

Just

as

in

§14,

I

set[4]

àguv

=

61guv

+

d2guv.

The

S1guv’s

must

meet

the

condition

81Bu

=

0,

which in

the

special

case

considered

by you

takes

on

the

form

y,--(ypi9^)

=

0,

(comp.

equation

(5)

of

your

letter),

where

Q

is

the

Laplacian operator.

From this

follows, as

is

known,

because

of

the

boundary

conditions,[5]

^

dx

This

is

characteristic

of

your special

case.

If

you multiply

this

equation by xo

and

integrate

it

over

the

entire

region,

then,

upon

partial integration

of each index

combination,

ƒ

6ig^dr

=

0

...

(I)

follows.

This

is

a

consequence

of the

peculiar degeneration

of the

equation

61Bu

=

0

in

the

special

case you

have raised.

It

follows

furthermore

from formula

(63)

of

the

paper[6]

and the definition

of

the

ô2guv’s

for

an

infinitesimal

region

in

general,

ƒ

hsTdr

=

0

...

(II)

Thus from

(I)

and

(II)

ƒ

Sgudr

=

0

follows

for each index combination.

Therefore,

it

is

inherent to

the

specialization you

have

introduced

that

for-

mulas

(I)

of

this letter

are

not

adequate

conditions to lend

a

tensor

character

to

Euv/9

under

an

infinitesimal

transformation.

Generally, however, equation

61Bu

=

0

cannot be reduced to

a

first-order

equation

for

the

S1guv’s.

Then

my proof

indeed holds true for all finite transfor-

mations.