DOC.

405

DECEMBER

1917 415

is obtained,

or

approximately

enough

9#44

•

,

%44

.

,

\

,

d2xA

X

+

-7T-y+-

+

dx

dy

u )

dsI

2

It

is

this

expression

which indicates

the

vanishing

of

the

last

equation

of motion.[5]

For

dx4/ds,

one

obtains,

if

one

sets for

the

guv's

(incorrectly,

by

the

way)

-1

0 0 0

0

-1

0 0

0 0

-1

0

0 0 0

g44

dx4

1

ds

=

1

V2

The

root of

your

error

lies

in

that

1/1-v2

cannot be inserted in its

place,[6] since

d/dt(g44)

and

d/dt(v2)

are

of

the

same

order of

magnitude.-

I probably

do

not

understand

your

first

difficulty.[7]

You

say entirely correctly:

If

I

were

within

the interior of

a

rotating, hollow

rotational

body, mechanically

I

would have

to

find

myself

to be in

the

state of

a

rotational

system

when

I

am

“at rest.”

A point of

mass

must

be able

to

move

in

a

circle

free of forces

when

it

is

moving

at

a

suitable

angular velocity

around

the

Z-axis

(in

the rotational

body’s

sense

of

motion).

Now, you

require, however,

that

d2x/ds2

vanish

for

such

a

point

when it

is

on

the

X-Z

plane.

This does

not

apply

to

rotational

motion,

though.

What

must

rather

apply

is

cPx

ds2

=

-w'2r

=

-w'2x,

whereby

w' is

the

point’s

rot.

velocity,

which

ought

to relate to

your

w

as

w'2

=

Cw2.

Your

equation

then

yields, as

it

should,[8]

2C

=

-C'.

The

fact that this

cannot be otherwise

is, incidentally,

guaranteed

by

the

general

covariance of all

equation systems

in

the

theory.-[9]