DOC.

41

HAMILTON'S PRINCIPLE 243

which

we

shall

now

derive. For this

purpose

we

carry

out

an

infinitesimal

transforma-

tion of the coordinates

by setting

x'v

=

xv

+ Axv, (10)

where the

Axv

are

arbitrarily eligible, infinitesimally

small functions of the

coordinates.

x'v are

the coordinates of the world

point

in the

new system,

the

same

point

whose coordinates

were

xv

in the

original system.

Just

as

for

the

coordinates,

there

is

a

transformation law for

any

other

quantity

u, of the

type

u'

=

u

+

Au,

where

Au

must

always

be

expressible

in

terms

of

the

Axv.

From the covariant

properties

of

the

guv one

derives

easily

for the

guv

and the

guv

the

transformation

laws:

dAx,

dAxu

Ag^v

=

g/ux

+

8va

ox_

ox

a a

(11)

Agouv

aana.

oxa dxa

(12)

AR

can

be calculated with the

help

of

(11)

and

(12),

since

R

depends only upon [8]

{5}

the

guv

and the

gouv.

Thus,

one gets

the

equation

(13)

Iv^J

ÖXv

dgauv

where

we

used the abbreviation

S;

=

2|®

8?

+

©

si-

(14)

[9]

^

dgtr

From these two

equations

we

draw two conclusions that

are important

in the

following.

We

know

R/-g

to

be

an

invariant under

arbitrary

substitutions but

not

It

is,

however,

easy

to

show that the latter

quantity

is invariant under

linear

substitutions

of the coordinates.

Consequently,

the

right-hand

side

of

(13)

must

always

vanish when all

d2Axo/dxvdxa

do.

Then it

follows

that

R*

must

satisfy

the

identity

Sov =

0.

(15)

If

we

furthermore choose the

Ax

v

such that

they

differ from

zero only

inside the

domain

considered,

but vanish in

an

infinitesimal

neighborhood

of

the

boundary,

then

[p.1115]