DOC.

185

JANUARY

1916 183

all

on

the

papers

but

shall calculate

everything

out for

you.[2]

Then,

if

anything

should remain

incomprehensible,

the

gap

can

be

easily

filled.-

1)

Lagrangian

form

of

the

equations.

Statement:

Let

V-g

=

1.

In

addition,

let L

=

gor{aBa}{TaB}.

Then

follows,

if

£

is

conceived

of

as a

function

of

the

gaT's

and

goT

=

dgoT/dxo:[3]

8L

dg°

aß

a

ra

ß

dL

dgoT

aß

a

(1)

Comment: I

always

omit

the

summation

symbol.

An index must

always

be

summed when it

appears

twice.[4]

Proof: From

differentiating

£-always considered

as a

funct.

of

gor

&

goT-follows

aß

I

I

ra

dL

=

a

ß

dgaT

+

2g°

i\A

t

(Two

terms

differing only

in index

name are summarized).

From

this

follows

furthermore

from

goTd

ra Ta

Ta

ß

=

d

\g

ß ß

dg0

dL

=

-dg"

(

f

Ta

+ 2

aß

Ta

ß

a

ß

=

-dgCT{

} {

}

+

2

aß

d

(

g°TgßX

Ta

a A

In

addition

aß

a

ßa

a

(a)

Taking

into

consideration

that the

second term does

not

change

when the

sum

of

the

indices

a

& ß

are

exchanged simultaneously

with

A

and

r,

the

second term

is

then

also

equal

to

aß

a

d

gaTg‘

or

ß\

Ta

A

+

A

a

T