DOC. 30 FOUNDATION OF GENERAL

RELATIVITY

167

(the "complements"

of

covariant and contravariant tensors

respectively),

and

Buv

=

guvgabAaß.

We

call

Buv

the

reduced

tensor

associated

with

Auv.

Similarly,

Buv

=

guvgaßA.

It

may

be

noted that

guv

is

nothing more

than the

comple-

ment

of

guv,

since

g^g^gaß

=

g^B"a

=

gw.

§

9.

The

Equation of

the Geodetic Line. The Motion

of

a

Particle

As

the linear element ds is

defined

independently

of

the

system

of co-ordinates,

the

line

drawn between two

points

P

and

P'

of

the four-dimensional continuum in

such

a

way

that

Jds

is stationary-a

geodetic

line-has

a

meaning

which

also

is

independent

of the choice of co-ordinates.

Its

equation

is

8

ds

=

0

....

(20)

p

Carrying

out the variation

in

the

usual

way, we

obtain

from

this

equation

four differential

equations

which

define

the

geodetic

line;

this

operation

will be

inserted here

for

the

sake

of

completeness.

Let

A

be

a

function

of

the

co-ordinates

xv,

and let this

define

a

family

of surfaces

which intersect the

required

geodetic

line

as

well

as

all

the

lines

in immediate

proximity

to

it which

are

drawn

through

the

points

P

and P'.

Any

such

line

may

then

be

supposed

to be

given

by

expres-

sing

its co-ordinates

xv

as

functions

of

A.

Let the

symbol S

indicate the transition

from

a

point

of

the

required

geodetic

to the

point corresponding

to

the

same A on a

neighbouring

line.

Then

for

(20) we

may

substitute

fAaSweZX

=

0

JAj

n

dXn

d/Xy

I

...(20a)

But

since