268

THE

RELATIVITY

PRINCIPLE

Here

V'

should be viewed

as a

function

of

w'

known

from

the optics

of

stationary

bodies.

Dividing

these

equations,

one

obtains

V

=

V'

~

i

+

V'v

I +

cz

7J

This

equation

could also

have been

obtained directly

by

applying

the addition

theorem

for velocities.1 If

V'

is

to

be

considered

as

known,

the last

equation

solves the

problem

completely.

However,

if

only

the

frequency

(w)

referred

to

the "stationary"

system

S

is

to

be

considered

as

known, as

for

example

in the

well-known experiment

by

Fizeau, then the

two foregoing

equations

have to be used

in

conjunction

with the relation

between

w'

and

V'

in

order

to

determine the three

unknowns

w',

V', and

V.

Further, if

G

or G'

is the

group

velocity

referred

to

S

or

S',

respectively,

then,

according

to

the addition

theorem

for velocities,

e"T7^v+G]n

Since

the

relation

between

G'

and

w'

can

be

obtained

from

the optics

of

stationary bodies,2

and

w'

can

be

calculated

from

w

according

to

the

foregoing,

the

group

velocity

G

can

be

calculated

even

if

only

the

frequency

of

light relative

to S and

the

body's

velocity of motion

are

given.

1Cf.

M.

Laue, Ann. d.

Phys.

23

(1907):

989.

[33]

V1 2Because

G'

=

j-Jp~~'+

1

V

w

[34]