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]