366
THE
RADIATION
PROBLEM
not
interfering
with each
other
at
all.
We
could thus exactly
determine the
quantity
E2
by a
consideration that is
mathematically
somewhat
complicated.
We
shall
content
ourselves here with
a
simple
dimensional consideration.
The
following
conditions
must
be
satisfied:
1.
The
magnitude
of
the
mean
fluctuation
depends
only
on
X
(wave-
length),
dX, a,
and
v,
where
a
denotes the radiation
density
related
to
[37]
the
wavelengths
(adX
=
pdv).
2.
Since the radiation
energies
of
adjacent
wavelength
ranges and
volumes1
are
simply
additive,
and
the
corresponding
fluctuations
are
indepen-
dent of each
other,
at
a
given
X
and
p, E2
must
be
proportional
to
the
quantities
dX
and
v.
3.
e2
has
the
dimension of
the
square
of
an
energy.
The
expression
for
E2
is
thereby completely
determined
up
to
a
numerical factor
(of
order
of
magnitude
1). In
this
way
one
arrives
at
the
expression
a2X4vdX,
which
upon
introduction
of
the variables
used above
[38]
reduces
to
the
second term
of
the formula for
E2
just
developed.
But
we
would
have
obtained solely
this
second term
for
E2
had
we
started
out
with
the Jeans formula.
One
would
then also
have
to
put
R/NK
equal
to
a
constant
of
order
of
magnitude
1, which corresponds
to
Planck's determination
of
the
[39] elementary
quantum2.
Thus,
the first
term of
the
above
expression
for
E2,
which
for the visible radiation
surrounding
us
everywhere
makes
a
far
greater
contribution than
the second
one,
is
not compatible
with the
current
theory.
If
one
would
put,
with Planck,
R/NK
=
1,
then the first
term,
if
present
alone, would yield
a
fluctuation
of the
radiation
energy
equal to
that
produced
if the radiation consisted
of
point quanta
of
energy hv
moving
[40] independently
of each other.
This
can
be
shown
by a
simple
calculation.
One
should remember
that the contribution of the first
term to
the
average percent
fluctuation of
energy
E2
1Only
if
these
are
large
enough,
of
course.
2By
carrying out
the interference consideration indicated
above,
one
would
obtain
R/Nk
=
1.