DOC.
8
ANALYSIS
OF A RESONATOR'S MOTION
229
over
all
angles
of
incidence,
hence
8
(13)
B\
t
.
T
= A\T sin2p =
_ upVo.
In
the
same
way
we
get
(14)
^2
rp
(2ltV Y
.2
Tr
.4 2
64
^v0
Cvt7
=
sm4pcos2o)
=
- -
J
v°r
^
15
c2
rv
By
inserting
(13)
and
(14)
into
(12) we finally
obtain
(15) A2
=
-£l^LLP2
40TC2Vq
#5.
The
Radiation Law
Now
we
only
have
to
insert the
values
(9)
and
(15)
found
above into
our
equation
(2),
and
we
arrive at
the differential
equation containing
the radiation
law:
c3N
v
dp
P2
=
P
" t
24ni?0v2
3
dv
which
yields by
integration
/i*\
8-rcRQv2
(16)
P
=
3
M
•
c
N
This
is
the
well-known
Rayleigh
radiation
law,
which
is
in
the
most
glaring
contradiction
with
experiment.
The foundations of
our
derivation
must
therefore contain
an
assertion
that
does not
agree
with what
really
takes
place
in
thermal radiation.
Let
us
therefore
place
these foundations under closer
critical
scrutiny.
One
has
wanted
to find
the
reason
why
all exact statistical
analyses
in the field
of
radiation
theory
lead
to
Rayleigh's
law in
the
application
of
this
approach
to
the radiation
itself.
With
some justification,
Planck10
brings
up
this
argument
against
Jeans's
derivation.
However,
in
the
above
derivation there
is
no
question
whatsoever of
a
somehow
arbitrary
transference of
statistical
considerations
to radiation;
the
energy
equipartition
theorem
was
applied
only
to
the
translatory
motion of
oscillators. But
the
successes
of the kinetic
theory
of
gases
demonstrate that
this
law
can
be
considered
as
thoroughly
proved
for
translatory
motion.
The theoretical foundation
we
used
in
our derivation,
which
is
certain
to
contain
an
unfounded
assumption,
is
thus
nothing
else
but that
underlying
the
theory
of
light
dispersion
in
completely transparent
bodies.
The
actual
phenomena
differ from
the
results deducible from
this
foundation
owing
to
the
fact
that additional
kinds
of
10 M.
Planck,
l.c., p.
178.
[22]