DOC.
85 MAY 1915 99
function in
p
and
tn
to
the
oscillation
originating
from individual resonators
is
very simple
and
does not
change
much.[4]
In the
addendum
to
section II of
my
manuscript,
in which
probability density p
is
no longer regarded as a
constant
but
as a
function of
tn, you already
see
related calculations.
I
want
to
think this
through
more
carefully, though,
in
the
near
future.
Your
second
objection
that
in
my
rendition
the
radiation at the
intersection
point
“successively
stems constantly
from different
layers,”
I
really
cannot
ac-
knowledge
as
correct,
in contrast to
the
first.
Within the
time
range
[T]
=
r,
to
which
I
have
restricted the
validity
of
my
expansion,
it
stems from all
layers;
what
happens
beyond
this
time
range
is
not
supposed
to be contained in
my
Fourier
series at
all.
Thus
your
subsequent
comment
is
dealt with:
“If I
choose
the
den-
sity
as
a
function of
a freely
estimated
location,
which is
surely permissible
for
real
radiation
paths,
then
I
obtain
a
radiation that
has
a
variable
mean
intensity
against
time;
such
radiation
is
not unordered.”
Apart
from
this
I
cannot concede
to
you
that
temporally
variable
radiation
is
not unordered. For since
virtually
all
radiation
changes intensity
in
time,
accordingly hardly any
unordered
radiation
would exist at
all.
But
now
to
your
main
objection.
You
say:
In
a
Fourier series
that
is
sup-
posed
to be
zero
at
one
section
(or more sections)
of
its
development range,
the
coefficients
are
not
independent.
You
may
gather how
much
this
expresses my
own
sentiments from
the
fact
that
I
myself
have stressed
the
degree
of freedom
of
a
beam
in oral discussions with Lenz and
Sommerfeld.[5]
It
just
seems
to
me
that
these relations must have
the
form
of
equations,
not
the
much looser form
of
statistical
relations
as
I
have set them
up.
Moreover,
this
statement
of
my
considerations does
not
apply.
For
I
do not
require
that
my
Fourier series vanish
in the
section of the
development range
not
within
the
valid
range;
I
do
not
require anything
at
all from them.
They
are
rather
entirely
irrelevant
to
me
there,
as
they
have
nothing
to
do with
reality.
Such Fourier series
are
used
frequently
in
physics.
When
you develop, e.g.,
the
electrical
field strength at
a
particular point
in
space
from
-T
to
+T
according
to
Fourier,
and then
derive in the usual
manner
of
resonance
theory
the
forced
swing
of
an
oscillator,
then
you
arrive
at
a
series that
naturally
has
a
development
range
of
2T,
but
is
valid
only
from
-T+d
to +T, where
d
is
the
time in which
the
resonator fades
away.
For from
-T
to
-T+d the
resonator
oscillation
depends
in
part
on
the
electrical oscillation before time
-T,
about
which
nothing is
revealed
by
the
Fourier series.
Despite
this difference between
development
and
applicable
ranges,
no
one
has misused
the
coefficients of the
resonator
series
more
than
those
of the
field
strength series,
and in
my
opinion
justifiably
so.
In order to knock
the bottom
completely
out
of your
main
objection,
I
can–
something
I
had overlooked
up
to
now-omit
entirely
in
the last
section of
my