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