DOC. 7

PROBABILITY

CALCULUS

219

IE

Sn,F

+

f2

3F

dS^

f2

31ogF

dS(n

dSlK..

dS(Hl)

=

J

flogF

Y,

d

ds(n)

S(n)F

+

f

dF

dS^

dS(1\..

dS{n

[8]

which, according

to

(12),

also vanishes.

This

proves

that

integral

(13) vanishes; however,

because

of the

quadratic

character

of the

integrand,

this

is

possible only

if

we

have

everywhere

for

every n

(14)

SpF

=

0.

dS-n

Thus,

we

arrive at

a

statistical law

for F that

is

identical

with

Gauss's

law

of

errors

with

respect

to

every S(n):

(15)

F

=

const.e

2f* .e

2/*. [9]

The

probability

of

a

combination of

values

S(n)

is

thus

simply

the

product

of the

probabilities

of the

individual

S(n).

It

is

clear

that,

if

equation

(15)

holds

for

S(1),

S(2)

...,

then the

same

equation is

satisfied

for

a

combination of

quantities

S(n)

=

a

.

n

In that

case,

instead of

f2, we

have

the

quantities

af2 in

the

exponents.

But the

coefficients

An,

Bn

in

our

physical

problem are

of the

type

S(n)';

and,

indeed,

we

have

to set

A

S(n)

=

[10]

and hence

Un

"

'.fr

Therewith

is

also

proved

the

validity

of

equation

(1)

and

the

impossibility

of

constructing

a

probability-theoretical

relation between the

coefficients

of the Fourier

series

that

describes

the thermal radiation.

(Received on

29

August

1910)