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)
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