DOC.
32
THEORY
OF BROWNIAN
MOTION
345
nential distribution law
to
include the
case
of
ex-
ternal forces
acting on
the
gas
molecules
(Boltz-
mann
1868; see
also Boltzmann
1896,
§
19).
[11]
A
square
root
sign over
the second
term
of
the
equation
is
missing.
[12]
The
development
of
the
ultramicroscope
by
Siedentopf
and
Zsigmondy
had,
in
fact,
shifted the lower limit
of
observability
to
ca.
10-6
cm
(Siedentopf
and
Zsigmondy
1903).
For
a contemporary
discussion
of
microscopic
ob-
servability, see
Cotton
and
Mouton
1906,
chap.
1.
[13]
For
an
elaboration
of Einstein's
study
of
Brownian motion under the influence
of
an
elas-
tic
force,
including a proposed experimental
verification, see, e.g.,
Smoluchowski
1913;
see
also
Fürth
1922,
p.
65,
fn.
15.
[14]
The
following argument
is
presented
in
greater
detail in Einstein 1905i
(Doc. 14),
§
1
and
§
2.
[15]
Einstein 1905k
(Doc.
16).
Einstein
pre-
sumably
meant to
refer
to
Einstein 1905i
(Doc.
14),
§
1
and
§
2, as
corrected
in
Einstein
1922,
and,
in
particular,
to
p.
136.
[16]
Planck
1900a.
This
paper
does
not,
how-
ever, give
the formula for
black-body
radiation
to
which Einstein referred. This formula
is
given,
e.g., in
Planck
1901a, which
is
cited
by
Einstein elsewhere for this formula
(see, e.g.,
Einstein
1905i
[Doc. 14],
p.
136).
[17]
See
Planck
1901b.
[18]
For
a
discussion
of Einstein's
views
on
this fundamental
imperfection, see
the editorial
note,
"Einstein's
Early
Work
on
the
Quantum
Hypothesis,"
pp.
139-141.
[19]
In
1908,
Perrin
reported on experiments
showing
that the vertical distribution
of
granules
of
gamboge
in
a liquid
is
exponential
(Perrin
1908a).
Perrin mentioned
Einstein's
name,
but
gave a
derivation
of
the
exponential
distribution
that differs from
Einstein's.
[20]
The basic results
of
this section
are gener-
alizations
of
those derived in Einstein 1905k
(Doc. 16),
§
4.
[21]
The
integral
should extend from 0
to oo.
[22]
This value for N
is
close
to
the value de-
rived in
Einstein
1906c
(Doc. 33);
in Einstein
1922,
the
more
accurate value
6
x
1023 per
mole
is
given
instead. For
a
discussion
of
the
discrep-
ancy
between these
values,
and its
origin,
see
the editorial
note,
"Einstein's
Dissertation
on
the Determination
of
Molecular Dimensions,"
§
V,
pp.
179-182.
[23] Kirchhoff
1897
(see,
in
particular, p. 380);
for
Einstein's
previous use
of Stokes's
formula
for the derivation
of
the
mean square displace-
ment in Brownian
motion,
see
Einstein 1905k
(Doc.
16),
§
3
and
§
5.
[24]
See Kirchhoff
1897,
in
particular,
pp.
375-376.
[25]
Using
the values for
R
and N
given
on
p.
378,
and the value
of
k
given
in Einstein
1905j
(Doc. 15),
p.
21,
one
obtains the result
given by
Einstein. In
1909,
Perrin
performed an experi-
mental
test
of Einstein's
formula
(see
Perrin
1909a).
He worked with
granules
of
resin,
hav-
ing a
diameter
of
ca.
13
jll,
which contained
small inclusions that enabled him to follow their
rotational motion. The
experimental
values he
found
are
in
good agreement
with those
pre-
dicted
by
Einstein's
formula. See also
note 6.
[26]
See the
previous
discussion
on p.
375.
For
an
account
of
this and other
closely
related
prob-
lems,
see
De Haas-Lorentz
1913, chap.
7
(in
particular, pp.
87-88). Fluctuations
of
the
po-
tential difference between the
plates
of
a con-
denser
are
treated
in
Einstein 1907b
(Doc. 39).
Einstein's
study
of
charge
and
potential
fluctua-
tions
was
the
starting point
of his
attempt
to
de-
velop
methods for the
measurements
of
small
quantities
of
electricity
(see
Einstein 1908a
[Doc. 48]
and Vol.
5,
the editorial
note,
"Ein-
stein's 'Maschinchen'
for the Measurement of
Small
Quantities
of
Electricity").
[27]
For
a closely
related
discussion,
see
Ein-
stein 1907c
(Doc. 40).
[28]
The
equation
should be
"da/dz
=
ß0".
[29]
This restriction
was
removed later
by
in-
cluding
the threshold value
\lB
of
the time in
a
description
of
Brownian motion valid for all
time intervals
(see
the discussion in
Fürth
1922,
pp.
60-61, fn.
8).