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).