190

REVIEW OF BROWNIAN

MOTION

times

t

chosen, the less this

assumption applies. For

if

at

time

z

=

0

the

instantaneous value of the

velocity

of

change

were

dot

_

a

[28]

It

~

P0

s

and

if in

some

subsequent

time interval the

velocity of

change ß

were

not

influenced

by

the

random

thermal

process

but the

change

of

ß

were

determined

by

the passive resistance

(1/B)

alone,

dß/dz would

obey

the relation

i

ß

dz

~ B

u

is

defined here

by

the stipulation that

u(ß2/2)

should

be

the

energy

that

corresponds

to

the velocity

of

change

ß.

Thus,

in the

case

of

translational

motion of

a

suspended

sphere,

e.g.,

u(ß2/2) would

be

the kinetic

energy

of

the

sphere plus

the kinetic

energy

of

the

co-moving

liquid. Integrating,

we

get

ß

= ß0e~

*B

From

this result

one

concludes that formula (II) holds

only

for

time

[29]

intervals that

are

large

compared

with

B.

For

corpuscles

with

a

diameter of

1

micron and density

p

=

1

in

water

at

room

temperature,

the

lower

limit of validity of formula

(II) is

about

10-7

seconds;

this

lower

limit for time intervals increases

as

the

square

of the

corpuscle

radius.

Both

facts hold

true

for the translational

as

well

as

the

rotational

motion

of particles.

Bern,

December

1905.

(Received

on

19

December

1905)