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)