126
MOVEMENT
OF SMALL PARTICLES
be
small
compared
with
V*.
In accordance with the
theory
mentioned,
this
system
shall
be completely
described
by
the
state
variables
P1...Pl.
Even
if the molecular picture
were
established
down
to
the smallest
detail, the calculation
of
the
integal
B
would
be
so
difficult
as
to
make
an
exact
calculation of
F
all
but inconceivable.
However,
here
we
only
have
to
know how
F
depends
on
the size of the
volume
V*
in
which
all the dissolved
molecules
or
suspended
bodies (hereafter briefly called
"particles")
are
contained.
Let
us
denote
by
x1, y1,
z1
the
rectangular
coordinates of the
center
of gravity
of the first particle,
by
x2, y2,
Z2
those
of
the
second, etc.,
and
by xn,
yn,
zn
those
of
the last particle,
and
assign
to
the
centers of
gravity
of the
particles
the infinitesimally small parallelepiped-shaped
regions
dx1dy1dz1,
dx2dy2dz2...dxndyndzn,
all of
which
shall lie in
V*.
We
now
seek
the value
of
the
integral
occurring
in the
expression
for
F,
with
the restriction that the
centers
of gravity
of the particles shall lie in the
regions
just
assigned to
them. In
any
case,
this
integral
can
be
put
into the
form
dB
=
dx1dy1...dzn.J,
where
J is
independent
of
dx1dy1,
etc.,
as
well
as
of
V*,
i.e.,
of the
position of the
semipermeable
wall.
But
J is also
independent
of the
particular
choice of the positions
of
the
center-of-gravity
regions
and
of the
value
of
V*, as we
will
show immediately.
For
if
a
second
system
of
infin-
itesimally
small
regions
were
assigned
to
the
centers
of
gravity of
the
particles
and
denoted
by
dx'1dy'1dz'1, dx'2dy'2dz'2...dx'ndy'n
dz'n,
and
if these
regions
differed
from
the
originally
assigned
ones
by
their
position
alone,
but
not
by
their size,
and
if,
likewise, all of
them
were
contained in
V*, we
would
similarly have
where
Hence,
dB'
=
dx^dy^..-dz],
dx^dyy
.
.dzn =
dx^dyy
.
.dz*n
dB
_
J
~
Tr'