36
THEORY OF
THERMAL
EQUILIBRIUM
If
we
choose
h
in
this
way,
the
integral
reduces
to
a
quantity
inde-
pendent
of
E,
so
that
we
obtain the
following expression
for the
number
of
systems
whose
variables
P1...qn
lie
within the indicated limits:
[15]
dN'
= A"e-2hE.dp1...dqn
.
Thus,
also for
a
different
meaning
of
A",
this is the
expression
for the
probability
that the
state
variables of
a
system
mechanically
linked with
a
system
of
relatively
infinite
energy
lie
between infinitesimally
close limits
[16] when
the
state has
become
stationary.
§4.
Proof
that
the quantity
h
is positive
Let
(p(x)
be
a homogeneous
quadratic
function
of
the variables
x1...xn. We
consider the
quantity
z
=
l
dx1...dxn,
where
the limits of
integration
shall
be
determined
by
the condition that
p(x)
lies
between
a
certain value
y
and
y+A,
where
A
is
a
constant.
We
assert that
z,
which
is
a
function
of
y
only,
always
increases with
increasing
y
when
n
2.
If
we
introduce the
new
variables
x1 =
ax'1...xn
=
ax'n,
where
a
=
const.,
then
we
have
z
=
an
dx'1...dx'n
.
Further,
we
obtain
ip(x)
=
a2ip(x').
Hence,
the limits
of integration
of the
integral
obtained for
p(x')
are
y
and
y+A
a2
a2
Further, if
we assume
that
A
is
infinitesimally small,
we
obtain
z
=
an-2 dx'1...dx'n
.
[17]
Here
y'
lies
between
the limits