54
FOUNDATIONS OF THERMODYNAMICS
where
the
integral
is
to
be
extended
over
all values
of
II
to
which
correspond
values
of
the
energy H
lying
between
E*
-
n
and
E*
+
ÖE*
-
n.
Had
the
integration been
carried
out,
we
would
have found
the
distribution
of
the
state
of the
systems
o.
This is in fact
possible.
We
put
e'2hH-da1...dax
=
xW
.
where
the
integral
on
the
left-hand
side is
to be
extended
over
all
values
of
the variables for
which
H
lies
between
the
definite values
E and
E
+
8E*.
The
integral
that
appears
in the
expression
dN2
then
assumes
the
form
X(E*
-
n)
,
or,
since
n
is
infinitesimally small
compared
with
E*,
x(E*)
-
x'(E*).n
.
Thus,
if
h
can
be
chosen such
that
x'(E*)
=
0,
the
integral
reduces
to
a
quantity
that is
independent of
the
state
of
a.
It is possible
to
put,
up
to
the
infinitesimally small,
X(E)
=
e
-2
hE
=
e'2hE-w(E)
,
1 x
[9]
where
the
integration
limits
are
the
same as
above,
and
where
w
denotes
a
new
function
of E.
The
condition for
h
now assumes
the
form
consequently:
X'(E*)
=
e~2hE*-{u(E*)
-
2hu(E*)}
=
0
;
*4 «im.
1 u(P)
If
h
is
chosen
in this
way,
the
expression
for
dN2
will
assume
the
form