DOC.
3
35
be chosen
in
one
and
only
one
such
way
that the
integral
in
our
equation
becomes
independent
of
E.
It
is obvious that
the
integral
e-2hHdx1...dxn,
for
which
the limits
of
integration
may
be
determined
by
the limits
E
and
E +
3E,
will for
a
specific
5E
be
a
function
of
E
alone;
let
us
call the latter
X(E). The
integral
in the
expression
for
dN'
can
then
be
written in the
form
[14]
X(E
-
E)
.
Since
E
is infinitesimally small
compared
with
E,
this
can
be
written,
up
to
quantities
which
are
infinitesimally small of
higher
order, in
the
form
X(E
-
E)
=
X(E)
-
EX'(E)
.
The
necessary
and
sufficient condition for this
integral
to
be
independent of
E
is
hence
X'(E)
=
0
.
But
then
we can
put
X(E)
=
e-2hE.w(E)
,
where
w(E)
=
J
dx1...dxn,
extended
over
all values of the variables
whose
energy
function lies
between
E
and
E+SE.
Hence
the condition
found
for
h assumes
the
form
e~2ÄE.«(E).-
-2
h
+
«'(E)
«(E)
=
0
,
or
4
=
*^
«(E)
Thus,
there
always
exists
one
and
only
one
value for
h
that
satisfies
the
conditions found.
Further,
since
w(E)
and
w'(E) are always
positive,
as
shall
be
shown
in the
next
section,
h
is also
always
a
positive
quantity.