34
THEORY
OF
THERMAL
EQUILIBRIUM
action of the
masses
of
one
partial
system
on
the
masses
of the other partial
system are
negligible
in
comparison
with the
energy E
of the partial
system
[12]
S. Further,
the
energy
H
of
the partial
system
E
shall
be
infinitely
large
compared
with
E.
Up
to
the
infinitesimally
small
of
higher
order,
one
might
then
put
E
=
H
+
E
.
We now
choose
a
region
g
that is
infinitesimally
small in
all
state
var-
iables
p1...qn,
J1...Xn
and
is
so
constituted that
E
lies
between the
constant
values
E
and
E +
$E.
The number
dN
of
systems
whose state
variables
belong to
the
region
g
is then
according to
the
results
of
the
preceding
section
dN
= A
dp
*
••-dxnU
9
1
We
note
now
that
we are
free
to
replace
A
with
any
continuous function
of
the
energy
that
assumes
the value
A
for
E
=
E,
as
this will
only
infinitesimally
change our
result.
For
this
function
we
choose
A'.e-2hE,
where
h
denotes
a
constant which is arbitrary for the time
being,
and which
we
will
specify
soon.
We
write,
then,
[13]
dN
=
1
-2ÄE,
,
e
dpv..
dxnn
9
1
We
now
ask:
How many
systems are
in
states
in
which
p1 is
between
p1
+
dp1,
and, respectively,
p2
between
p2
+
dp2...
qn
between
qn
and
qn
+
dqn,
but
f1...Xn
have
arbitrary
values
compatible
with the conditions
of
our system?
If
we
call this
number
dN',
we
obtain
-2
hE
dN'
-
Ax
e
dpy..dqn e di^...dxn
2
hH
The
integration extends
over
those values of the
state
variables for
which
H
lies
between
E-
E
and
E-
E+
5E. We
now
claim that the
value of
h
can