DOC.
3
33
variables
belonged
to
the
region
G
at t
=
0,
and
these
systems only,
will
belong
to
a
specific
region
g
at
time
t
=
t,
so
that the
following equation
applies
dN
=
i(pt,...qn)
9
However,
for
each such
system
Liouville's
theorem
holds,
which has
the
form
J
dPv...dQn
= J
dp1,...dqn
From
the last three
equations
it follows that
ii(P1,...Qn)
=
i(pv.--qn)
.1
Thus,
ij)
is
an
invariant
of
the
system, which from
the
above must have
the
form
u(p1,...qn)
=
i*(E).
However,
for all
systems
considered,
i/)*(E)
differs
only
infinitesimally
from
u*(E) =
const.,
and
our
equation
of
state
will then
simply
be
dN
=
dpv...dqn,
where
A
is
a
quantity
independent
of the p's
and
q's.
[9]
§3.
On
the (stationary) probability
of
the states
of
a
system
S
that is
mechanically
linked
with
a
system
E
whose
energy
is
relatively infinite
We
again
consider
an
infinite
number
(N)
of
mechanical
systems
whose
energy
shall lie
between
two
infinitesimally
different limits
E
and
E
+
£E.
Let each such mechanical
system
be,
again,
a
mechanical
link between
a
system
S
with
state
variables
p1,...qn
and
a
system
S
with
state
variables [11]
x1,...Xn.
The expression
for the total
energy
of
both
systems
shall
be
con-
stituted
such
that
those
terms of
the
energy
that
accrue
through
1Cf. L. Boltzmann,
Gastheorie
[Theory
of gases],
Part
2,
§32
and
§37.
[10]