244
DOC.
9
CRITICAL OPALESCENCE
*2
_
R2 7
Dpax
pax
However,
this
is
not
yet
the
average
value
we
seek. We must also
take the
average
value
with
respect
to time. This
appears only
in
the
last
factor of the
expression
for
Jpot.
If
we
take into
account
that the
time
average
of
this
factor
has
the value
1/2,
and
put
for
brevity
(16)
2
(M
-V)/
2
(v
-
v')Z
=
r\,
2
we
obtain the
following expression
for the
final
average
value
of
ey2:
sin2£
sin2t|
sin2{
pav
?2
12
C2
?•hi
Further, according
to
(7),
B2fm
is independent
of
p
a x,
and
can
therefore be
placed
before the summation
signs. Also,
according
to
(16)
and
(15a),
the
5
belonging
to
consecutive values
of
p
differ
from
each other
by
l- -
and, thus, by an infinitesimally
2
L
small
quantity.
The
triple
sum
that
appears
can
therefore be
turned
into
a
triple
integral.
Since,
according
to
the
aforesaid,
the
interval
A
5
between
two consecutive
$-values
in
the
triple sum
is
described
by
the relation
A$~
=1,
it
/
we
have
sin2$ sin2t| sin2{
5•2
C
(It)
sin2g
sin2ii
sin2t
A
,
2
C2
where the
last
sum can
immediately
be
written
as a
triple
integral.
From
(16)
and
(15a)
we can
conclude that for
all
practical
purposes
this
integral
can
be
taken between the
limits
-
00
and
+
», so
that
it
decomposes
into
a
product
of three
integrals,
each of
which
has
the
value
n.
With
this
taken into
account, we finally
obtain for
ey2,
with
the
help
of
(7)
and
substituting
the
expression
for
A,