214
DOC.
7
PROBABILITY CALCULUS
§ 3. The
Statistical Law of
the
Individual
S
Before
investigating
a
combination of
all
quantities
$(n)
=
zfn(a)
v/Z
we
will
formulate the
probability
law
for
one
single
such
quantity.
We
consider
a
manifold
of
N-systems
of the
kind
defined
above.
To
each
system
belongs a
numerical
value
S.
Because
of the
statistical
distribution of the
a,
these
quantities
obey
a
specific probability
law, so
that the number of
systems
whose numerical
values lies
between
S
and
S
+
dS:
(4)
dN
=
F(S)dS.
[7]
If
we now
add
one more
element
to
each
system consisting
of Z
elements, i.e.,
if
we
pass
from
Sz
to
SZ+1,
the individual
members of
our
manifold
will
change
their
numerical
values
and
will enter into
another
region
dS.
If
it
is
to be
possible,
nevertheless,
to arrive at
a
statistical law
that
is
independent
of
Z,
then the number dN
must not
change
in this
transition.
Thus,
the number of
systems
entering
a given (in our
simplest
case, one-dimensional) region
dS
must
be the
same as
the number
of
systems
leaving
it.
If $ denotes the number of
systems passing
through
a given
numerical
value
S0
in
their transition
from
Z
to
Z
+
1
elements,
both
as regards magnitude
and
direction,
then
we
must have
(5)
hence
div
&
=
0,
dt
dS
=
0
and, since,
indeed,
$
must
always
be
zero
for
S
=
®°,
we
will also have
(6)
®
=
0.
Now
we
have
S
(Z*
1)
=
*W(g)
=
s
v/zTT
(Z)
z
+
/(«)
N
z
+
l
^/zTT
or,
since
Z
is
to be
a
very large
number,
(7)
s
(Z+1) (Z)