the usual arrangement, the separation is
said to have a _species_ denoted by the partition (p1q1r1...) of the
number n.

_Definition II._--If in any distribution of n objects into n parcels
(one object in each parcel), we write down a number [xi], whenever we
observe [xi] similar objects in similar parcels we will obtain a
succession of numbers [xi]1, [xi]2, [xi]3, ..., where ([xi]1, [xi]2,
[xi]3 ...) is some partition of n. The distribution is then said to have
a _specification_ denoted by the partition ([xi]1[xi]2[xi]3...).

Now it is clear that P consists of an aggregate of terms, each of which,
to a numerical factor _pres_, is a separation of the partition

( s1^{[sigma]1} s2^{[sigma]2} s3^{[sigma]3} ...)

of species (p1q1r1...). Further, P is the distribution function of
objects into parcels denoted by (p1q1r1...), subject to the restriction
that the distributions have each of them the specification denoted by
the partition

( s1^{[sigma]1} s2^{[sigma]2} s3^{[sigma]3} ...).

Employing a more general notation we may write

X_p1^[pi]1 X_p2^[pi]2 X_p3^[pi]3 ... =
[Sigma]P x_s1^[sigma]1 x_s2^[sigma]2 x_s3^[sigma]3 ...

and then P is the distribution function of objects into parcels

(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...),

the distributions being such as to have the specification

(s1^[sigma]1 s2^[sigma]2 s3^[sigma]3 ...),

Multiplying out P so as to exhibit it as a sum of monomials, we get a
result--

X_p1^[pi]1 X_p2^[pi]2 X_p3^[pi]3 ... =
[Sigma][Sigma][theta] ([lambda]1^l1 [lambda]2^l2 [lambda]3^l3)
x_s1^[sigma]1 x_s2^[sigma]2 x_s3^[sigma]3 ...

indicating that for distributions of specification

(s1^[sigma]1 s2^[sigma]2 s3^[sigma]3 ...)

there are [theta] ways of distributing n objects denoted by

([lambda]1^l1 [lambda]2^l2 [lambda]3^l3 ...)

amongst n parcels denoted by

(p1^[pi]1 p2^[pi]2 p3^[pi]3 ...),

one object in each parcel. Now observe that as before we may interchange
parcel and object, and that this operation leaves the specification of
the distribution unchanged. Hence the number of distributions must be
the same, and if

X_p1^[pi]1 X_p2^[pi]2 X_p3^[pi]3 ... =
= ... + [theta]([lambda]1^l1 [lambda]2^l2 [lambda]3^l3)
x_s1^[sigma]1 x_s2^[sigma]2 x_s3^[sigma]3 ... + ...

then also

X_[lambda]1^l1 X_[lambda]2^l2 X_[lambda]3^l3 ... =
= ... + [theta](p1^[pi]1 p2^[pi]2 p3^[pi]3)
x_s1^[sigma]1 x_s