h_(q1)h_(r1) ... is equal to the
coefficient of symmetric function (p1q1r1 ...) in the development of the
product h_p.h_q.h_r...."

The problem of Case I. may be considered when the distributions are
subject to various restrictions. If the restriction be to the effect
that an aggregate of similar parcels is not to contain more than one
object of a kind, we have clearly to deal with the elementary symmetric
functions a1, a2, a3, ... or (1), (1^2), (1^3), ... in lieu of the
quantities h1, h2, h3, ... The distribution function has then the value
a_(p1)a_(q1)a_(r1)... or (1^p1) (1^q1) (1^r1) ..., and by interchange of
object and parcel we arrive at the well-known theorem of symmetry in
symmetric functions, which states that the coefficient of symmetric
function (pqr ...) in the development of the product ap1aq1ar1 ... in
a series of monomial symmetric functions, is equal to the coefficient of
the function (p1q1r1 ...) in the similar development of the product
a_p.a_q.a_r....

The general result of Case I. may be further analysed with important
consequences.

Write X1 = (1)x1,
X2 = (2)x2 + (1^2)x1^2,
X3 = (3)x3 + (21)x2x1 + (1^3){x1}^3

. . . . .

and generally

X_s = [Sigma]([lambda][mu][nu] ...)x_[lambda]x_[mu]x_[nu] ...

the summation being in regard to every partition of s. Consider the
result of the multiplication--

X_p1 X_q1 X_r1 ... =
[Sigma]P(x_s1)^[sigma]1 (x_s2)^[sigma]2 (x_s3)^[sigma]3 ...

To determine the nature of the symmetric function P a few definitions
are necessary.

_Definition I._--Of a number n take any partition
([lambda]1[lambda]2[lambda]3 ... [lambda]s) and separate it into
component partitions thus:--

([lambda]1[lambda]2) ([lambda]3[lambda]4[lambda]5) ([lambda]6) ...

in any manner. This may be termed a _separation_ of the partition, the
numbers occurring in the separation being identical with those which
occur in the partition. In the theory of symmetric functions the
separation denotes the product of symmetric functions--

[Sigma] [alpha]^[lambda]1 [beta]^[lambda]2 [Sigma][alpha]^[lambda]3
[beta]^[lambda]4 [gamma]^[lambda]5 [Sigma][alpha]^[lambda]6 ...

The portions ([lambda]1[lambda]2), ([lambda]3[lambda]4[lambda]5),
([lambda]6)... are termed _separates_, and if [lambda]1 + [lambda]2 =
p1, [lambda]3 + [lambda]4 + [lambda]5 = q1, [lambda]6 = r1... be in
descending order of magnitude,