pecies.

Case I.

1. _One-to-One Distribution. Parcels m in number (i.e. m = n)._--Let hs
be the homogeneous product-sum of degree s of the quantities [alpha],
[beta], [gamma], ... so that

(1 - [alpha]x. 1 - [beta]x. 1 - [gamma]x. ...)^-1 =
1 + h1x + h2x^2 + h3x^3 + ...

h1 = [Sigma][alpha] = (1)
h2 = [Sigma][alpha]^2 + [Sigma][alpha][beta] = (2) + (1^2)
h3 = [Sigma][alpha]^3 + [Sigma][alpha]^2[beta] +
[Sigma][alpha][beta][gamma] = (3) + (21) + (1^3).

Form the product h_(p1)h_(q1)h_(r1)...

Any term in h_(p1) may be regarded as derived from p1 objects distributed
into p1 similar parcels, one object in each parcel, since the order of
occurrence of the letters [alpha], [beta], [gamma], ... in any term is
immaterial. Moreover, every selection of p1 letters from the letters in
[alpha]^p[beta]^q[gamma]^r ... will occur in some term of h_(p1), every
further selection of q1 letters will occur in some term of h_(q1), and so
on. Therefore in the product h_(p1)h_(q1)h_(r1) ... the term
[alpha]^p[beta]^q[gamma]^r ..., and therefore also the symmetric function
(pqr ...), will occur as many times as it is possible to distribute
objects defined by (pqr ...) into parcels defined by (p1q1r1 ...) one
object in each parcel. Hence

[Sigma]A_[(pqr...), (p1q1r1...)].(pqr ...) = h_(p1)h_(q1)h_(r1)....

This theorem is of algebraic importance; for consider the simple
particular case of the distribution of objects (43) into parcels (52),
and represent objects and parcels by small and capital letters
respectively. One distribution is shown by the scheme

A A A A A B B
a a a a b b b

wherein an object denoted by a small letter is placed in a parcel
denoted by the capital letter immediately above it. We may interchange
small and capital letters and derive from it a distribution of objects
(52) into parcels (43); viz.:--

A A A A B B B
a a a a a b b.

The process is clearly of general application, and establishes a
one-to-one correspondence between the distribution of objects (pqr ...)
into parcels (p1q1r1 ...) and the distribution of objects (p1q1r1 ...)
into parcels (pqr ...). It is in fact, in Case I., an intuitive
observation that we may either consider an object placed in or attached
to a parcel, or a parcel placed in or attached to an object.
Analytically we have

_Theorem._--"The coefficient of symmetric function (pqr ...) in the
development of the product h_(p1)