If we add all such functions which have the same weight we obtain,
algebraically speaking, all the products $w$ together of the quantities
$\alpha ,\beta ,\gamma ,\ldots \nu$, repetitions permissible.

Such a sum is called the Homogeneous Product-Sum of weight $w$ of
the $n$ quantities.

It is usually denoted by $h_{w}$.

We have

${\begin{aligned}\qquad \qquad h_{1}&=(1)=\textstyle \sum {\alpha }{\text{,}}\\\qquad \qquad h_{2}&=(2)+(1^{2})=\textstyle \sum {\alpha ^{2}}+\sum {\alpha \beta }{\text{,}}\\\qquad \qquad h_{3}&=(3)+(21)+(1^{3})=\textstyle \sum {\alpha ^{3}}+\sum {\alpha ^{2}\beta }+\sum {\alpha \beta \gamma }{\text{,}}\end{aligned}}$

and so forth.

We have before us the three sets of functions

${\begin{aligned}&s_{1},s_{2},s_{3},\ldots s_{\nu },\ldots {\text{,}}\\&a_{1},a_{2},a_{3},\ldots a_{\nu }{\text{,}}\\&h_{1},h_{2},h_{3},\ldots h_{\nu },\ldots {\text{.}}\end{aligned}}$

The first and third sets contain an infinite number of members, but
the second set only involves $n$ members where $n$ is the number of the
quantities $\alpha ,\beta ,\gamma ,\ldots$.

6. The identity of Art. 4 which connects the functions $a_{1},a_{2},a_{3},\ldots$
with $\alpha ,\beta ,\gamma ,\ldots$ may be written, by putting ${\tfrac {1}{y}}$ for ,
$1-a_{1}y+a_{2}y^{2}-\ldots +(-)^{n}a_{n}y^{n}\equiv (1-\alpha y)(1-\beta y)\ldots (1-\nu y)$,
or in the form
${\frac {1}{1-a_{1}y+a_{2}y^{2}-\ldots +(-)^{n}a_{n}y^{n}}}\equiv {\frac {1}{(1-\alpha y)(1-\beta y)\ldots (1-\nu y)}}$.
If we expand the last fraction in ascending powers of $y$, we obtain, in
the first place,
$1$
$+(\alpha +\beta +\gamma +\ldots +\nu )y$
$+(\alpha ^{2}\!+\beta ^{2}\!+\gamma ^{2}\!+\ldots +\nu ^{2}\!+\alpha \beta +\alpha \gamma +\beta \gamma +\ldots +\mu \nu )y^{2}$
$+(\alpha ^{3}\!\!\!+\!\beta ^{3}\!\!\!+\!\gamma ^{3}\!\!\!+\!\ldots \!+\!\nu ^{3}\!\!\!+\!\alpha ^{2}\!\beta \!+\!\alpha \beta ^{2}\!\!\!+\!\ldots \!+\!\mu ^{2}\!\nu \!+\!\mu \nu ^{2}\!\!\!+\!\alpha \beta \gamma \!+\!\alpha \beta \delta \!+\!\ldots \!+\!\!\lambda \mu \nu )y^{3}$
$+\ldots$.

It is clear that the coefficient of $y^{w}$ is the homogeneous product-sum
of weight $w$, so that we may write
${\frac {1}{1-a_{1}y+a_{2}y^{2}-\ldots +(-)^{n}a_{n}y^{n}}}\equiv 1+h_{1}y+h_{2}y^{2}+\ldots +h_{w}y^{w}+\ldots$
an identity.