Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/25

Chapter II

Opening of the Theory of Distributions

14. From the principles set forth in the concluding articles of Chapter I we can realise a definite way of expressing the result of algebraical multiplication.

Suppose that we have to form the product of a number of algebraical expressions each of which involves (say) three terms. The expressions are supposed to be given in a definite order from left to right. This order will be determined, usually, by the circumstances.

Let the factors be $n$ in number, and, in the given definite order, denoted by

$$(a_{1}+b_{1}+c_{1})(a_{2}+b_{2}+c_{2})(a_{3}+b_{3}+c_{3})\ldots$$

$(a_{n-1}+b_{n-1}+c_{n-1})(a_{1}+b_{1}+c_{1})$

where three terms are involved in each factor merely for the sake of simplicity.

To obtain a term of the product we select a term (any term) from each factor and place them in contact in the order in which they have been selected; the factors being dealt with in order from left to right. The term of the product, thus reached, may involve one, two or three of the symbols $a$, $b$, $c$ according to the way that the selection is carried out. To place the terms ($3^{n}$ in number) thus arrived at in a definite order we make our selections in such wise that the terms produced are in dictionary order. Thus the first three terms will be $a_{1}a_{2}a_{3}\ldots a_{n-1}a_{n}$,
$a_{1}a_{2}a_{3}\ldots a_{n-1}b_{n}$,
$a_{1}a_{2}a_{3}\ldots a_{n-1}c_{n}$, and the last three $c_{1}c_{2}c_{3}\ldots c_{n-1}a_{n}$,
$c_{1}c_{2}c_{3}\ldots c_{n-1}b_{n}$,
$c_{1}c_{2}c_{3}\ldots c_{n-1}c_{n}$.

15. As an example, consider the development of $(\alpha +\beta )^{n}$. Writing down the $n$ factors $(\alpha +\beta )(\alpha +\beta )(\alpha +\beta )\ldots (\alpha +\beta )(\alpha +\beta )$,