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



14. From the principles set forth in the concluding articles of Chapter 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 $(a_1+b_1+c_1)(a_2+b_2+c_2)(a_3+b_3+c_3)\ldots$ $(a_{n-1}+b_{n-1}+c_{n-1})(a_n+b_n+c_n)$ 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 and the last three

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