Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/126

indeterminate product,^[1] is a distributive function of $\alpha ,\beta ,$ and $\gamma .$ It is evidently not aflTected by changing the order of the letters. It is, therefore, an algebraic product in the sense in which the term has been defined.

So, again, if we prefix $\textstyle \sum \displaystyle \pm$ to an indeterminate product to denote the sum of all terms obtained by giving the factors every possible order, those terms being taken negatively which are obtained by an odd number of simple permutations,

$\textstyle \sum \displaystyle \pm \alpha \left\vert \beta \right\vert \gamma ,$

for instance, will be a distributive function of $\alpha ,\beta ,\gamma ,$ which is multiplied by —1 when two of these letters change places. It will therefore be a combinatorial product.

It is a characteristic and very important property of an indeterminate product that every product of all its factors with any other quantities is also a product of the indeterminate product and the other quantities. We need not stop for a formal proof of this proposition, which indeed is an immediate consequence of the definitions of the terms.

These considerations bring us naturally to what Grassmann calls regressive multiplication, which I will first illustrate by a very simple example. If $n,$ the degree of multiplicity of our original quantities, is 4, the combinatorial product of $\alpha \times \beta \times \gamma$ and $\delta \times \epsilon ,$ viz.,

$\alpha \times \beta \times \gamma \times \delta \times \epsilon ,$

is necessarily zero, since the number of factors exceeds four. But if for $\delta \times \epsilon$ we set its equivalent

$\delta \left\vert \epsilon -\epsilon \right\vert \delta ,$

we may multiply the fiirst factor in each of these indeterminate products combinatorially by $\alpha \times \beta \times \gamma ,$ and prefix the result, which is a numerical quantity, as coefficient to the second factor. This will give

$(\alpha \times \beta \times \gamma \times \delta )\epsilon -(\alpha \times \beta \times \gamma \times \delta )\delta .$

Now, the first term of this expression is a product of $\alpha \times \beta \times \gamma ,\delta$ and $\epsilon ,$ and therefore, by the principle just stated, a product of $\alpha \times \beta \times \gamma ,\delta$ and $\delta \mid \epsilon .$ The second term is a similar product of $\alpha \times \beta \times \gamma$ and $\epsilon \mid \delta .$ Therefore the whole expression is a product of $\alpha \times \beta \times \gamma$ and $\delta \left\vert \epsilon -\epsilon \right\vert \delta ,$ that is, of $\alpha \times \beta \times \gamma$ and $\delta \times \epsilon .$ That is, except in sign, what Grassmann calls the regressive product of $\alpha \times \beta \times \gamma$ and $\delta \times \epsilon .$

To generalize this process, we first observe that an expression of the form

$\textstyle \sum \displaystyle \pm \alpha \times \beta \mid \gamma \times \delta ,$

in which each term is an indeterminate product of two combinatorial products, and in which $\textstyle \sum \displaystyle \pm$ denotes the sum of all terms obtained by

1. This notation must not be confounded with Grassmann's use of the vertical line.