Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/127

putting every different pair of the letters before the dividing line, the negative sign being used for any terms which may be obtained by an odd number of simple permutations of the letters,—in other words, the expression

$\alpha \times \beta \mid \gamma \times \delta -\alpha \times \gamma \mid \beta \times \delta -\alpha \times \delta \mid \gamma \times \beta +\beta \times \gamma \mid \alpha \times \delta -\beta \times \delta \mid \alpha \times \gamma +\gamma \times \delta \mid \alpha \times \beta ,$

is a distributive function of $\alpha ,\beta ,\gamma ,$ and $\delta ,$ which is multiplied by —1 when two of these letters change places, and may, therefore, be regarded as equivalent to the combinatorial product $\alpha \times \beta \times \gamma \times \delta .$Now, if $n=5,$ the combinatorial product of

$\rho \times \sigma \times \tau$⁠and⁠$\alpha \times \beta \times \gamma \times \delta$

is zero. But if we multiply the first member of each of the above indeterminate products by $\rho \times \sigma \times \tau ,$ and prefix the result as coefficient to the second member, we obtain

$(\rho \times \sigma \times \tau \times \alpha \times \beta )\gamma \times \delta -(\rho \times \sigma \times \tau \times \alpha \times \gamma )\beta \times \delta +{\text{etc.,}}$

which is what Grassmann calls the regressive product of $\rho \times \sigma \times \tau$ and $\alpha \times \beta \times \gamma \times \delta .$ It is easy to see that the principle may be extended so as to give a regressive product in any case in which the total number of factors of two combinatorial products is greater than $n.$ Also, that we might form a regressive product by treating the first of the given combinatorials as we have treated the second. It may easily be shown that this would give the same result, except in some cases with a difference of sign. To avoid this inconvenience, we may make the rule, that whenever in the substitution of a sum of indeterminate products for a combinatorial, both factors of the indeterminate products are of odd degree, we change the sign of the whole expression. With this understanding, the results which we obtain will be identical with Grassmann's regressive product. The propriety of the name consists in the fact that the product is of less degree than either of the factors. For the contrary reason, the ordinary external or combinatorial multiplication is sometimes called by Grassmann progressive.

Regressive multiplication is associative and exhibits a very remarkable analogy with the progressive. This analogy I have not time here to develop, but will only remark that in this analogy lies in its most general form that celebrated principle of duality, which appears in various forms in geometry and certain branches of analysis.

To fix our ideas, I may observe that in geometry the progressive multiplication of points gives successively lines, planes and volumes; the regressive multiplication of planes gives successively lines, points and scalar quantities.

The indeterminate product affords a natural key to the subject of matrices. In fact, a sum of indeterminate products of the second