Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/129

is the determinant of the matrix $\Phi .$ A lower power, as the $m^{\text{th}},$ with the divisor $n(n-1)\dots (n-m+1)$ would express as multiple quantity all the subdeterminants of order $m.$^[1]

It is evident that by the combination of the operations of indeterminate, algebraic, and combinatorial multiplication we obtain multiple quantities of a more complicated nature than by the use of only one of these kinds of multiplication. The indeterminate product of combinatorial products we have already mentioned. The combinatorial product of algebraic products, and the indeterminate product of algebraic products, are also of great importance, especially in the theory of quantics. These three multiplications, with the internal, especially in connection with the general property of the indeterminate product given above, and the derivation of the algebraic and combinatorial products from the indeterminate, which affords a generalization of that property, give rise to a great wealth of multiplicative relations between these multiple quantities I say "wealth of multiplicative relations" designedly, for there is hardly any kind of relations between things which are the objects of mathematical study, which add so much to the resources of the student as those which we call multiplicative, except perhaps the simpler class which we call additive, and which are presupposed in the multiplicative. This is a truth quite independent of our using any of the notations of multiple algebra, although a suitable notation for such relations will of course increase their value.

Perhaps, before closing, I ought to say a few words on the applications of multiple algebra.

First of all, geometry, and the geometrical sciences which treat of things having position in space, kinematics, mechanics, astronomy physics, crystallography, seem to demand a method of this kind, for position in space is essentially a multiple quantity and can only be represented by simple quantities in an arbitrary and cumbersome manner. For this reason, and because our spatial intuitions are more developed than those of any other class of mathematical relations, these subjects are especially adapted to introduce the student to the methods of multiple algebra. Here, Nature herself takes us by the hand and leads us along by easy steps, as a mother teaches her child to walk. In the contemplation of such subjects, Möbius, Hamilton,

1. Quadratic matrices may alao be represented by a sum of indeterminate products of a quantity of the first degree with a combinatorial product of (n - 1)st degree, as, for example, when $n=4,$ by a sum of products of the form

   $\alpha \vert \beta \times \gamma \delta .$

   The theory of such matrices is almost identical with that of those of the other form, except that the external multiplication takes the place of the internal, in the multiplication of the matrices with each other and with quantities of the first degree.