MULTIPLE ALGEBRA. 113 



is the determinant of the matrix <1>. A lower power, as the m th , with 

 the divisor n(n l) ... (n m+1) would express as multiple quantity 

 all the subdeterminants of order m* 



It is evident that by the combination of the operations of inde- 

 terminate, 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 multi- 

 plicative 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 appli- 

 cations 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, Mobius, Hamilton, 



* Quadratic matrices may also be represented by a sum of indeterminate products of 

 a quantity of the first degree with a combinatorial product of (?i-l)st degree, as, for 

 example, when n = 4, by a sum of products of the form 



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 multipli- 

 cation of the matrices with each other and with quantities of the first degree. 

 G. II. H 



