SURFACES IN HYPERSPACE. 277 



later, we come to differentiation wc shall have to take into account the 

 variability of the coefficients of the transformation. 



6. Sets of elements. We deal with sets of elements of different 

 orders; thus our system is a generalization of matrical as well as of 

 vectorial analysis. The fundamental elements are sets of quantities 

 X with in indices, each of which may take all values from 1 to n. 

 For example, if m = 2, 



m = 0, X, no index; 



m = 1, A'l, X2, one index; 



m = 2, Xu, X21, Xoi, X22, two indices; 



wi = 3, yYiii, X112, X121, X122, A''2n, A"2i2, A''22i, A'222, three indices. 



In general there are n"" quantities in the system of order m with 1 to w 

 as the range for each index. 



These systems of successive orders are analogous to the scalars, 

 vectors, dyadics, triadics, .... of Gibbs, and to the scalars, 

 vectors, open products with 1, 2, . . . openings of Grassmann; 

 the matrical analogy would take us to matrices of higher dimensions 

 than the usual two. The addition of two systems of the same order 

 and the multiplication of a system by a constant are according to 

 definitions obviously suggested by the analogies, i. e., the systems are 

 linear. 



Multiplication ^ of a system of order m into a system of order ni' 

 consists in multiplying each element of the first system into each 

 element of the second and gives a system of order m -\- m' . For 

 example, 



(Ai, X2) (Xii, X12, X22, ^"22) = X\Xn, XiXy2, XiX'll, XiA'^22, X2X\\, 



A2A12, A2A21, A2A22, 



which is a system of order 3 and may be written Xijk, i, j, k = 1,2. 

 By following the method of Gibbs ^° we may construct an outer or 

 " combinatory " product of two systems of order 1, as 



(Xu X2, X3) X (Fi, F2, F3) = Z2F3-F3F2, Z3FX-Z1F3, 



JLi 1 2 — iT-i' 1 



9 Grossmann calls the multiplication "outer" from analogy with Grass- 

 mann's outer product with which it has little in common; the real analogy 

 is with Gibbs' indeterminate and Grassmann's open product. 



10 On Multiple Algebra, Scientific Papers, Vol. II, pp. 91-117. 



