278 WILSON AND MOORE. 



for the case n — 3; and for the general case the elements of the pro- 

 duct would be Xii = XiYj—XjYi, This system of the second order 

 is skew symmetric, that is, X,/ = —Xji, Xjj = 0. We could likewise 

 form an " algebraic " product Xij = XiYj + XjYi, which is sym- 

 metric. And in general we could form the combinatory^ and algebraic 

 products of k systems. 



If we wish actually to write the systems as hypercomplex numbers 

 with " units " attached, we have 



Xiei + X2^2, 



Xneid + XnCiCi + XiiCiBi + X^ieie-i, 



and so on. The product of these two systems would be similarly 

 expressed with the units eiei^i, ^1^162, • • • exactly as the triadic 

 which arises from the product of a vector and a dyadic. 



If we wish to consider the units e\, 62, or eie\, eieo, e^ei, e^e^, etc., 

 replaced by the set of independent variables, X\, x^, or Xiyi, Xiy-i, x^yu 

 ^lyi, etc., the expressions become 



X\Xx + XiXi, XnXiyi + Xi2Xiy2 + Xzix^yi + Xi^xiy^, 



and so on, — that is, they become linear, bilinear, trilinear, . . ., 

 forms. Ricci's system of the mth order with range 1 to n is therefore 

 analogous to an ??i-linear form in n variables. 



7. Transformations of sets of elements. Consider next the 

 linear transformation ■'^■'■ 



Xi = ^jCij-yj. (3) 



These equations may be solved by multiplying by the cof actors Cik 

 each divided by the determinant lct/|, that is, by yn: = Ca-/|ciy|, 

 and summing with respect to i. Then 



yk = ^x^ikXi or yi = ^ffiiXj. (3') 



If Ui, Vi are variables contragredient to Xi and yi, the transformation 

 upon the w's and v's is 



Vi = '2jCj{Uj or tii = ^rYiiOj- (4) 



11 We may refer to Bocher's Introduction to Higher Algebra for the theory of 

 Hnear transformations, linear dependence, cogredient and contragredient 

 variables, bilinear forms, square matrices, etc. 



