278 WILSON AND MOORE. 



for the case ?i = 3; and for the general case the elements of the pro- 

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

 is skew symmetric, that is, Xa = —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 



Xi^i + X2e2, 



Xneiei + Xi2eie2 + XuCiei + Xooe^ei, 



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

 expressed with the units eiCiCi, 61^1^2, • • • exactly as the triadic 

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



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

 replaced by the set of independent variables, Xi, Xi, or x\y\, xxyi, xiy\y 

 xiyi, etc., the expressions become 



X\Xi -\- X1X2, XnXiyi -{- XnXiy2 + X-iiX^yi + X-i^x^yi, 



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 7?i-linear form in w variables. 



7. Transformations of sets of elements. Consider next the 

 linear transformation ^^ 



Xi = ^jCijyj. (3) 



These equations may be solved by multiplying by the cofactors dk 

 each divided by the determinant |ctj|, that is, by jik = Cik/\cij\, 

 and summing with respect to i. Then 



yk = '^rVikXi or yt = 'LfiiiXj. (3') 



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

 upon the w's and ij's is 



Vi 



. = v., 



jCjiUj or Ui = 'Sf/iiVj. (4) 



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

 linear transformations, hnear dependence, cogredient and contragredient 

 variables, bilinear forms, square matrices, etc. 



