288 WILSON AND MOORE. 



As the X's thus form a basis for the expression of systems in general 

 we may set up readily the progressive product of Grassmann ; for 



Yr Ys 



^ijCiCj 



iA a jA 



l'» s ;'* s 



In the system of Grassmann the progressi\-e product represents the 

 space determined by the elements (a parallelogram in the case of two 

 ^^ectors); but the interpretation here is not so direct because the 

 •systems Xr of the first order are not components of a vector, — they 

 have to be multiplied by certain factors to obtain components of a 

 vector. -^^ In like manner the terms XrYs — A^IV are not components 

 of a plane but may be converted into such by proper factors. 



13. Orthogonal unit n-tuples. We may define orthogonality 

 relative to a given quadratic form as in non-euclidean geometry. We 

 shall now howe\'er take the form as differential, namely, as 



Since the elements dxr form a contravariant system (H) a direction 

 in space may be defined by any contravariant system X^""^ if we set 

 up the simultaneous differential equations 



x(i) xt2) \<?) X(") ' ^ ^ 



and it is in this way that the contravariant systems used above, and 

 pre^•iously defined as contravariant systems, are associated with 

 special directions. 



If we have two systems iX'''^ , /X^*^ we define as is customary in differ- 



17 It is shown by Ricci and Levi-Civita {Math. Ann., 60) that if two dual 

 systems of the first order Xr, X''") are di\dded by Vorr and -^j ai""'^, the resulting 

 expressions Xr/ ^j arr, X('"V Va^'"''^ may be regarded respectively as the orthog- 

 onal projections of one and the same vector upon the tangents to the coordi- 

 nate Lines Xr and upon the normals to the coordinate surfaces; whereas the 

 expressions Xi''^ '\J an and XrVa^'"'') represent respectively the components of 

 the same vector along the same hnes and the same normals. This process 

 of rendering a system vectorial might be called vectorization and could be 

 ■extended to vectors of higher order (Stufe). 



